A survey of community search over big graphs
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paper_title: Extracting Analyzing and Visualizing Triangle K-Core Motifs within Networks

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Cliques are topological structures that usually provide important information for understanding the structure of a graph or network. However, detecting and extracting cliques efficiently is known to be very hard. In this paper, we define and introduce the notion of a Triangle K-Core, a simpler topological structure and one that is more tractable and can moreover be used as a proxy for extracting clique-like structure from large graphs. Based on this definition we first develop a localized algorithm for extracting Triangle K-Cores from large graphs. Subsequently we extend the simple algorithm to accommodate dynamic graphs (where edges can be dynamically added and deleted). Finally, we extend the basic definition to support various template pattern cliques with applications to network visualization and event detection on graphs and networks. Our empirical results reveal the efficiency and efficacy of the proposed methods on many real world datasets.
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paper_title: Extracting Communities from Complex Networks by the k-Dense Method

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To understand the structural and functional properties of large-scale complex networks, it is crucial to efficiently extract a set of cohesive subnetworks as communities. There have been proposed several such community extraction methods in the literature, including the classical k-core decomposition method and, more recently, the k-clique based community extraction method. The k-core method, although computationally efficient, is often not powerful enough for uncovering a detailed community structure and it produces only coarse-grained and loosely connected communities. The k-clique method, on the other hand, can extract fine-grained and tightly connected communities but requires a substantial amount of computational load for large-scale complex networks. In this paper, we present a new notion of a subnetwork called k-dense, and propose an efficient algorithm for extracting k-dense communities. We applied our method to the three different types of networks assembled from real data, namely, from blog trackbacks, word associations and Wikipedia references, and demonstrated that the k-dense method could extract communities almost as efficiently as the k-core method, while the qualities of the extracted communities are comparable to those obtained by the k-clique method.
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paper_title: Effective and Efficient Community Search Over Large Directed Graphs

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Communities are prevalent in social networks, knowledge graphs, and biological networks. Recently, the topic of community search (CS), extracting a dense subgraph containing a query vertex $q$q from a graph, has received great attention. However, existing CS solutions are designed for undirected graphs, and overlook directions of edges which potentially lose useful information carried on directions. In many applications (e.g., Twitter), users’ relationships are often modeled as directed graphs (e.g., if a user $a$a follows another user $b$b, then there is an edge from $a$a to $b$b). In this paper, we study the problem of CS on directed graph. Given a vertex $q$q of a graph $G$G, we aim to find a densely connected subgraph containing $q$q from $G$G, in which vertices have strong interactions and high similarities, by using the minimum in/out-degrees metric. We first develop a baseline algorithm based on the concept of D-core. We further propose three index structures and corresponding query algorithms. Our experimental results on seven real graphs show that our solutions are very effective and efficient. For example, on a graph with over 1 billion of edges, we only need around 40mins to index it and 1$\sim$∼2sec to answer a query.
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paper_title: Community Detection in Temporal Multilayer Networks, with an Application to Correlation Networks

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Networks are a convenient way to represent complex systems of interacting entities. Many networks contain “communities” of nodes that are more densely connected to each other than to nodes in the rest of the network. In this paper, we investigate the detection of communities in temporal networks represented as multilayer networks. As a focal example, we study time-dependent financial-asset correlation networks. We first argue that the use of the “modularity” quality function---which is defined by comparing edge weights in an observed network to expected edge weights in a “null network''---is application-dependent. We differentiate between “null networks” and “null models” in our discussion of modularity maximization, and we highlight that the same null network can correspond to different null models. We then investigate a multilayer modularity-maximization problem to identify communities in temporal networks. Our multilayer analysis depends only on the form of the maximization problem and not on the specific quality function that one chooses. We introduce a diagnostic to measure persistence of community structure in a multilayer network partition. We prove several results that describe how the multilayer maximization problem measures a trade-off between static community structure within layers and larger values of persistence across layers. We also discuss some computational issues that the popular “Louvain” heuristic faces with temporal multilayer networks and suggest ways to mitigate them.
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paper_title: Network structure and minimum degree

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Abstract Social network researchers have long sought measures of network cohesion, Density has often been used for this purpose, despite its generally admitted deficiencies. An approach to network cohesion is proposed that is based on minimum degree and which produces a sequence of subgraphs of gradually increasing cohesion. The approach also associates with any network measures of local density which promise to be useful both in characterizing network structures and in comparing networks.
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paper_title: Index-based Optimal Algorithms for Computing Steiner Components with Maximum Connectivity

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With the proliferation of graph applications, the problem of efficiently computing all $k$-edge connected components of a graph G for a user-given k has been recently investigated. In this paper, we study the problem of efficiently computing the steiner component with the maximum connectivity; that is, given a set q of query vertices in a graph G, we aim to find the maximum induced subgraph g of G such that g contains q and g has the maximum connectivity, where g is denoted as SMCC. To accommodate online query processing, we present an efficient algorithm based on a novel index such that the algorithm runs in linear time regarding the result size; thus, the algorithm is optimal since it needs at least linear time to output the result. Moreover, in this paper we also investigate variations of the above problem. We show that such a problem with the constraint that the size of the SMCC is not smaller than a given size can also be solved in linear time regarding the result size (thus, optimal). We also show that the problem of computing the connectivity (rather than the graph details) of SMCC can be solved in linear time regarding the query size (thus, optimal). To build the index, we extend the techniques in [7] to accommodate batch processing and computation sharing. To efficiently support the applications with graph updates, we also present novel increment techniques. Finally, we conduct extensive performance studies on large real and synthetic graphs, which demonstrate that our index-based algorithms significantly outperform baseline algorithms by several orders of magnitude and our indexing algorithms are efficient.
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paper_title: Finding influential communities in massive networks

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Community search is a problem of finding densely connected subgraphs that satisfy the query conditions in a network, which has attracted much attention in recent years. However, all the previous studies on community search do not consider the influence of a community. In this paper, we introduce a novel community model called k-influential community based on the concept of k-core to capture the influence of a community. Based on this community model, we propose a linear time online search algorithm to find the top-r k-influential communities in a network. To further speed up the influential community search algorithm, we devise a linear space data structure which supports efficient search of the top-r k-influential communities in optimal time. We also propose an efficient algorithm to maintain the data structure when the network is frequently updated. Additionally, we propose a novel I/O-efficient algorithm to find the top-r k-influential communities in a disk-resident graph under the assumption of \({{\mathcal {U}}}=O(n)\), where \({{\mathcal {U}}}\) and n denote the size of the main memory and the number of nodes, respectively. Finally, we conduct extensive experiments on six real-world massive networks, and the results demonstrate the efficiency and effectiveness of the proposed methods.
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paper_title: Querying Intimate-Core Groups in Weighted Graphs

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The semantic intimacy of relations in many real-world networks (e.g., social, biological, and communication networks) can be modeled by weighted edges in which the more semantically intimate relations between the nodes translate to smaller edge weights. Recently, the problem of community search that aims to find the cohesive groups containing a given set of query nodes has attracted a great deal of attention. However, the bulk of literature on community search problem assumes a simple unweighted input graph, ignoring how semantically intimate the nodes have in retrieved communities. The discovered communities may have a highly cohesive structure, while they perform poorly in the semantic of intimate connections. In this paper, we investigate a novel problem of Querying Intimate-Core Groups (QICG): given a weighted undirected graph G, a set of query nodes Q and a positive integer k, to find a connected subgraph of G in which each node has at least k neighbors, and the sum of weights on its edges is minimum among all such subgraphs. We show that the QICG problem is NP-hard. We develop efficient algorithms based on several practical heuristic strategies to enhance the retrieval efficiency. Extensive experiments are conducted on real-world datasets to evaluate efficiency and effectiveness of proposed algorithms. The results confirm that our intimate-core group model outperforms state-of-the-art models in weighted graphs.
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paper_title: Skyline Community Search in Multi-valued Networks

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Given a scientific collaboration network, how can we find a group of collaborators with high research indicator (e.g., h-index) and diverse research interests? Given a social network, how can we identify the communities that have high influence (e.g., PageRank) and also have similar interests to a specified user? In such settings, the network can be modeled as a multi-valued network where each node has d ($d \ge 1$) numerical attributes (i.e., h-index, diversity, PageRank, similarity score, etc.). In the multi-valued network, we want to find communities that are not dominated by the other communities in terms of d numerical attributes. Most existing community search algorithms either completely ignore the numerical attributes or only consider one numerical attribute of the nodes. To capture d numerical attributes, we propose a novel community model, called skyline community, based on the concepts of k-core and skyline. A skyline community is a maximal connected k-core that cannot be dominated by the other connected k-cores in the d-dimensional attribute space. We develop an elegant space-partition algorithm to efficiently compute the skyline communities. Two striking advantages of our algorithm are that (1) its time complexity relies mainly on the size of the answer s (i.e., the number of skyline communities), thus it is very efficient if s is small; and (2) it can progressively output the skyline communities, which is very useful for applications that only require part of the skyline communities. Extensive experiments on both synthetic and real-world networks demonstrate the efficiency, scalability, and effectiveness of the proposed algorithm.
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paper_title: The community-search problem and how to plan a successful cocktail party

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A lot of research in graph mining has been devoted in the discovery of communities. Most of the work has focused in the scenario where communities need to be discovered with only reference to the input graph. However, for many interesting applications one is interested in finding the community formed by a given set of nodes. In this paper we study a query-dependent variant of the community-detection problem, which we call the community-search problem: given a graph G, and a set of query nodes in the graph, we seek to find a subgraph of G that contains the query nodes and it is densely connected. We motivate a measure of density based on minimum degree and distance constraints, and we develop an optimum greedy algorithm for this measure. We proceed by characterizing a class of monotone constraints and we generalize our algorithm to compute optimum solutions satisfying any set of monotone constraints. Finally we modify the greedy algorithm and we present two heuristic algorithms that find communities of size no greater than a specified upper bound. Our experimental evaluation on real datasets demonstrates the efficiency of the proposed algorithms and the quality of the solutions we obtain.
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paper_title: Community membership identification from small seed sets

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In many applications we have a social network of people and would like to identify the members of an interesting but unlabeled group or community. We start with a small number of exemplar group members -- they may be followers of a political ideology or fans of a music genre -- and need to use those examples to discover the additional members. This problem gives rise to the seed expansion problem in community detection: given example community members, how can the social graph be used to predict the identities of remaining, hidden community members? In contrast with global community detection (graph partitioning or covering), seed expansion is best suited for identifying communities locally concentrated around nodes of interest. A growing body of work has used seed expansion as a scalable means of detecting overlapping communities. Yet despite growing interest in seed expansion, there are divergent approaches in the literature and there still isn't a systematic understanding of which approaches work best in different domains. Here we evaluate several variants and uncover subtle trade-offs between different approaches. We explore which properties of the seed set can improve performance, focusing on heuristics that one can control in practice. As a consequence of this systematic understanding we have found several opportunities for performance gains. We also consider an adaptive version in which requests are made for additional membership labels of particular nodes, such as one finds in field studies of social communities. This leads to interesting connections and contrasts with active learning and the trade-offs of exploration and exploitation. Finally, we explore topological properties of communities and seed sets that correlate with algorithm performance, and explain these empirical observations with theoretical ones. We evaluate our methods across multiple domains, using publicly available datasets with labeled, ground-truth communities.
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paper_title: Linear-time enumeration of maximal K-edge-connected subgraphs in large networks by random contraction

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Capturing sets of closely related vertices from large networks is an essential task in many applications such as social network analysis, bioinformatics, and web link research. Decomposing a graph into k-core components is a standard and efficient method for this task, but obtained clusters might not be well-connected. The idea of using maximal k-edge-connected subgraphs was recently proposed to address this issue. Although we can obtain better clusters with this idea, the state-of-the-art method is not efficient enough to process large networks with millions of vertices. In this paper, we propose a new method to decompose a graph into maximal k-edge-connected components, based on random contraction of edges. Our method is simple to implement but improves performance drastically. We experimentally show that our method can successfully decompose large networks and it is thousands times faster than the previous method. Also, we theoretically explain why our method is efficient in practice. To see the importance of maximal k-edge-connected subgraphs, we also conduct experiments using real-world networks to show that many k-core components have small edge-connectivity and they can be decomposed into a lot of maximal k-edge-connected subgraphs.
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paper_title: Survey on Social Community Detection

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Community detection is a growing field of interest in the area of social network applications. Many community detection methods and surveys have been introduced in recent years, with each such method being classified according to its algorithm type. This chapter presents an original survey on this topic, featuring a new approach based on both semantics and type of output. Semantics opens up new perspectives and allows interpreting high-order social relations. A special focus is also given to community evaluation since this step becomes important in social data mining.
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paper_title: Efficient Computation of Importance Based Communities in Web-Scale Networks Using a Single Machine

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Finding decompositions of a graph into a family of communities is crucial to understanding its underlying structure. Algorithms for finding communities in networks often rely only on structural information and search for cohesive subsets of nodes. In practice however, we would like to find communities that are not only cohesive, but also influential or important. In order to capture such communities, Li, Qin, Yu, and Mao introduced a novel community model called "k-influential community" based on the concept of $k$-core, with numerical values representing "influence" assigned to the nodes. They formulate the problem of finding the top-r most important communities as finding r connected k-core subgraphs ordered by the lower-bound of their importance. In this paper, our goal is to scale-up the computation of top-r, k-core communities to web-scale graphs of tens of billions of edges. We feature several fast new algorithms for this problem. With our implementations, we show that we can efficiently handle massive networks using a single consumer-level machine within a reasonable amount of time.
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paper_title: Functional cartography of complex metabolic networks

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High-throughput techniques are leading to an explosive growth in the size of biological databases and creating the opportunity to revolutionize our understanding of life and disease. Interpretation of these data remains, however, a major scientific challenge. Here, we propose a methodology that enables us to extract and display information contained in complex networks. Specifically, we demonstrate that one can (i) find functional modules in complex networks, and (ii) classify nodes into universal roles according to their pattern of intra- and inter-module connections. The method thus yields a ``cartographic representation'' of complex networks. Metabolic networks are among the most challenging biological networks and, arguably, the ones with more potential for immediate applicability. We use our method to analyze the metabolic networks of twelve organisms from three different super-kingdoms. We find that, typically, 80% of the nodes are only connected to other nodes within their respective modules, and that nodes with different roles are affected by different evolutionary constraints and pressures. Remarkably, we find that low-degree metabolites that connect different modules are more conserved than hubs whose links are mostly within a single module.
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paper_title: Effective Community Search over Large Spatial Graphs

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Communities are prevalent in social networks, knowledge graphs, and biological networks. Recently, the topic of community search (CS) has received plenty of attention. Given a query vertex, CS looks for a dense subgraph that contains it. Existing CS solutions do not consider the spatial extent of a community. They can yield communities whose locations of vertices span large areas. In applications that facilitate the creation of social events (e.g., finding conference attendees to join a dinner), it is important to find groups of people who are physically close to each other. In this situation, it is desirable to have a spatial-aware community (or SAC), whose vertices are close structurally and spatially. Given a graph G and a query vertex q, we develop exact solutions for finding an SAC that contains q. Since these solutions cannot scale to large datasets, we have further designed three approximation algorithms to compute an SAC. We have performed an experimental evaluation for these solutions on both large real and synthetic datasets. Experimental results show that SAC is better than the communities returned by existing solutions. Moreover, our approximation solutions can find SACs accurately and efficiently.
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paper_title: Community Search over Big Graphs: Models, Algorithms, and Opportunities

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Communities serve as basic structures for understanding the organization of many real-world networks, such as social, biological, collaboration, and communication networks. Recently, community search over large graphs has attracted significantly increasing attention, from simple and static graphs to evolving, attributed, location-based graphs. Different from the well-studied problem of community detection that finds all communities in an entire network, community search is to find the cohesive communities w.r.t. the query nodes. In this tutorial, we survey the state-of-the-art of community search on various kinds of networks across different application areas such as densely-connected community search, attributed community search, social circle discovery, and querying geosocial groups. We first highlight the challenges posed by the community search problems. We continue the presentation of their principles, methodologies, algorithms, and applications, and give a comprehensive comparison of the state-of-the-art techniques. This tutorial finally concludes by offering future directions for research in this important and growing area.
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paper_title: Approximate Closest Community Search in Networks

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Recently, there has been significant interest in the study of the community search problem in social and information networks: given one or more query nodes, find densely connected communities containing the query nodes. However, most existing studies do not address the "free rider" issue, that is, nodes far away from query nodes and irrelevant to them are included in the detected community. Some state-of-the-art models have attempted to address this issue, but not only are their formulated problems NP-hard, they do not admit any approximations without restrictive assumptions, which may not always hold in practice. ::: ::: In this paper, given an undirected graph G and a set of query nodes Q, we study community search using the k-truss based community model. We formulate our problem of finding a closest truss community (CTC), as finding a connected k-truss subgraph with the largest k that contains Q, and has the minimum diameter among such subgraphs. We prove this problem is NP-hard. Furthermore, it is NP-hard to approximate the problem within a factor (2-e), for any e > 0. However, we develop a greedy algorithmic framework, which first finds a CTC containing Q, and then iteratively removes the furthest nodes from Q, from the graph. The method achieves 2-approximation to the optimal solution. To further improve the efficiency, we make use of a compact truss index and develop efficient algorithms for k-truss identification and maintenance as nodes get eliminated. In addition, using bulk deletion optimization and local exploration strategies, we propose two more efficient algorithms. One of them trades some approximation quality for efficiency while the other is a very efficient heuristic. Extensive experiments on 6 real-world networks show the effectiveness and efficiency of our community model and search algorithms.
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paper_title: Finding weighted k-truss communities in large networks

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Community search is a fundamental problem in social network mining, which has attracted much attention in recent years. However, most previous community models only consider the link structure and ignore the link weights of the community, which may miss some useful properties of the community. In this paper, we propose a novel community model, called weightedk-truss community, based on the concept of k-truss. The proposed model takes the edge weight into consideration, thus can better characterize the properties of a community. Based on the new community model, we design a BFS-based online search algorithm to find the top-rweightedk-truss communities in O(m1.5) time, where m denotes the number of edges in a network. To speed up the online search algorithm, we devise a space-efficient index structure, namely KEP-Index, to support efficient community search. We propose two algorithms to construct the index structure in an offline manner. Based on KEP-Index, the time complexity for finding the top-rweightedk-truss communities is linear to the size of these communities, thus it is optimal. We conduct extensive experiments on six large real-world networks, as well as a case study over a co-authorship network. The results demonstrate the efficiency and effectiveness of the proposed community model and algorithms.
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paper_title: An Optimal and Progressive Approach to Online Search of Top-K Influential Communities

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Community search over large graphs is a fundamental problem in graph analysis. Recent studies propose to compute top-k influential communities, where each reported community not only is a cohesive subgraph but also has a high influence value. The existing approaches to the problem of top-k influential community search can be categorized as index-based algorithms and online search algorithms without indexes. The index-based algorithms, although being very efficient in conducting community searches, need to pre-compute a special-purpose index and only work for one built-in vertex weight vector. In this paper, we investigate on-line search approaches and propose an instance-optimal algorithm LocalSearch whose time complexity is linearly proportional to the size of the smallest subgraph that a correct algorithm needs to access without indexes. In addition, we also propose techniques to make LocalSearch progressively compute and report the communities in decreasing influence value order such that k does not need to be specified. Moreover, we extend our framework to the general case of top-k influential community search regarding other cohesiveness measures. Extensive empirical studies on real graphs demonstrate that our algorithms outperform the existing online search algorithms by several orders of magnitude.
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paper_title: On Spatial-Aware Community Search

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Communities are prevalent in social networks, knowledge graphs, and biological networks. Recently, the topic of community search (CS) has received plenty of attention. The CS problem aims to look for a dense subgraph that contains a query vertex. Existing CS solutions do not consider the spatial extent of a community. They can yield communities whose locations of vertices span large areas. In applications that facilitate setting social events (e.g., finding conference attendees to join a dinner), it is important to find groups of people who are physically close to each other, so it is desirable to have a spatial-aware community (or SAC), whose vertices are close structurally and spatially. Given a graph $G$ and a query vertex $q$ , we develop an exact solution to find the SAC containing $q$ , but it cannot scale to large datasets, so we design three approximation algorithms. We further study the problem of continuous SAC search on a “dynamic spatial graph,” whose vertices’ locations change with time, and propose three fast solutions. We evaluate the solutions on both real and synthetic datasets, and the results show that SACs are better than communities returned by existing solutions. Moreover, our approximation solutions perform accurately and efficiently.
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paper_title: Efficient Computing of Radius-Bounded k-Cores

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Driven by real-life applications in geo-social networks, in this paper, we investigate the problem of computing the radius-bounded k-cores (RB-k-cores) that aims to find cohesive subgraphs satisfying both social and spatial constraints on large geo-social networks. In particular, we use k-core to ensure the social cohesiveness and we use a radius-bounded circle to restrict the locations of users in a RB-k-core. We explore several algorithmic paradigms to compute RB-k-cores, including a triple vertex-based paradigm, a binary-vertex-based paradigm, and a paradigm utilizing the concept of rotating circles. The rotating circle-based paradigm is further enhanced with several pruning techniques to achieve better efficiency. The experimental studies conducted on both real and synthetic datasets demonstrate that our proposed rotating-circle-based algorithms can compute all RB-k-cores very efficiently. Moreover, it can also be used to compute the minimum-circle-bounded k-core and significantly outperforms the existing techniques for computing the minimum circle-bounded k-core.
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paper_title: Online search of overlapping communities

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A great deal of research has been conducted on modeling and discovering communities in complex networks. In most real life networks, an object often participates in multiple overlapping communities. In view of this, recent research has focused on mining overlapping communities in complex networks. The algorithms essentially materialize a snapshot of the overlapping communities in the network. This approach has three drawbacks, however. First, the mining algorithm uses the same global criterion to decide whether a subgraph qualifies as a community. In other words, the criterion is fixed and predetermined. But in reality, communities for different vertices may have very different characteristics. Second, it is costly, time consuming, and often unnecessary to find communities for an entire network. Third, the approach does not support dynamically evolving networks. In this paper, we focus on online search of overlapping communities, that is, given a query vertex, we find meaningful overlapping communities the vertex belongs to in an online manner. In doing so, each search can use community criterion tailored for the vertex in the search. To support this approach, we introduce a novel model for overlapping communities, and we provide theoretical guidelines for tuning the model. We present several algorithms for online overlapping community search and we conduct comprehensive experiments to demonstrate the effectiveness of the model and the algorithms. We also suggest many potential applications of our model and algorithms.
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paper_title: Geo-social group queries with minimum acquaintance constraints

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The prosperity of location-based social networking services enables geo-social group queries for group-based activity planning and marketing. This paper proposes a new family of geo-social group queries with minimum acquaintance constraint (GSGQs), which are more appealing than existing geo-social group queries in terms of producing a cohesive group that guarantees the worst-case acquaintance level. GSGQs, also specified with various spatial constraints, are more complex than conventional spatial queries; particularly, those with a strict $k$NN spatial constraint are proved to be NP-hard. For efficient processing of general GSGQ queries on large location-based social networks, we devise two social-aware index structures, namely SaR-tree and SaR*-tree. The latter features a novel clustering technique that considers both spatial and social factors. Based on SaR-tree and SaR*-tree, efficient algorithms are developed to process various GSGQs. Extensive experiments on real-world Gowalla and Dianping datasets show that our proposed methods substantially outperform the baseline algorithms based on R-tree.
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paper_title: Algorithmic Graph Theory

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Preface 1. Introducing graphs and algorithmic complexity 2. Spanning-trees, branchings and connectivity 3. Planar graphs 4. Networks and flows 5. Matchings 6. Eulerian and Hamiltonian tours 7. Colouring graphs 8. Graph problems and intractability Appendix Author Index Subject Index.
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paper_title: Influential Community Search in Large Networks

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Community search is a problem of finding densely connected subgraphs that satisfy the query conditions in a network, which has attracted much attention in recent years. However, all the previous studies on community search do not consider the influence of a community. In this paper, we introduce a novel community model called k-influential community based on the concept of k-core, which can capture the influence of a community. Based on the new community model, we propose a linear-time online search algorithm to find the top-r k-influential communities in a network. To further speed up the influential community search algorithm, we devise a linear-space index structure which supports efficient search of the top-r k-influential communities in optimal time. We also propose an efficient algorithm to maintain the index when the network is frequently updated. We conduct extensive experiments on 7 real-world large networks, and the results demonstrate the efficiency and effectiveness of the proposed methods.
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paper_title: Local search of communities in large graphs

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Community search is important in social network analysis. For a given vertex in a graph, the goal is to find the best community the vertex belongs to. Intuitively, the best community for a given vertex should be in the vicinity of the vertex. However, existing solutions use \emph{global search} to find the best community. These algorithms, although straight-forward, are very costly, as all vertices in the graph may need to be visited. In this paper, we propose a \emph{local search} strategy, which searches in the neighborhood of a vertex to find the best community for the vertex. We show that, because the minimum degree measure used to evaluate the goodness of a community is not \emph{monotonic}, designing efficient local search solutions is a very challenging task. We present theories and algorithms of local search to address this challenge. The efficiency of our local search strategy is verified by extensive experiments on both synthetic networks and a variety of real networks with millions of nodes.
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paper_title: A Classification for Community Discovery Methods in Complex Networks

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Many real-world networks are intimately organized according to a community structure. Much research effort has been devoted to develop methods and algorithms that can efficiently highlight this hidden structure of a network, yielding a vast literature on what is called today community detection. Since network representation can be very complex and can contain different variants in the traditional graph model, each algorithm in the literature focuses on some of these properties and establishes, explicitly or implicitly, its own definition of community. According to this definition, each proposed algorithm then extracts the communities, which typically reflect only part of the features of real communities. The aim of this survey is to provide a ‘user manual’ for the community discovery problem. Given a meta definition of what a community in a social network is, our aim is to organize the main categories of community discovery methods based on the definition of community they adopt. Given a desired definition of community and the features of a problem (size of network, direction of edges, multidimensionality, and so on) this review paper is designed to provide a set of approaches that researchers could focus on. The proposed classification of community discovery methods is also useful for putting into perspective the many open directions for further research. © 2011 Wiley Periodicals, Inc. Statistical Analysis and Data Mining 4: 512–546, 2011 © 2011 Wiley Periodicals, Inc.
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paper_title: Network Community Detection: A Review and Visual Survey

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Community structure is an important area of research. It has received a considerable attention from the scientific community. Despite its importance, one of the key problems in locating information about community detection is the diverse spread of related articles across various disciplines. To the best of our knowledge, there is no current comprehensive review of recent literature which uses a scientometric analysis using complex networks analysis covering all relevant articles from the Web of Science (WoS). Here we present a visual survey of key literature using CiteSpace. The idea is to identify emerging trends besides using network techniques to examine the evolution of the domain. Towards that end, we identify the most influential, central, as well as active nodes using scientometric analyses. We examine authors, key articles, cited references, core subject categories, key journals, institutions, as well as countries. The exploration of the scientometric literature of the domain reveals that Yong Wang is a pivot node with the highest centrality. Additionally, we have observed that Mark Newman is the most highly cited author in the network. We have also identified that the journal, "Reviews of Modern Physics" has the strongest citation burst. In terms of cited documents, an article by Andrea Lancichinetti has the highest centrality score. We have also discovered that the origin of the key publications in this domain is from the United States. Whereas Scotland has the strongest and longest citation burst. Additionally, we have found that the categories of "Computer Science" and "Engineering" lead other categories based on frequency and centrality respectively.
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paper_title: Index-Based Densest Clique Percolation Community Search in Networks

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Community search is important in graph analysis and can be used in many real applications. In the literature, various community models have been proposed. However, most of them cannot well identify the overlaps between communities which is an essential feature of real graphs. To address this issue, the $k$ -clique percolation community model was proposed and has been proven effective in many applications. Motivated by this, in this paper, we adopt the $k$ -clique percolation community model and study the densest clique percolation community search problem which aims to find the $k$ -clique percolation community with the maximum $k$ value that contains a given set of query nodes. We adopt an index-based approach to solve this problem. Based on the observation that a $k$ -clique percolation community is a union of maximal cliques, we devise a novel compact index, $\mathsf {DCPC}$ - $\mathsf {Index}$ , to preserve the maximal cliques and their connectivity information of the input graph. With $\mathsf {DCPC}$ - $\mathsf {Index}$ , we can answer the densest clique percolation community query efficiently. Besides, we also propose an index construction algorithm based on the definition of $\mathsf {DCPC}$ - $\mathsf {Index}$ and further improve the algorithm in terms of efficiency and memory consumption. We conduct extensive performance studies on real graphs and the experimental results demonstrate the efficiency of our index-based query processing algorithm and index construction algorithm.
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paper_title: Robust Local Community Detection: On Free Rider Effect and Its Elimination

paper_content:

Given a large network, local community detection aims at finding the community that contains a set of query nodes and also maximizes (minimizes) a goodness metric. This problem has recently drawn intense research interest. Various goodness metrics have been proposed. However, most existing metrics tend to include irrelevant subgraphs in the detected local community. We refer to such irrelevant subgraphs as free riders. We systematically study the existing goodness metrics and provide theoretical explanations on why they may cause the free rider effect. We further develop a query biased node weighting scheme to reduce the free rider effect. In particular, each node is weighted by its proximity to the query node. We define a query biased density metric to integrate the edge and node weights. The query biased densest subgraph, which has the largest query biased density, will shift to the neighborhood of the query nodes after node weighting. We then formulate the query biased densest connected subgraph (QDC) problem, study its complexity, and provide efficient algorithms to solve it. We perform extensive experiments on a variety of real and synthetic networks to evaluate the effectiveness and efficiency of the proposed methods.
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paper_title: An O(m) Algorithm for Cores Decomposition of Networks

paper_content:

The structure of large networks can be revealed by partitioning them to smaller parts, which are easier to handle. One of such decompositions is based on $k$--cores, proposed in 1983 by Seidman. In the paper an efficient, $O(m)$, $m$ is the number of lines, algorithm for determining the cores decomposition of a given network is presented.
---
paper_title: On Social-Temporal Group Query with Acquaintance Constraint

paper_content:

Three essential criteria are important for activity planning, including: (1) finding a group of attendees familiar with the initiator, (2) ensuring each attendee in the group to have tight social relations with most of the members in the group, and (3) selecting an activity period available for all attendees. Therefore, this paper proposes Social-Temporal Group Query to find the activity time and attendees with the minimum total social distance to the initiator. Moreover, this query incorporates an acquaintance constraint to avoid finding a group with mutually unfamiliar attendees. Efficient processing of the social-temporal group query is very challenging. We show that the problem is NP-hard via a proof and formulate the problem with Integer Programming. We then propose two efficient algorithms, SGSelect and STGSelect, which include effective pruning techniques and employ the idea of pivot time slots to substantially reduce the running time, for finding the optimal solutions. Experimental results indicate that the proposed algorithms are much more efficient and scalable. In the comparison of solution quality, we show that STGSelect outperforms the algorithm that represents manual coordination by the initiator.
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paper_title: Query Optimal k-Plex Based Community in Graphs

paper_content:

Community search problem, which is to find good communities given a set of query nodes in a graph, has attracted increasing research interest recently. Though various measurement models have been proposed to define and solve community search problem. Few of them could define a community concisely and have good quality of query results. They either involve additional constraints for modeling communities, such as size and diameter, or suffer from the free rider effect, i.e., include irrelevant subgraphs. In this paper, we propose a new k-plex based community model for community search. We show that our model not only is simple and clear, but also meets with basic requirements of defining a community search problem. We formulate the maximum k-plex community query (MCKPQ) problem, that is, given a set of query nodes Q, searching for optimal k-plex containing Q. We prove that MCKPQ is NP-hard, and it is hard to approximate in any constant factor. We first give exact solutions. Then, we propose an efficient branch-and-bound (B&B) method and design an effective upper bound function and a pruning strategy. Furthermore, we optimize the basic B&B by fast candidate generation. We also give a fast heuristic solution, which produces high-quality results in practice. The effectiveness of our model of community and the efficiency of our methods are verified by elaborate experiments.
---
paper_title: Communities in Networks

paper_content:

We survey some of the concepts, methods, and applications of community detection, which has become an increasingly important area of network science. To help ease newcomers into the field, we provide a guide to available methodology and open problems, and discuss why scientists from diverse backgrounds are interested in these problems. As a running theme, we emphasize the connections of community detection to problems in statistical physics and computational optimization.
---
paper_title: Keyword searching and browsing in databases using BANKS

paper_content:

With the growth of the Web, there has been a rapid increase in the number of users who need to access online databases without having a detailed knowledge of the schema or of query languages; even relatively simple query languages designed for non-experts are too complicated for them. We describe BANKS, a system which enables keyword-based search on relational databases, together with data and schema browsing. BANKS enables users to extract information in a simple manner without any knowledge of the schema or any need for writing complex queries. A user can get information by typing a few keywords, following hyperlinks, and interacting with controls on the displayed results. BANKS models tuples as nodes in a graph, connected by links induced by foreign key and other relationships. Answers to a query are modeled as rooted trees connecting tuples that match individual keywords in the query. Answers are ranked using a notion of proximity coupled with a notion of prestige of nodes based on inlinks, similar to techniques developed for Web search. We present an efficient heuristic algorithm for finding and ranking query results.
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paper_title: Most Influential Community Search over Large Social Networks

paper_content:

Detecting social communities in large social networks provides an effective way to analyze the social media users' behaviors and activities. It has drawn extensive attention from both academia and industry. One essential aspect of communities in social networks is outer influence which is the capability to spread internal information of communities to external users. Detecting the communities of high outer influence has particular interest in a wide range of applications, e.g., Ads trending analytics, social opinion mining and news propagation pattern discovery. However, the existing detection techniques largely ignore the outer influence of the communities. To fill the gap, this work investigates the Most Influential Community Search problem to disclose the communities with the highest outer influences. We firstly propose a new community model, maximal kr-Clique community, which has desirable properties, i.e., society, cohesiveness, connectivity, and maximum. Then, we design a novel tree-based index structure, denoted as C-Tree, to maintain the offline computed r-cliques. To efficiently search the most influential communities, we also develop four advanced index-based algorithms which improve the search performance of non-indexed solution by about 200 times. The efficiency and effectiveness of our solution have been extensively verified using six real datasets and a small case study.
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paper_title: Effective and efficient attributed community search

paper_content:

Given a graph G and a vertex $$q \in G$$ ::: , the community search query returns a subgraph of G that contains vertices related to q. Communities, which are prevalent in attributed graphs such as social networks and knowledge bases, can be used in emerging applications such as product advertisement and setting up of social events. In this paper, we investigate the attributed community query (or ACQ), which returns an attributed community (AC) for an attributed graph. The AC is a subgraph of G, which satisfies both structure cohesiveness (i.e., its vertices are tightly connected) and keyword cohesiveness (i.e., its vertices share common keywords). The AC enables a better understanding of how and why a community is formed (e.g., members of an AC have a common interest in music, because they all have the same keyword “music”). An AC can be “personalized”; for example, an ACQ user may specify that an AC returned should be related to some specific keywords like “research” and “sports”. To enable efficient AC search, we develop the CL-tree index structure and three algorithms based on it. We further propose efficient algorithms for maintaining the index on dynamic graphs. Moreover, we study two problems that are related to the ACQ problem. We evaluate our solutions on six large graphs. Our results show that ACQ is more effective and efficient than existing community retrieval approaches. Moreover, an AC contains more precise and personalized information than that of existing community search and detection methods.
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paper_title: Finding local community structure in networks.

paper_content:

Although the inference of global community structure in networks has recently become a topic of great interest in the physics community, all such algorithms require that the graph be completely known. Here, we define both a measure of local community structure and an algorithm that infers the hierarchy of communities that enclose a given vertex by exploring the graph one vertex at a time. This algorithm runs in time O(k2d) for general graphs when d is the mean degree and k is the number of vertices to be explored. For graphs where exploring a new vertex is time consuming, the running time is linear, O(k). We show that on computer-generated graphs the average behavior of this technique approximates that of algorithms that require global knowledge. As an application, we use this algorithm to extract meaningful local clustering information in the large recommender network of an online retailer.
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paper_title: Efficiently computing k-edge connected components via graph decomposition

paper_content:

Efficiently computing k-edge connected components in a large graph, G = (V, E), where V is the vertex set and E is the edge set, is a long standing research problem. It is not only fundamental in graph analysis but also crucial in graph search optimization algorithms. Consider existing techniques for computing k-edge connected components are quite time consuming and are unlikely to be scalable for large scale graphs, in this paper we firstly propose a novel graph decomposition paradigm to iteratively decompose a graph G for computing its k-edge connected components such that the number of drilling-down iterations h is bounded by the "depth" of the k-edge connected components nested together to form G, where h usually is a small integer in practice. Secondly, we devise a novel, efficient threshold-based graph decomposition algorithm, with time complexity O(l × |E|), to decompose a graph G at each iteration, where l usually is a small integer with l « |V|. As a result, our algorithm for computing k-edge connected components significantly improves the time complexity of an existing state-of-the-art technique from O(|V|2|E| + |V|3 log |V|) to O(h × l × |E|). Finally, we conduct extensive performance studies on large real and synthetic graphs. The performance studies demonstrate that our techniques significantly outperform the state-of-the-art solution by several orders of magnitude.
---
paper_title: Effective Community Search for Large Attributed Graphs

paper_content:

Given a graph G and a vertex q ∈ G, the community search query returns a subgraph of G that contains vertices related to q. Communities, which are prevalent in attributed graphs such as social networks and knowledge bases, can be used in emerging applications such as product advertisement and setting up of social events. In this paper, we investigate the attributed community query (or ACQ), which returns an attributed community (AC) for an attributed graph. The AC is a subgraph of G, which satisfies both structure cohesiveness (i.e., its vertices are tightly connected) and keyword cohesiveness (i.e., its vertices share common keywords). The AC enables a better understanding of how and why a community is formed (e.g., members of an AC have a common interest in music, because they all have the same keyword "music"). An AC can be "personalized"; for example, an ACQ user may specify that an AC returned should be related to some specific keywords like "research" and "sports". ::: ::: To enable efficient AC search, we develop the CL-tree index structure and three algorithms based on it. We evaluate our solutions on four large graphs, namely Flickr, DBLP, Tencent, and DBpedia. Our results show that ACs are more effective and efficient than existing community retrieval approaches. Moreover, an AC contains more precise and personalized information than that of existing community search and detection methods.
---
paper_title: Community discovery in social networks: Applications, methods and emerging trends

paper_content:

Data sets originating from many different real world domains can be represented in the form of interaction networks in a very natural, concise and meaningful fashion. This is particularly true in the social context, especially given recent advances in Internet technologies and Web 2.0 applications leading to a diverse range of evolving social networks. Analysis of such networks can result in the discovery of important patterns and potentially shed light on important properties governing the growth of such networks.
---
paper_title: A Tutorial on Spectral Clustering

paper_content:

In recent years, spectral clustering has become one of the most popular modern clustering algorithms. It is simple to implement, can be solved efficiently by standard linear algebra software, and very often outperforms traditional clustering algorithms such as the k-means algorithm. On the first glance spectral clustering appears slightly mysterious, and it is not obvious to see why it works at all and what it really does. The goal of this tutorial is to give some intuition on those questions. We describe different graph Laplacians and their basic properties, present the most common spectral clustering algorithms, and derive those algorithms from scratch by several different approaches. Advantages and disadvantages of the different spectral clustering algorithms are discussed.
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paper_title: Graph structure in the Web

paper_content:

The study of the web as a graph is not only fascinating in its own right, but also yields valuable insight into web algorithms for crawling, searching and community discovery, and the sociological phenomena which characterize its evolution. We report on experiments on local and global properties of the web graph using two Altavista crawls each with over 200 million pages and 1.5 billion links. Our study indicates that the macroscopic structure of the web is considerably more intricate than suggested by earlier experiments on a smaller scale.
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paper_title: Exploring Local Community Structures in Large Networks

paper_content:

In this paper, we extend the concept of degree from single vertex to sub-graph, and present a formal definition of module/community in a network based on this extension. A new locally optimized algorithm is designed to find the module for a given source vertex in a network. Our analysis shows that the complexity of this algorithm is O(K2d), where K is the number of vertices to be explored in the sub-graph and d is the average degree of the vertices in the sub-graph. Based on this algorithm, we implement a JAVA tool, MoNet, for exploring local community structures in large networks. Using this tool to analyze a co-purchase network from Amazon shows that there are local community structures in this network. Further analyses on these local community structures demonstrate that media items are much easier to form compact local modules than book items do, indicating that recommending digital media items to customers based on co-purchasing information in the online store will be more efficient than recommending books.
---
paper_title: Uncovering the overlapping community structure of complex networks in nature and society

paper_content:

Many complex systems in nature and society can be described in terms of networks capturing the intricate web of connections among the units they are made of. A key question is how to interpret the global organization of such networks as the coexistence of their structural subunits (communities) associated with more highly interconnected parts. Identifying these a priori unknown building blocks (such as functionally related proteins, industrial sectors and groups of people) is crucial to the understanding of the structural and functional properties of networks. The existing deterministic methods used for large networks find separated communities, whereas most of the actual networks are made of highly overlapping cohesive groups of nodes. Here we introduce an approach to analysing the main statistical features of the interwoven sets of overlapping communities that makes a step towards uncovering the modular structure of complex systems. After defining a set of new characteristic quantities for the statistics of communities, we apply an efficient technique for exploring overlapping communities on a large scale. We find that overlaps are significant, and the distributions we introduce reveal universal features of networks. Our studies of collaboration, word-association and protein interaction graphs show that the web of communities has non-trivial correlations and specific scaling properties.
---
paper_title: Truss Decomposition in Massive Networks

paper_content:

The k-truss is a type of cohesive subgraphs proposed recently for the study of networks. While the problem of computing most cohesive subgraphs is NP-hard, there exists a polynomial time algorithm for computing k-truss. Compared with k-core which is also efficient to compute, k-truss represents the "core" of a k-core that keeps the key information of, while filtering out less important information from, the k-core. However, existing algorithms for computing k-truss are inefficient for handling today's massive networks. We first improve the existing in-memory algorithm for computing k-truss in networks of moderate size. Then, we propose two I/O-efficient algorithms to handle massive networks that cannot fit in main memory. Our experiments on real datasets verify the efficiency of our algorithms and the value of k-truss.
---
paper_title: Spatial Networks

paper_content:

Complex systems are very often organized under the form of networks where nodes and edges are embedded in space. Transportation and mobility networks, Internet, mobile phone networks, power grids, social and contact networks, neural networks, are all examples where space is relevant and where topology alone does not contain all the information. Characterizing and understanding the structure and the evolution of spatial networks is thus crucial for many different fields ranging from urbanism to epidemiology. An important consequence of space on networks is that there is a cost associated to the length of edges which in turn has dramatic effects on the topological structure of these networks. We will expose thoroughly the current state of our understanding of how the spatial constraints affect the structure and properties of these networks. We will review the most recent empirical observations and the most important models of spatial networks. We will also discuss various processes which take place on these spatial networks, such as phase transitions, random walks, synchronization, navigation, resilience, and disease spread.
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paper_title: Listing $k$-cliques in Sparse Real-World Graphs

paper_content:

Motivated by recent studies in the data mining community, we develop the most efficient parallel algorithm for listing all k-cliques in a graph. Our theoretical analysis shows that our algorithm boasts the best asymptotic upper bound on the running time for the case when the input graph is sparse. Our experimental evaluation on large real-world graphs demonstrates that our parallel algorithm is faster than state-of-the-art algorithms, while boasting an excellent degree of parallelism. In particular, we are able to list all k-cliques (for any value of k) in graphs containing up to tens of millions of edges as well as all 10-cliques in graphs containing billions of edges, within a few minutes and a few hours respectively. We show how it can be employed as an effective subroutine for finding the k-clique core decomposition and an approximate k-clique densest subgraphs in very large real-world graphs.
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paper_title: Linear-time enumeration of maximal K-edge-connected subgraphs in large networks by random contraction

paper_content:

Capturing sets of closely related vertices from large networks is an essential task in many applications such as social network analysis, bioinformatics, and web link research. Decomposing a graph into k-core components is a standard and efficient method for this task, but obtained clusters might not be well-connected. The idea of using maximal k-edge-connected subgraphs was recently proposed to address this issue. Although we can obtain better clusters with this idea, the state-of-the-art method is not efficient enough to process large networks with millions of vertices. In this paper, we propose a new method to decompose a graph into maximal k-edge-connected components, based on random contraction of edges. Our method is simple to implement but improves performance drastically. We experimentally show that our method can successfully decompose large networks and it is thousands times faster than the previous method. Also, we theoretically explain why our method is efficient in practice. To see the importance of maximal k-edge-connected subgraphs, we also conduct experiments using real-world networks to show that many k-core components have small edge-connectivity and they can be decomposed into a lot of maximal k-edge-connected subgraphs.
---
paper_title: An O(m) Algorithm for Cores Decomposition of Networks

paper_content:

The structure of large networks can be revealed by partitioning them to smaller parts, which are easier to handle. One of such decompositions is based on $k$--cores, proposed in 1983 by Seidman. In the paper an efficient, $O(m)$, $m$ is the number of lines, algorithm for determining the cores decomposition of a given network is presented.
---
paper_title: Efficiently computing k-edge connected components via graph decomposition

paper_content:

Efficiently computing k-edge connected components in a large graph, G = (V, E), where V is the vertex set and E is the edge set, is a long standing research problem. It is not only fundamental in graph analysis but also crucial in graph search optimization algorithms. Consider existing techniques for computing k-edge connected components are quite time consuming and are unlikely to be scalable for large scale graphs, in this paper we firstly propose a novel graph decomposition paradigm to iteratively decompose a graph G for computing its k-edge connected components such that the number of drilling-down iterations h is bounded by the "depth" of the k-edge connected components nested together to form G, where h usually is a small integer in practice. Secondly, we devise a novel, efficient threshold-based graph decomposition algorithm, with time complexity O(l × |E|), to decompose a graph G at each iteration, where l usually is a small integer with l « |V|. As a result, our algorithm for computing k-edge connected components significantly improves the time complexity of an existing state-of-the-art technique from O(|V|2|E| + |V|3 log |V|) to O(h × l × |E|). Finally, we conduct extensive performance studies on large real and synthetic graphs. The performance studies demonstrate that our techniques significantly outperform the state-of-the-art solution by several orders of magnitude.
---
paper_title: Truss Decomposition in Massive Networks

paper_content:

The k-truss is a type of cohesive subgraphs proposed recently for the study of networks. While the problem of computing most cohesive subgraphs is NP-hard, there exists a polynomial time algorithm for computing k-truss. Compared with k-core which is also efficient to compute, k-truss represents the "core" of a k-core that keeps the key information of, while filtering out less important information from, the k-core. However, existing algorithms for computing k-truss are inefficient for handling today's massive networks. We first improve the existing in-memory algorithm for computing k-truss in networks of moderate size. Then, we propose two I/O-efficient algorithms to handle massive networks that cannot fit in main memory. Our experiments on real datasets verify the efficiency of our algorithms and the value of k-truss.
---
paper_title: Defining and evaluating network communities based on ground-truth

paper_content:

Nodes in real-world networks, such as social, information or technological networks, organize into communities where edges appear with high concentration among the members of the community. Identifying communities in networks has proven to be a challenging task mainly due to a plethora of definitions of a community, intractability of algorithms, issues with evaluation and the lack of a reliable gold-standard ground-truth. We study a set of 230 large social, collaboration and information networks where nodes explicitly define group memberships. We use these groups to define the notion of ground-truth communities. We then propose a methodology which allows us to compare and quantitatively evaluate different definitions of network communities on a large scale. We choose 13 commonly used definitions of network communities and examine their quality, sensitivity and robustness. We show that the 13 definitions naturally group into four classes. We find that two of these definitions, Conductance and Triad-participation-ratio, consistently give the best performance in identifying ground-truth communities.
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paper_title: Community Detection in Multi-Layer Graphs: A Survey

paper_content:

Community detection, also known as graph clustering, has been extensively studied in the literature. The goal of community detection is to partition vertices in a complex graph into densely-connected components socalled communities. In recent applications, however, an entity is associated with multiple aspects of relationships, which brings new challenges in community detection. The multiple aspects of interactions can be modeled as a multi-layer graph comprised of multiple interdependent graphs, where each graph represents an aspect of the interactions. Great efforts have therefore been made to tackle the problem of community detection in multi-layer graphs. In this survey, we provide readers with a comprehensive understanding of community detection in multi-layer graphs and compare the state-of-the-art algorithms with respect to their underlying properties.
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paper_title: Comparing community structure identification

paper_content:

We compare recent approaches to community structure identification in terms of sensitivity and computational cost. The recently proposed modularity measure is revisited and the performance of the methods as applied to ad hoc networks with known community structure, is compared. We find that the most accurate methods tend to be more computationally expensive, and that both aspects need to be considered when choosing a method for practical purposes. The work is intended as an introduction as well as a proposal for a standard benchmark test of community detection methods.
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paper_title: Discovering Communities from Social Networks: Methodologies and Applications

paper_content:

A network community refers to a group of network nodes within which the links connecting them are dense but between which they are sparse. A network community mining problem (NCMP for short) can be stated as the problem of finding all such communities from a given network. A large variety of problems can be translated into NCMPs, ranging from graph partition applications, such as VLSI layout [1, 2], large-scale scientific computing [3, 4], load balance for distributed computing [5, 6], and image segmentation [19], to complex network analysis, such as social network analysis [7, 8, 9], biological network analysis[11, 12, 24], and Web pages clustering [13, 14, 15]. Network communities in different application contexts may imply different meanings and serve different purposes. For example, they may be sets of electronic units closely placed on a VLSI circuit board, collections of processes that frequently communicate with each other, segments of an image, circles of a society within which people share common interests and keep frequent contacts, classes of Web pages related to common topics, or groups of proteins exhibiting similar functions. Discovering such hidden patterns from networks will enable us to better understand the structural and/or functional characteristics of networks and to more effectively utilize them. The remainder of this chapter is organized as follows: Section 2 introduces some of the existing methods for solving NCMPs. Section 3 presents several interesting applications of NCMPs as related to social network analysis. Finally, Section 4 concludes the chapter.
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paper_title: The art of community detection

paper_content:

Networks in nature possess a remarkable amount of structure. Via a series of data-driven discoveries, the cutting edge of network science has recently progressed from positing that the random graphs of mathematical graph theory might accurately describe real networks to the current viewpoint that networks in nature are highly complex and structured entities. The identification of high order structures in networks unveils insights into their functional organization. Recently, Clauset, Moore, and Newman, introduced a new algorithm that identifies such heterogeneities in complex networks by utilizing the hierarchy that necessarily organizes the many levels of structure. Here, we anchor their algorithm in a general community detection framework and discuss the future of community detection.
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paper_title: Community Discovery in Dynamic Networks: A Survey

paper_content:

Several research studies have shown that complex networks modeling real-world phenomena are characterized by striking properties: (i) they are organized according to community structure, and (ii) their structure evolves with time. Many researchers have worked on methods that can efficiently unveil substructures in complex networks, giving birth to the field of community discovery. A novel and fascinating problem started capturing researcher interest recently: the identification of evolving communities. Dynamic networks can be used to model the evolution of a system: nodes and edges are mutable, and their presence, or absence, deeply impacts the community structure that composes them. ::: ::: This survey aims to present the distinctive features and challenges of dynamic community discovery and propose a classification of published approaches. As a “user manual,” this work organizes state-of-the-art methodologies into a taxonomy, based on their rationale, and their specific instantiation. Given a definition of network dynamics, desired community characteristics, and analytical needs, this survey will support researchers to identify the set of approaches that best fit their needs. The proposed classification could also help researchers choose in which direction to orient their future research.
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paper_title: Survey on Social Community Detection

paper_content:

Community detection is a growing field of interest in the area of social network applications. Many community detection methods and surveys have been introduced in recent years, with each such method being classified according to its algorithm type. This chapter presents an original survey on this topic, featuring a new approach based on both semantics and type of output. Semantics opens up new perspectives and allows interpreting high-order social relations. A special focus is also given to community evaluation since this step becomes important in social data mining.
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paper_title: A Classification for Community Discovery Methods in Complex Networks

paper_content:

Many real-world networks are intimately organized according to a community structure. Much research effort has been devoted to develop methods and algorithms that can efficiently highlight this hidden structure of a network, yielding a vast literature on what is called today community detection. Since network representation can be very complex and can contain different variants in the traditional graph model, each algorithm in the literature focuses on some of these properties and establishes, explicitly or implicitly, its own definition of community. According to this definition, each proposed algorithm then extracts the communities, which typically reflect only part of the features of real communities. The aim of this survey is to provide a ‘user manual’ for the community discovery problem. Given a meta definition of what a community in a social network is, our aim is to organize the main categories of community discovery methods based on the definition of community they adopt. Given a desired definition of community and the features of a problem (size of network, direction of edges, multidimensionality, and so on) this review paper is designed to provide a set of approaches that researchers could focus on. The proposed classification of community discovery methods is also useful for putting into perspective the many open directions for further research. © 2011 Wiley Periodicals, Inc. Statistical Analysis and Data Mining 4: 512–546, 2011 © 2011 Wiley Periodicals, Inc.
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paper_title: Network Community Detection: A Review and Visual Survey

paper_content:

Community structure is an important area of research. It has received a considerable attention from the scientific community. Despite its importance, one of the key problems in locating information about community detection is the diverse spread of related articles across various disciplines. To the best of our knowledge, there is no current comprehensive review of recent literature which uses a scientometric analysis using complex networks analysis covering all relevant articles from the Web of Science (WoS). Here we present a visual survey of key literature using CiteSpace. The idea is to identify emerging trends besides using network techniques to examine the evolution of the domain. Towards that end, we identify the most influential, central, as well as active nodes using scientometric analyses. We examine authors, key articles, cited references, core subject categories, key journals, institutions, as well as countries. The exploration of the scientometric literature of the domain reveals that Yong Wang is a pivot node with the highest centrality. Additionally, we have observed that Mark Newman is the most highly cited author in the network. We have also identified that the journal, "Reviews of Modern Physics" has the strongest citation burst. In terms of cited documents, an article by Andrea Lancichinetti has the highest centrality score. We have also discovered that the origin of the key publications in this domain is from the United States. Whereas Scotland has the strongest and longest citation burst. Additionally, we have found that the categories of "Computer Science" and "Engineering" lead other categories based on frequency and centrality respectively.
---
paper_title: Communities in Networks

paper_content:

We survey some of the concepts, methods, and applications of community detection, which has become an increasingly important area of network science. To help ease newcomers into the field, we provide a guide to available methodology and open problems, and discuss why scientists from diverse backgrounds are interested in these problems. As a running theme, we emphasize the connections of community detection to problems in statistical physics and computational optimization.
---
paper_title: Empirical comparison of algorithms for network community detection

paper_content:

Detecting clusters or communities in large real-world graphs such as large social or information networks is a problem of considerable interest. In practice, one typically chooses an objective function that captures the intuition of a network cluster as set of nodes with better internal connectivity than external connectivity, and then one applies approximation algorithms or heuristics to extract sets of nodes that are related to the objective function and that "look like" good communities for the application of interest. In this paper, we explore a range of network community detection methods in order to compare them and to understand their relative performance and the systematic biases in the clusters they identify. We evaluate several common objective functions that are used to formalize the notion of a network community, and we examine several different classes of approximation algorithms that aim to optimize such objective functions. In addition, rather than simply fixing an objective and asking for an approximation to the best cluster of any size, we consider a size-resolved version of the optimization problem. Considering community quality as a function of its size provides a much finer lens with which to examine community detection algorithms, since objective functions and approximation algorithms often have non-obvious size-dependent behavior.
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paper_title: Community detection in large‐scale networks: a survey and empirical evaluation

paper_content:

Community detection is a common problem in graph data analytics that consists of finding groups of densely connected nodes with few connections to nodes outside of the group. In particular, identifying communities in large-scale networks is an important task in many scientific domains. In this review, we evaluated eight state-of-the-art and five traditional algorithms for overlapping and disjoint community detection on large-scale real-world networks with known ground-truth communities. These 13 algorithms were empirically compared using goodness metrics that measure the structural properties of the identified communities, as well as performance metrics that evaluate these communities against the ground-truth. Our results show that these two types of metrics are not equivalent. That is, an algorithm may perform well in terms of goodness metrics, but poorly in terms of performance metrics, or vice versa. © 2014 The Authors. WIREs Computational
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paper_title: Community discovery in social networks: Applications, methods and emerging trends

paper_content:

Data sets originating from many different real world domains can be represented in the form of interaction networks in a very natural, concise and meaningful fashion. This is particularly true in the social context, especially given recent advances in Internet technologies and Web 2.0 applications leading to a diverse range of evolving social networks. Analysis of such networks can result in the discovery of important patterns and potentially shed light on important properties governing the growth of such networks.
---
paper_title: Community detection in graphs

paper_content:

The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i.e. the organization of vertices in clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Such clusters, or communities, can be considered as fairly independent compartments of a graph, playing a similar role like, e.g., the tissues or the organs in the human body. Detecting communities is of great importance in sociology, biology and computer science, disciplines where systems are often represented as graphs. This problem is very hard and not yet satisfactorily solved, despite the huge effort of a large interdisciplinary community of scientists working on it over the past few years. We will attempt a thorough exposition of the topic, from the definition of the main elements of the problem, to the presentation of most methods developed, with a special focus on techniques designed by statistical physicists, from the discussion of crucial issues like the significance of clustering and how methods should be tested and compared against each other, to the description of applications to real networks.
---
paper_title: Network structure and minimum degree

paper_content:

Abstract Social network researchers have long sought measures of network cohesion, Density has often been used for this purpose, despite its generally admitted deficiencies. An approach to network cohesion is proposed that is based on minimum degree and which produces a sequence of subgraphs of gradually increasing cohesion. The approach also associates with any network measures of local density which promise to be useful both in characterizing network structures and in comparing networks.
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paper_title: The community-search problem and how to plan a successful cocktail party

paper_content:

A lot of research in graph mining has been devoted in the discovery of communities. Most of the work has focused in the scenario where communities need to be discovered with only reference to the input graph. However, for many interesting applications one is interested in finding the community formed by a given set of nodes. In this paper we study a query-dependent variant of the community-detection problem, which we call the community-search problem: given a graph G, and a set of query nodes in the graph, we seek to find a subgraph of G that contains the query nodes and it is densely connected. We motivate a measure of density based on minimum degree and distance constraints, and we develop an optimum greedy algorithm for this measure. We proceed by characterizing a class of monotone constraints and we generalize our algorithm to compute optimum solutions satisfying any set of monotone constraints. Finally we modify the greedy algorithm and we present two heuristic algorithms that find communities of size no greater than a specified upper bound. Our experimental evaluation on real datasets demonstrates the efficiency of the proposed algorithms and the quality of the solutions we obtain.
---
paper_title: Local search of communities in large graphs

paper_content:

Community search is important in social network analysis. For a given vertex in a graph, the goal is to find the best community the vertex belongs to. Intuitively, the best community for a given vertex should be in the vicinity of the vertex. However, existing solutions use \emph{global search} to find the best community. These algorithms, although straight-forward, are very costly, as all vertices in the graph may need to be visited. In this paper, we propose a \emph{local search} strategy, which searches in the neighborhood of a vertex to find the best community for the vertex. We show that, because the minimum degree measure used to evaluate the goodness of a community is not \emph{monotonic}, designing efficient local search solutions is a very challenging task. We present theories and algorithms of local search to address this challenge. The efficiency of our local search strategy is verified by extensive experiments on both synthetic networks and a variety of real networks with millions of nodes.
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paper_title: Greedy approximation algorithms for finding dense components in a graph

paper_content:

We study the problem of finding highly connected subgraphs of undirected and directed graphs. For undirected graphs, the notion of density of a subgraph we use is the average degree of the subgraph. For directed graphs, a corresponding notion of density was introduced recently by Kannan and Vinay. This is designed to quantify highly connectedness of substructures in a sparse directed graph such as the web graph. We study the optimization problems of finding subgraphs maximizing these notions of density for undirected and directed graphs. This paper gives simple greedy approximation algorithms for these optimization problems. We also answer an open question about the complexity of the optimization problem for directed graphs.
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paper_title: A linear-time algorithm for a special case of disjoint set union

paper_content:

This paper presents a linear-time algorithm for the special case of the disjoint set union problem in which the structure of the unions (defined by a “union tree”) is known in advance. The algorithm executes an intermixed sequence of m union and find operations on n elements in O(m+n) time and O(n) space. This is a slight but theoretically significant improvement over the fastest known algorithm for the general problem, which runs in O(mα(m+n, n)+n) time and O(n) space, where a is a functional inverse of Ackermann's function. Used as a subroutine, the algorithm gives similar improvements in the efficiency of algorithms for solving several other problems, including two-processor scheduling, matching on convex graphs, finding nearest common ancestors off-line, testing a flow graph for reducibility, and finding two disjoint directed spanning trees. The algorithm obtains its efficiency by combining the fast algorithm for the general problem with table look-up on small sets, and requires a random access machine for its implementation. The algorithm extends to the case in which single-node additions to the union tree are allowed. The extended algorithm is useful in finding maximum cardinality matchings in nonbipartite graphs.
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paper_title: Local search of communities in large graphs

paper_content:

Community search is important in social network analysis. For a given vertex in a graph, the goal is to find the best community the vertex belongs to. Intuitively, the best community for a given vertex should be in the vicinity of the vertex. However, existing solutions use \emph{global search} to find the best community. These algorithms, although straight-forward, are very costly, as all vertices in the graph may need to be visited. In this paper, we propose a \emph{local search} strategy, which searches in the neighborhood of a vertex to find the best community for the vertex. We show that, because the minimum degree measure used to evaluate the goodness of a community is not \emph{monotonic}, designing efficient local search solutions is a very challenging task. We present theories and algorithms of local search to address this challenge. The efficiency of our local search strategy is verified by extensive experiments on both synthetic networks and a variety of real networks with millions of nodes.
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paper_title: Effective Community Search for Large Attributed Graphs

paper_content:

Given a graph G and a vertex q ∈ G, the community search query returns a subgraph of G that contains vertices related to q. Communities, which are prevalent in attributed graphs such as social networks and knowledge bases, can be used in emerging applications such as product advertisement and setting up of social events. In this paper, we investigate the attributed community query (or ACQ), which returns an attributed community (AC) for an attributed graph. The AC is a subgraph of G, which satisfies both structure cohesiveness (i.e., its vertices are tightly connected) and keyword cohesiveness (i.e., its vertices share common keywords). The AC enables a better understanding of how and why a community is formed (e.g., members of an AC have a common interest in music, because they all have the same keyword "music"). An AC can be "personalized"; for example, an ACQ user may specify that an AC returned should be related to some specific keywords like "research" and "sports". ::: ::: To enable efficient AC search, we develop the CL-tree index structure and three algorithms based on it. We evaluate our solutions on four large graphs, namely Flickr, DBLP, Tencent, and DBpedia. Our results show that ACs are more effective and efficient than existing community retrieval approaches. Moreover, an AC contains more precise and personalized information than that of existing community search and detection methods.
---
paper_title: The community-search problem and how to plan a successful cocktail party

paper_content:

A lot of research in graph mining has been devoted in the discovery of communities. Most of the work has focused in the scenario where communities need to be discovered with only reference to the input graph. However, for many interesting applications one is interested in finding the community formed by a given set of nodes. In this paper we study a query-dependent variant of the community-detection problem, which we call the community-search problem: given a graph G, and a set of query nodes in the graph, we seek to find a subgraph of G that contains the query nodes and it is densely connected. We motivate a measure of density based on minimum degree and distance constraints, and we develop an optimum greedy algorithm for this measure. We proceed by characterizing a class of monotone constraints and we generalize our algorithm to compute optimum solutions satisfying any set of monotone constraints. Finally we modify the greedy algorithm and we present two heuristic algorithms that find communities of size no greater than a specified upper bound. Our experimental evaluation on real datasets demonstrates the efficiency of the proposed algorithms and the quality of the solutions we obtain.
---
paper_title: A faster approximation algorithm for the Steiner problem in graphs

paper_content:

Abstract We present a new implementation of the Kou, Markowsky and Berman algorithm for finding a Steiner tree for a connected, undirected distance graph with a specified subset S of the set of vertices V . The total distance of all edges of this Steiner tree is at most 2(1-1/ l ) times that of a Steiner minimal tree, where l is the minimum number of leaves in any Steiner minimal tree for the given graph. The algorithm runs in O(| E |+| V |log| V |) time in the worst case, where E is the set of all edges and V the set of all vertices in the graph.
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paper_title: The community-search problem and how to plan a successful cocktail party

paper_content:

A lot of research in graph mining has been devoted in the discovery of communities. Most of the work has focused in the scenario where communities need to be discovered with only reference to the input graph. However, for many interesting applications one is interested in finding the community formed by a given set of nodes. In this paper we study a query-dependent variant of the community-detection problem, which we call the community-search problem: given a graph G, and a set of query nodes in the graph, we seek to find a subgraph of G that contains the query nodes and it is densely connected. We motivate a measure of density based on minimum degree and distance constraints, and we develop an optimum greedy algorithm for this measure. We proceed by characterizing a class of monotone constraints and we generalize our algorithm to compute optimum solutions satisfying any set of monotone constraints. Finally we modify the greedy algorithm and we present two heuristic algorithms that find communities of size no greater than a specified upper bound. Our experimental evaluation on real datasets demonstrates the efficiency of the proposed algorithms and the quality of the solutions we obtain.
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paper_title: A fast algorithm for Steiner trees

paper_content:

Given an undirected distance graph G=(V, E, d) and a set S, where V is the set of vertices in G, E is the set of edges in G, d is a distance function which maps E into the set of nonnegative numbers and S?V is a subset of the vertices of V, the Steiner tree problem is to find a tree of G that spans S with minimal total distance on its edges. In this paper, we analyze a heuristic algorithm for the Steiner tree problem. The heuristic algorithm has a worst case time complexity of O(¦S¦¦V¦ 2) on a random access computer and it guarantees to output a tree that spans S with total distance on its edges no more than 2(1?1/l) times that of the optimal tree, where l is the number of leaves in the optimal tree.
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paper_title: Effective and Efficient Community Search Over Large Directed Graphs

paper_content:

Communities are prevalent in social networks, knowledge graphs, and biological networks. Recently, the topic of community search (CS), extracting a dense subgraph containing a query vertex $q$q from a graph, has received great attention. However, existing CS solutions are designed for undirected graphs, and overlook directions of edges which potentially lose useful information carried on directions. In many applications (e.g., Twitter), users’ relationships are often modeled as directed graphs (e.g., if a user $a$a follows another user $b$b, then there is an edge from $a$a to $b$b). In this paper, we study the problem of CS on directed graph. Given a vertex $q$q of a graph $G$G, we aim to find a densely connected subgraph containing $q$q from $G$G, in which vertices have strong interactions and high similarities, by using the minimum in/out-degrees metric. We first develop a baseline algorithm based on the concept of D-core. We further propose three index structures and corresponding query algorithms. Our experimental results on seven real graphs show that our solutions are very effective and efficient. For example, on a graph with over 1 billion of edges, we only need around 40mins to index it and 1$\sim$∼2sec to answer a query.
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paper_title: Data Structures and Algorithms

paper_content:

From the Publisher: ::: This book presents the data structures and algorithms that underpin much of today's computer programming. The basis of this book is the material contained in the first six chapters of our earlier work, The Design and Analysis of Computer Algorithms. We have expanded that coverage and have added material on algorithms for external storage and memory management. As a consequence, this book should be suitable as a text for a first course on data structures and algorithms. The only prerequisite we assume is familiarity with some high-level programming language such as Pascal.
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paper_title: Effective Community Search for Large Attributed Graphs

paper_content:

Given a graph G and a vertex q ∈ G, the community search query returns a subgraph of G that contains vertices related to q. Communities, which are prevalent in attributed graphs such as social networks and knowledge bases, can be used in emerging applications such as product advertisement and setting up of social events. In this paper, we investigate the attributed community query (or ACQ), which returns an attributed community (AC) for an attributed graph. The AC is a subgraph of G, which satisfies both structure cohesiveness (i.e., its vertices are tightly connected) and keyword cohesiveness (i.e., its vertices share common keywords). The AC enables a better understanding of how and why a community is formed (e.g., members of an AC have a common interest in music, because they all have the same keyword "music"). An AC can be "personalized"; for example, an ACQ user may specify that an AC returned should be related to some specific keywords like "research" and "sports". ::: ::: To enable efficient AC search, we develop the CL-tree index structure and three algorithms based on it. We evaluate our solutions on four large graphs, namely Flickr, DBLP, Tencent, and DBpedia. Our results show that ACs are more effective and efficient than existing community retrieval approaches. Moreover, an AC contains more precise and personalized information than that of existing community search and detection methods.
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paper_title: D-cores: measuring collaboration of directed graphs based on degeneracy

paper_content:

Community detection and evaluation is an important task in graph mining. In many cases, a community is defined as a subgraph characterized by dense connections or interactions between its nodes. A variety of measures are proposed to evaluate different quality aspects of such communities—in most cases ignoring the directed nature of edges. In this paper, we introduce novel metrics for evaluating the collaborative nature of directed graphs—a property not captured by the single node metrics or by other established community evaluation metrics. In order to accomplish this objective, we capitalize on the concept of graph degeneracy and define a novel D-core framework, extending the classic graph-theoretic notion of \(k\)-cores for undirected graphs to directed ones. Based on the D-core, which essentially can be seen as a measure of the robustness of a community under degeneracy, we devise a wealth of novel metrics used to evaluate graph collaboration features of directed graphs. We applied the D-core approach on large synthetic and real-world graphs such as Wikipedia, DBLP, and ArXiv and report interesting results at the graph as well at the node level.
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paper_title: Community structure in directed networks

paper_content:

We consider the problem of finding communities or modules in directed networks. In the past, the most common approach to this problem has been to ignore edge direction and apply methods developed for community discovery in undirected networks, but this approach discards potentially useful information contained in the edge directions. Here we show how the widely used community finding technique of modularity maximization can be generalized in a principled fashion to incorporate information contained in edge directions. We describe an explicit algorithm based on spectral optimization of the modularity and show that it gives demonstrably better results than previous methods on a variety of test networks, both real and computer generated.
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paper_title: Clustering large attributed information networks: an efficient incremental computing approach

paper_content:

In recent years, many information networks have become available for analysis, including social networks, road networks, sensor networks, biological networks, etc. Graph clustering has shown its effectiveness in analyzing and visualizing large networks. The goal of graph clustering is to partition vertices in a large graph into clusters based on various criteria such as vertex connectivity or neighborhood similarity. Many existing graph clustering methods mainly focus on the topological structures, but largely ignore the vertex properties which are often heterogeneous. Recently, a new graph clustering algorithm, SA-cluster, has been proposed which combines structural and attribute similarities through a unified distance measure. SA-Cluster performs matrix multiplication to calculate the random walk distances between graph vertices. As part of the clustering refinement, the graph edge weights are iteratively adjusted to balance the relative importance between structural and attribute similarities. As a consequence, matrix multiplication is repeated in each iteration of the clustering process to recalculate the random walk distances which are affected by the edge weight update. In order to improve the efficiency and scalability of SA-cluster, in this paper, we propose an efficient algorithm In-Cluster to incrementally update the random walk distances given the edge weight increments. Complexity analysis is provided to estimate how much runtime cost Inc-Cluster can save. We further design parallel matrix computation techniques on a multicore architecture. Experimental results demonstrate that Inc-Cluster achieves significant speedup over SA-Cluster on large graphs, while achieving exactly the same clustering quality in terms of intra-cluster structural cohesiveness and attribute value homogeneity.
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paper_title: Combining link and content for community detection: a discriminative approach

paper_content:

In this paper, we consider the problem of combining link and content analysis for community detection from networked data, such as paper citation networks and Word Wide Web. Most existing approaches combine link and content information by a generative model that generates both links and contents via a shared set of community memberships. These generative models have some shortcomings in that they failed to consider additional factors that could affect the community memberships and isolate the contents that are irrelevant to community memberships. To explicitly address these shortcomings, we propose a discriminative model for combining the link and content analysis for community detection. First, we propose a conditional model for link analysis and in the model, we introduce hidden variables to explicitly model the popularity of nodes. Second, to alleviate the impact of irrelevant content attributes, we develop a discriminative model for content analysis. These two models are unified seamlessly via the community memberships. We present efficient algorithms to solve the related optimization problems based on bound optimization and alternating projection. Extensive experiments with benchmark data sets show that the proposed framework significantly outperforms the state-of-the-art approaches for combining link and content analysis for community detection.
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paper_title: Efficient community detection in large networks using content and links

paper_content:

In this paper we discuss a very simple approach of combining content and link information in graph structures for the purpose of community discovery, a fundamental task in network analysis. Our approach hinges on the basic intuition that many networks contain noise in the link structure and that content information can help strengthen the community signal. This enables ones to eliminate the impact of noise (false positives and false negatives), which is particularly prevalent in online social networks and Web-scale information networks. Specifically we introduce a measure of signal strength between two nodes in the network by fusing their link strength with content similarity. Link strength is estimated based on whether the link is likely (with high probability) to reside within a community. Content similarity is estimated through cosine similarity or Jaccard coefficient. We discuss a simple mechanism for fusing content and link similarity. We then present a biased edge sampling procedure which retains edges that are locally relevant for each graph node. The resulting backbone graph can be clustered using standard community discovery algorithms such as Metis and Markov clustering. Through extensive experiments on multiple real-world datasets (Flickr, Wikipedia and CiteSeer) with varying sizes and characteristics, we demonstrate the effectiveness and efficiency of our methods over state-of-the-art learning and mining approaches several of which also attempt to combine link and content analysis for the purposes of community discovery. Specifically we always find a qualitative benefit when combining content with link analysis. Additionally our biased graph sampling approach realizes a quantitative benefit in that it is typically several orders of magnitude faster than competing approaches.
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paper_title: Online community detection in social sensing

paper_content:

The proliferation of location and GPS data streams which are collected in a wide variety of participatory sensing applications has created numerous possibilities for analysis of the underlying patterns of activity. Typically, the spatio-temporal patterns arising from such activity can be analyzed in order to determine the latent community structure in the underlying data. In this paper, we will examine the problem of online community detection from the location data collected from such social sensing applications in real time. Such data brings numerous challenges associated with it, in that they can be of a relatively large scale, and can be extremely noisy from the perspective of both data representation and analysis. Furthermore, the community structure in the underlying data cannot be directly inferred from the shape of the underlying trajectories, since a considerable amount of variation may exist in terms of trajectories of individuals belonging to the same community. In this paper, we will design online algorithms for community detection in social sensing applications. Our algorithm uses a robust and efficiently updateable model with the use of Gibbs sampling, and we will show its effectiveness and efficiency for social sensing applications.
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paper_title: On Attributed Community Search

paper_content:

Communities, which are prevalent in attributed graphs (e.g., social networks and knowledge bases) can be used in emerging applications such as product advertisement and setting up of social events. Given a graph G and a vertex \(q \in G\), the community search (CS) query returns a subgraph of G that contains vertices related to q. In this article, we study CS over two common attributed graphs, where (1) vertices are associated with keywords; and (2) vertices are augmented with locations. For keyword-based attributed graphs, we investigate the keyword-based attributed community (or KAC) query, which returns a KAC for a query vertex. A KAC satisfies both structure cohesiveness (i.e., its vertices are tightly connected) and keyword cohesiveness (i.e., its vertices share common keywords). For spatial-based attributed graphs, we aim to find the spatial-aware community (or SAC), whose vertices are close structurally and spatially, for a query vertex in an online manner. To enable efficient KAC search and SAC search, we propose efficient query algorithms. We also perform experimental evaluation on large real datasets, and the results show that our methods achieve higher effectiveness than the state-of-the-art community retrieval algorithms. Moreover, our solutions are faster than baseline approaches. In addition, we develop the C-Explorer system to assist users in extracting, visualizing, and analyzing KACs.
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paper_title: Community Detection with Prior Knowledge

paper_content:

The problem of community detection is a challenging one because of the presence of hubs and noisy links, which tend to create highly imbalanced graph clusters. Often, these resulting clusters are not very intuitive and difficult to interpret. With the growing availability of network information, there is a significant amount of prior knowledge available about the communities in social, communication and several other networks. These community labels may be noisy and very limited, though they do help in community detection. In this paper, we explore the use of such noisy labeled information for finding high quality communities. We will present an adaptive density-based clustering which allows flexible incorporation of prior knowledge in to the community detection process. We use a random walk framework to compute the node densities and the level of supervision regulates the node densities and the quality of resulting density based clusters. Our framework is general enough to produce both overlapping and non-overlapping clusters. We empirically show that even with a tiny amount of supervision, our approach can produce superior communities compared to popular baselines.
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paper_title: Graph Clustering Based on Structural/Attribute Similarities

paper_content:

The goal of graph clustering is to partition vertices in a large graph into different clusters based on various criteria such as vertex connectivity or neighborhood similarity. Graph clustering techniques are very useful for detecting densely connected groups in a large graph. Many existing graph clustering methods mainly focus on the topological structure for clustering, but largely ignore the vertex properties which are often heterogenous. In this paper, we propose a novel graph clustering algorithm, SA-Cluster, based on both structural and attribute similarities through a unified distance measure. Our method partitions a large graph associated with attributes into k clusters so that each cluster contains a densely connected subgraph with homogeneous attribute values. An effective method is proposed to automatically learn the degree of contributions of structural similarity and attribute similarity. Theoretical analysis is provided to show that SA-Cluster is converging. Extensive experimental results demonstrate the effectiveness of SA-Cluster through comparison with the state-of-the-art graph clustering and summarization methods.
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paper_title: Joint latent topic models for text and citations

paper_content:

In this work, we address the problem of joint modeling of text and citations in the topic modeling framework. We present two different models called the Pairwise-Link-LDA and the Link-PLSA-LDA models. The Pairwise-Link-LDA model combines the ideas of LDA [4] and Mixed Membership Block Stochastic Models [1] and allows modeling arbitrary link structure. However, the model is computationally expensive, since it involves modeling the presence or absence of a citation (link) between every pair of documents. The second model solves this problem by assuming that the link structure is a bipartite graph. As the name indicates, Link-PLSA-LDA model combines the LDA and PLSA models into a single graphical model. Our experiments on a subset of Citeseer data show that both these models are able to predict unseen data better than the baseline model of Erosheva and Lafferty [8], by capturing the notion of topical similarity between the contents of the cited and citing documents. Our experiments on two different data sets on the link prediction task show that the Link-PLSA-LDA model performs the best on the citation prediction task, while also remaining highly scalable. In addition, we also present some interesting visualizations generated by each of the models.
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paper_title: Effective and efficient attributed community search

paper_content:

Given a graph G and a vertex $$q \in G$$ ::: , the community search query returns a subgraph of G that contains vertices related to q. Communities, which are prevalent in attributed graphs such as social networks and knowledge bases, can be used in emerging applications such as product advertisement and setting up of social events. In this paper, we investigate the attributed community query (or ACQ), which returns an attributed community (AC) for an attributed graph. The AC is a subgraph of G, which satisfies both structure cohesiveness (i.e., its vertices are tightly connected) and keyword cohesiveness (i.e., its vertices share common keywords). The AC enables a better understanding of how and why a community is formed (e.g., members of an AC have a common interest in music, because they all have the same keyword “music”). An AC can be “personalized”; for example, an ACQ user may specify that an AC returned should be related to some specific keywords like “research” and “sports”. To enable efficient AC search, we develop the CL-tree index structure and three algorithms based on it. We further propose efficient algorithms for maintaining the index on dynamic graphs. Moreover, we study two problems that are related to the ACQ problem. We evaluate our solutions on six large graphs. Our results show that ACQ is more effective and efficient than existing community retrieval approaches. Moreover, an AC contains more precise and personalized information than that of existing community search and detection methods.
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paper_title: Community Detection in Networks with Node Attributes

paper_content:

Community detection algorithms are fundamental tools that allow us to uncover organizational principles in networks. When detecting communities, there are two possible sources of information one can use: the network structure, and the features and attributes of nodes. Even though communities form around nodes that have common edges and common attributes, typically, algorithms have only focused on one of these two data modalities: community detection algorithms traditionally focus only on the network structure, while clustering algorithms mostly consider only node attributes. In this paper, we develop Communities from Edge Structure and Node Attributes (CESNA), an accurate and scalable algorithm for detecting overlapping communities in networks with node attributes. CESNA statistically models the interaction between the network structure and the node attributes, which leads to more accurate community detection as well as improved robustness in the presence of noise in the network structure. CESNA has a linear runtime in the network size and is able to process networks an order of magnitude larger than comparable approaches. Last, CESNA also helps with the interpretation of detected communities by finding relevant node attributes for each community.
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paper_title: Using content and interactions for discovering communities in social networks

paper_content:

In recent years, social networking sites have not only enabled people to connect with each other using social links but have also allowed them to share, communicate and interact over diverse geographical regions. Social network provide a rich source of heterogeneous data which can be exploited to discover previously unknown relationships and interests among groups of people. In this paper, we address the problem of discovering topically meaningful communities from a social network. We assume that a persons' membership in a community is conditioned on its social relationship, the type of interaction and the information communicated with other members of that community. We propose generative models that can discover communities based on the discussed topics, interaction types and the social connections among people. In our models a person can belong to multiple communities and a community can participate in multiple topics. This allows us to discover both community interests and user interests based on the information and linked associations. We demonstrate the effectiveness of our model on two real word data sets and show that it performs better than existing community discovery models.
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paper_title: Effective Community Search for Large Attributed Graphs

paper_content:

Given a graph G and a vertex q ∈ G, the community search query returns a subgraph of G that contains vertices related to q. Communities, which are prevalent in attributed graphs such as social networks and knowledge bases, can be used in emerging applications such as product advertisement and setting up of social events. In this paper, we investigate the attributed community query (or ACQ), which returns an attributed community (AC) for an attributed graph. The AC is a subgraph of G, which satisfies both structure cohesiveness (i.e., its vertices are tightly connected) and keyword cohesiveness (i.e., its vertices share common keywords). The AC enables a better understanding of how and why a community is formed (e.g., members of an AC have a common interest in music, because they all have the same keyword "music"). An AC can be "personalized"; for example, an ACQ user may specify that an AC returned should be related to some specific keywords like "research" and "sports". ::: ::: To enable efficient AC search, we develop the CL-tree index structure and three algorithms based on it. We evaluate our solutions on four large graphs, namely Flickr, DBLP, Tencent, and DBpedia. Our results show that ACs are more effective and efficient than existing community retrieval approaches. Moreover, an AC contains more precise and personalized information than that of existing community search and detection methods.
---
paper_title: A model-based approach to attributed graph clustering

paper_content:

Graph clustering, also known as community detection, is a long-standing problem in data mining. However, with the proliferation of rich attribute information available for objects in real-world graphs, how to leverage structural and attribute information for clustering attributed graphs becomes a new challenge. Most existing works take a distance-based approach. They proposed various distance measures to combine structural and attribute information. In this paper, we consider an alternative view and propose a model-based approach to attributed graph clustering. We develop a Bayesian probabilistic model for attributed graphs. The model provides a principled and natural framework for capturing both structural and attribute aspects of a graph, while avoiding the artificial design of a distance measure. Clustering with the proposed model can be transformed into a probabilistic inference problem, for which we devise an efficient variational algorithm. Experimental results on large real-world datasets demonstrate that our method significantly outperforms the state-of-art distance-based attributed graph clustering method.
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paper_title: Topic-link LDA: joint models of topic and author community

paper_content:

Given a large-scale linked document collection, such as a collection of blog posts or a research literature archive, there are two fundamental problems that have generated a lot of interest in the research community. One is to identify a set of high-level topics covered by the documents in the collection; the other is to uncover and analyze the social network of the authors of the documents. So far these problems have been viewed as separate problems and considered independently from each other. In this paper we argue that these two problems are in fact inter-dependent and should be addressed together. We develop a Bayesian hierarchical approach that performs topic modeling and author community discovery in one unified framework. The effectiveness of our model is demonstrated on two blog data sets in different domains and one research paper citation data from CiteSeer.
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paper_title: Uncovering space-independent communities in spatial networks

paper_content:

Many complex systems are organized in the form of a network embedded in space. Important examples include the physical Internet infrastucture, road networks, flight connections, brain functional networks, and social networks. The effect of space on network topology has recently come under the spotlight because of the emergence of pervasive technologies based on geolocalization, which constantly fill databases with people’s movements and thus reveal their trajectories and spatial behavior. Extracting patterns and regularities from the resulting massive amount of human mobility data requires the development of appropriate tools for uncovering information in spatially embedded networks. In contrast with most works that tend to apply standard network metrics to any type of network, we argue in this paper for a careful treatment of the constraints imposed by space on network topology. In particular, we focus on the problem of community detection and propose a modularity function adapted to spatial networks. We show that it is possible to factor out the effect of space in order to reveal more clearly hidden structural similarities between the nodes. Methods are tested on a large mobile phone network and computer-generated benchmarks where the effect of space has been incorporated.
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paper_title: Scalable Algorithms for Nearest-Neighbor Joins on Big Trajectory Data

paper_content:

Trajectory data are prevalent in systems that monitor the locations of moving objects. In a location-based service, for instance, the positions of vehicles are continuously monitored through GPS; the trajectory of each vehicle describes its movement history. We study joins on two sets of trajectories, generated by two sets $M$ and $R$ of moving objects. For each entity in $M$ , a join returns its $k$ nearest neighbors from $R$ . We examine how this query can be evaluated in cloud environments. This problem is not trivial, due to the complexity of the trajectory, and the fact that both the spatial and temporal dimensions of the data have to be handled. To facilitate this operation, we propose a parallel solution framework based on MapReduce. We also develop a novel bounding technique, which enables trajectories to be pruned in parallel. Our approach can be used to parallelize existing single-machine trajectory join algorithms. We also study a variant of the join, which can further improve query efficiency. To evaluate the efficiency and the scalability of our solutions, we have performed extensive experiments on large real and synthetic datasets.
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paper_title: Effective Community Search over Large Spatial Graphs

paper_content:

Communities are prevalent in social networks, knowledge graphs, and biological networks. Recently, the topic of community search (CS) has received plenty of attention. Given a query vertex, CS looks for a dense subgraph that contains it. Existing CS solutions do not consider the spatial extent of a community. They can yield communities whose locations of vertices span large areas. In applications that facilitate the creation of social events (e.g., finding conference attendees to join a dinner), it is important to find groups of people who are physically close to each other. In this situation, it is desirable to have a spatial-aware community (or SAC), whose vertices are close structurally and spatially. Given a graph G and a query vertex q, we develop exact solutions for finding an SAC that contains q. Since these solutions cannot scale to large datasets, we have further designed three approximation algorithms to compute an SAC. We have performed an experimental evaluation for these solutions on both large real and synthetic datasets. Experimental results show that SAC is better than the communities returned by existing solutions. Moreover, our approximation solutions can find SACs accurately and efficiently.
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paper_title: Efficient Computing of Radius-Bounded k-Cores

paper_content:

Driven by real-life applications in geo-social networks, in this paper, we investigate the problem of computing the radius-bounded k-cores (RB-k-cores) that aims to find cohesive subgraphs satisfying both social and spatial constraints on large geo-social networks. In particular, we use k-core to ensure the social cohesiveness and we use a radius-bounded circle to restrict the locations of users in a RB-k-core. We explore several algorithmic paradigms to compute RB-k-cores, including a triple vertex-based paradigm, a binary-vertex-based paradigm, and a paradigm utilizing the concept of rotating circles. The rotating circle-based paradigm is further enhanced with several pruning techniques to achieve better efficiency. The experimental studies conducted on both real and synthetic datasets demonstrate that our proposed rotating-circle-based algorithms can compute all RB-k-cores very efficiently. Moreover, it can also be used to compute the minimum-circle-bounded k-core and significantly outperforms the existing techniques for computing the minimum circle-bounded k-core.
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paper_title: Community structure in social and biological networks

paper_content:

A number of recent studies have focused on the statistical properties of networked systems such as social networks and the Worldwide Web. Researchers have concentrated particularly on a few properties that seem to be common to many networks: the small-world property, power-law degree distributions, and network transitivity. In this article, we highlight another property that is found in many networks, the property of community structure, in which network nodes are joined together in tightly knit groups, between which there are only looser connections. We propose a method for detecting such communities, built around the idea of using centrality indices to find community boundaries. We test our method on computer-generated and real-world graphs whose community structure is already known and find that the method detects this known structure with high sensitivity and reliability. We also apply the method to two networks whose community structure is not well known—a collaboration network and a food web—and find that it detects significant and informative community divisions in both cases.
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paper_title: Geo-social group queries with minimum acquaintance constraints

paper_content:

The prosperity of location-based social networking services enables geo-social group queries for group-based activity planning and marketing. This paper proposes a new family of geo-social group queries with minimum acquaintance constraint (GSGQs), which are more appealing than existing geo-social group queries in terms of producing a cohesive group that guarantees the worst-case acquaintance level. GSGQs, also specified with various spatial constraints, are more complex than conventional spatial queries; particularly, those with a strict $k$NN spatial constraint are proved to be NP-hard. For efficient processing of general GSGQ queries on large location-based social networks, we devise two social-aware index structures, namely SaR-tree and SaR*-tree. The latter features a novel clustering technique that considers both spatial and social factors. Based on SaR-tree and SaR*-tree, efficient algorithms are developed to process various GSGQs. Extensive experiments on real-world Gowalla and Dianping datasets show that our proposed methods substantially outperform the baseline algorithms based on R-tree.
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paper_title: Detecting hot topics from Twitter: A multiview approach

paper_content:

Twitter is widely used all over the world, and a huge number of hot topics are generated by Twitter users in real time. These topics are able to reflect almost every aspect of people's daily lives. Therefore, the detection of topics in Twitter can be used in many real applications, such as monitoring public opinion, hot product recommendation and incidence detection. However, the performance of traditional topic detection methods is still far from perfect largely owing to the tweets' features, such as their limited length and arbitrary abbreviations. To address these problems, we propose a novel framework (MVTD) for Twitter topic detection using multiview clustering, which can integrate multirelations among tweets, such as semantic relations, social tag relations and temporal relations. We also propose some methods for measuring relations among tweets. In particular, to better measure the semantic similarity of tweets, we propose a new document similarity measure based on a suffix tree (STVSM). In addition, a new keyword extraction method based on a suffix tree is proposed. Experiments on real datasets show that the performance of MVTD is much better than that of a single view, and it is useful for detecting topics from Twitter.
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paper_title: Finding community structure in spatially constrained complex networks

paper_content:

One feature discovered in the study of complex networks is community structure, in which vertices are gathered into several groups where more edges exist within groups than between groups. Many approaches have been developed for identifying communities; these approaches essentially segment networks based on topological structure or the attribute similarity of vertices, while few approaches consider the spatial character of the networks. Many complex networks are spatially constrained such that the vertices and edges are embedded in space. In geographical space, nearer objects are more related than distant objects. Thus, the relations among vertices are defined not only by the links connecting them but also by the distance between them. In this article, we propose a geo-distance-based method of detecting communities in spatially constrained networks to identify communities that are both highly topologically connected and spatially clustered. The algorithm is based on the fast modularity maximisation CNM algorithm. First, we modify the modularity to geo-modularity Qgeo by introducing an edge weight that is the inverse of the geographic distance to the power of n. Then, we propose the concept of a spatial clustering coefficient as a measure of clustering of the network to determine the power value n of the distance. The algorithm is tested with China air transport network and BrightKite social network data-sets. The segmentation of the China air transport network is similar to the seven economic regions of China. The segmentation of the BrightKite social network shows the regionality of social groups and identifies the dynamic social groups that reflect users’ location changes. The algorithm is useful in exploring the interaction and clustering properties of geographical phenomena and providing timely location-based services for a group of people.
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paper_title: Mining for geographically disperse communities in social networks by leveraging distance modularity

paper_content:

Social networks where the actors occupy geospatial locations are prevalent in military, intelligence, and policing operations such as counter-terrorism, counter-insurgency, and combating organized crime. These networks are often derived from a variety of intelligence sources. The discovery of communities that are geographically disperse stems from the requirement to identify higher-level organizational structures, such as a logistics group that provides support to various geographically disperse terrorist cells. We apply a variant of Newman-Girvan modularity to this problem known as distance modularity. To address the problem of finding geographically disperse communities, we modify the well-known Louvain algorithm to find partitions of networks that provide near-optimal solutions to this quantity. We apply this algorithm to numerous samples from two real-world social networks and a terrorism network data set whose nodes have associated geospatial locations. Our experiments show this to be an effective approach and highlight various practical considerations when applying the algorithm to distance modularity maximization. Several military, intelligence, and law-enforcement organizations are working with us to further test and field software for this emerging application.
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paper_title: Regionalization with dynamically constrained agglomerative clustering and partitioning (REDCAP)

paper_content:

Regionalization is to divide a large set of spatial objects into a number of spatially contiguous regions while optimizing an objective function, which is normally a homogeneity (or heterogeneity) measure of the derived regions. This research proposes and evaluates a family of six hierarchical regionalization methods. The six methods are based on three agglomerative clustering approaches, including the single linkage, average linkage (ALK), and the complete linkage (CLK), each of which is constrained with spatial contiguity in two different ways (i.e. the first-order constraining and the full-order constraining). It is discovered that both the Full-Order-CLK and the Full-Order-ALK methods significantly outperform existing methods across four quality evaluations: the total heterogeneity, region size balance, internal variation, and the preservation of data distribution. Moreover, the proposed algorithms are efficient and can find the solution in O(n 2log n) time. With such data scalability, for the first time it is possible to effectively regionalize large data sets that have 10 000 or more spatial objects. A detailed comparison and evaluation of the six methods are carried out with the 2004 US presidential election data.
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paper_title: Spatial Networks

paper_content:

Complex systems are very often organized under the form of networks where nodes and edges are embedded in space. Transportation and mobility networks, Internet, mobile phone networks, power grids, social and contact networks, neural networks, are all examples where space is relevant and where topology alone does not contain all the information. Characterizing and understanding the structure and the evolution of spatial networks is thus crucial for many different fields ranging from urbanism to epidemiology. An important consequence of space on networks is that there is a cost associated to the length of edges which in turn has dramatic effects on the topological structure of these networks. We will expose thoroughly the current state of our understanding of how the spatial constraints affect the structure and properties of these networks. We will review the most recent empirical observations and the most important models of spatial networks. We will also discuss various processes which take place on these spatial networks, such as phase transitions, random walks, synchronization, navigation, resilience, and disease spread.
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paper_title: Efficient Algorithms for Answering the m-Closest Keywords Query

paper_content:

As an important type of spatial keyword query, the m-closest keywords (mCK) query finds a group of objects such that they cover all query keywords and have the smallest diameter, which is defined as the largest distance between any pair of objects in the group. The query is useful in many applications such as detecting locations of web resources. However, the existing work does not study the intractability of this problem and only provides exact algorithms, which are computationally expensive. In this paper, we prove that the problem of answering mCK queries is NP-hard. We first devise a greedy algorithm that has an approximation ratio of 2. Then, we observe that an mCK query can be approximately answered by finding the circle with the smallest diameter that encloses a group of objects together covering all query keywords. We prove that the group enclosed in the circle can answer the mCK query with an approximation ratio of 2 over 3. Based on this, we develop an algorithm for finding such a circle exactly, which has a high time complexity. To improve efficiency, we propose another two algorithms that find such a circle approximately, with a ratio of 2 over √3 + e. Finally, we propose an exact algorithm that utilizes the group found by the 2 over √3 + e)-approximation algorithm to obtain the optimal group. We conduct extensive experiments using real-life datasets. The experimental results offer insights into both efficiency and accuracy of the proposed approximation algorithms, and the results also demonstrate that our exact algorithm outperforms the best known algorithm by an order of magnitude.
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paper_title: Geometrical Solutions for Some Minimax Location Problems

paper_content:

Four closely related minimax location problems are considered. Each involves locating a point in the plane to minimize the maximum distance (plus a possible constant) to a given finite set of points. The distance measures considered are the Euclidean and the rectilinear. In each case efficient, finite solution procedures are given. The arguments are geometrical.
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paper_title: On Spatial-Aware Community Search

paper_content:

Communities are prevalent in social networks, knowledge graphs, and biological networks. Recently, the topic of community search (CS) has received plenty of attention. The CS problem aims to look for a dense subgraph that contains a query vertex. Existing CS solutions do not consider the spatial extent of a community. They can yield communities whose locations of vertices span large areas. In applications that facilitate setting social events (e.g., finding conference attendees to join a dinner), it is important to find groups of people who are physically close to each other, so it is desirable to have a spatial-aware community (or SAC), whose vertices are close structurally and spatially. Given a graph $G$ and a query vertex $q$ , we develop an exact solution to find the SAC containing $q$ , but it cannot scale to large datasets, so we design three approximation algorithms. We further study the problem of continuous SAC search on a “dynamic spatial graph,” whose vertices’ locations change with time, and propose three fast solutions. We evaluate the solutions on both real and synthetic datasets, and the results show that SACs are better than communities returned by existing solutions. Moreover, our approximation solutions perform accurately and efficiently.
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paper_title: Geometrical Solutions for Some Minimax Location Problems

paper_content:

Four closely related minimax location problems are considered. Each involves locating a point in the plane to minimize the maximum distance (plus a possible constant) to a given finite set of points. The distance measures considered are the Euclidean and the rectilinear. In each case efficient, finite solution procedures are given. The arguments are geometrical.
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paper_title: Geo-social group queries with minimum acquaintance constraints

paper_content:

The prosperity of location-based social networking services enables geo-social group queries for group-based activity planning and marketing. This paper proposes a new family of geo-social group queries with minimum acquaintance constraint (GSGQs), which are more appealing than existing geo-social group queries in terms of producing a cohesive group that guarantees the worst-case acquaintance level. GSGQs, also specified with various spatial constraints, are more complex than conventional spatial queries; particularly, those with a strict $k$NN spatial constraint are proved to be NP-hard. For efficient processing of general GSGQ queries on large location-based social networks, we devise two social-aware index structures, namely SaR-tree and SaR*-tree. The latter features a novel clustering technique that considers both spatial and social factors. Based on SaR-tree and SaR*-tree, efficient algorithms are developed to process various GSGQs. Extensive experiments on real-world Gowalla and Dianping datasets show that our proposed methods substantially outperform the baseline algorithms based on R-tree.
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paper_title: Community Detection in Temporal Multilayer Networks, with an Application to Correlation Networks

paper_content:

Networks are a convenient way to represent complex systems of interacting entities. Many networks contain “communities” of nodes that are more densely connected to each other than to nodes in the rest of the network. In this paper, we investigate the detection of communities in temporal networks represented as multilayer networks. As a focal example, we study time-dependent financial-asset correlation networks. We first argue that the use of the “modularity” quality function---which is defined by comparing edge weights in an observed network to expected edge weights in a “null network''---is application-dependent. We differentiate between “null networks” and “null models” in our discussion of modularity maximization, and we highlight that the same null network can correspond to different null models. We then investigate a multilayer modularity-maximization problem to identify communities in temporal networks. Our multilayer analysis depends only on the form of the maximization problem and not on the specific quality function that one chooses. We introduce a diagnostic to measure persistence of community structure in a multilayer network partition. We prove several results that describe how the multilayer maximization problem measures a trade-off between static community structure within layers and larger values of persistence across layers. We also discuss some computational issues that the popular “Louvain” heuristic faces with temporal multilayer networks and suggest ways to mitigate them.
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paper_title: Persistent Community Detection in Dynamic Social Networks

paper_content:

While community detection is an active area of research in social network analysis, little effort has been devoted to community detection using time-evolving social network data. We propose an algorithm, Persistent Community Detection (PCD), to identify those communities that exhibit persistent behavior over time, for usage in such settings. Our motivation is to distinguish between steady-state network activity, and impermanent behavior such as cascades caused by a noteworthy event. The results of extensive empirical experiments on real-life big social networks data show that our algorithm performs much better than a set of baseline methods, including two alternative models and the state-of-the-art.
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paper_title: Community Discovery in Dynamic Networks: A Survey

paper_content:

Several research studies have shown that complex networks modeling real-world phenomena are characterized by striking properties: (i) they are organized according to community structure, and (ii) their structure evolves with time. Many researchers have worked on methods that can efficiently unveil substructures in complex networks, giving birth to the field of community discovery. A novel and fascinating problem started capturing researcher interest recently: the identification of evolving communities. Dynamic networks can be used to model the evolution of a system: nodes and edges are mutable, and their presence, or absence, deeply impacts the community structure that composes them. ::: ::: This survey aims to present the distinctive features and challenges of dynamic community discovery and propose a classification of published approaches. As a “user manual,” this work organizes state-of-the-art methodologies into a taxonomy, based on their rationale, and their specific instantiation. Given a definition of network dynamics, desired community characteristics, and analytical needs, this survey will support researchers to identify the set of approaches that best fit their needs. The proposed classification could also help researchers choose in which direction to orient their future research.
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paper_title: Local Community Detection in Dynamic Networks

paper_content:

Given a time-evolving network, how can we detect communities over periods of high internal and low external interactions? To address this question we generalize traditional local community detection in graphs to the setting of dynamic networks. Adopting existing static-network approaches in an "aggregated" graph of all temporal interactions is not appropriate for the problem as dynamic communities may be short-lived and thus lost when mixing interactions over long periods. Hence, dynamic community mining requires the detection of both the community nodes and an optimal time interval in which they are actively interacting. ::: We propose a filter-and-verify framework for dynamic community detection. To scale to long intervals of graph evolution, we employ novel spectral bounds for dynamic community conductance and employ them to filter suboptimal periods in near-linear time. We also design a time-and-graph-aware locality sensitive hashing family to effectively spot promising community cores. Our method PHASR discovers communities of consistently higher quality (2 to 67 times better) than those of baselines. At the same time, our bounds allow for pruning between $55\%$ and $95\%$ of the search space, resulting in significant savings in running time compared to exhaustive alternatives for even modest time intervals of graph evolution.
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paper_title: Literature survey on dynamic community detection and models of social networks

paper_content:

In this paper, we realize a literature survey on the issue of community detection over time, first we present some basic concepts about networks modeled as graphs, then we state in an non exhaustive way the research fields arising from social networks. We present some of the existing models and methods to track communities over time. Community detection in networks is a prevailing subject in the area of social network analysis. There exist numerous methods and algorithms in the literature that have been dealing with that issue.
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paper_title: Discovering Temporal Communities from Social Network Documents

paper_content:

This paper studies the discovery of communities from social network documents produced over time, addressing the discovery of temporal trends in community memberships. We first formulate static community discovery at a single time period as a tripartite graph partitioning problem. Then we propose to discover the temporal communities by threading the statically derived communities in different time periods using a new constrained partitioning algorithm, which partitions graphs based on topology as well as prior information regarding vertex membership. We evaluate the proposed approach on synthetic datasets and a real-world dataset prepared from the CiteSeer.
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paper_title: Multi-scale Community Detection in Temporal Networks Using Spectral Graph Wavelets

paper_content:

Spectral graph wavelets introduce a notion of scale in networks, and are thus used to obtain a local view of the network from each node. By carefully constructing a wavelet filter function for these wavelets, a multi-scale community detection method for monoplex networks has already been developed. This construction takes advantage of the partitioning properties of the network Laplacian. In this paper we elaborate on a novel method which uses spectral graph wavelets to detect multi-scale communities in temporal networks. To do this we extend the definition of spectral graph wavelets to temporal networks by adopting a multilayer framework. We use arguments from Perturbation Theory to investigate the spectral properties of the supra-Laplacian matrix for clustering purposes in temporal networks. Using these properties, we construct a new wavelet filter function, which attenuates the influence of uninformative eigenvalues and centres the filter around eigenvalues which contain information on the coarsest description of prevalent community structures over time. We use the spectral graph wavelets as feature vectors in a connectivity-constrained clustering procedure to detect multi-scale communities at different scales, and refer to this method as Temporal Multi-Scale Community Detection (TMSCD). We validate the performance of TMSCD and a competing methodology on various benchmarks. The advantage of TMSCD is the automated selection of relevant scales at which communities should be sought.
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paper_title: Persistent Community Search in Temporal Networks

paper_content:

Community search is a fundamental graph mining task. Unfortunately, most previous community search studies focus mainly on identifying communities in a network without temporal information. In this paper, we study the problem of finding persistent communities in a temporal network, in which every edge is associated with a timestamp. Our goal is to identify the communities that are persistent over time. To this end, we propose a novel persistent community model called (θ,⊺) community. We prove that the problem of identifying the maximum (θ,⊺) persistent k-core is NP-hard. To solve this problem, we propose a novel branch and bound algorithm with several carefully-designed pruning rules to find the maximum (θ,⊺)-persistent. We conduct k-cores efficiently. We conduct extensive experiments in several real-world temporal networks. The results demonstrate the efficiency, scalability, and effectiveness of the proposed solutions.
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paper_title: Influential Community Search in Large Networks

paper_content:

Community search is a problem of finding densely connected subgraphs that satisfy the query conditions in a network, which has attracted much attention in recent years. However, all the previous studies on community search do not consider the influence of a community. In this paper, we introduce a novel community model called k-influential community based on the concept of k-core, which can capture the influence of a community. Based on the new community model, we propose a linear-time online search algorithm to find the top-r k-influential communities in a network. To further speed up the influential community search algorithm, we devise a linear-space index structure which supports efficient search of the top-r k-influential communities in optimal time. We also propose an efficient algorithm to maintain the index when the network is frequently updated. We conduct extensive experiments on 7 real-world large networks, and the results demonstrate the efficiency and effectiveness of the proposed methods.
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paper_title: Persistent Community Search in Temporal Networks

paper_content:

Community search is a fundamental graph mining task. Unfortunately, most previous community search studies focus mainly on identifying communities in a network without temporal information. In this paper, we study the problem of finding persistent communities in a temporal network, in which every edge is associated with a timestamp. Our goal is to identify the communities that are persistent over time. To this end, we propose a novel persistent community model called (θ,⊺) community. We prove that the problem of identifying the maximum (θ,⊺) persistent k-core is NP-hard. To solve this problem, we propose a novel branch and bound algorithm with several carefully-designed pruning rules to find the maximum (θ,⊺)-persistent. We conduct k-cores efficiently. We conduct extensive experiments in several real-world temporal networks. The results demonstrate the efficiency, scalability, and effectiveness of the proposed solutions.
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paper_title: Influential Community Search in Large Networks

paper_content:

Community search is a problem of finding densely connected subgraphs that satisfy the query conditions in a network, which has attracted much attention in recent years. However, all the previous studies on community search do not consider the influence of a community. In this paper, we introduce a novel community model called k-influential community based on the concept of k-core, which can capture the influence of a community. Based on the new community model, we propose a linear-time online search algorithm to find the top-r k-influential communities in a network. To further speed up the influential community search algorithm, we devise a linear-space index structure which supports efficient search of the top-r k-influential communities in optimal time. We also propose an efficient algorithm to maintain the index when the network is frequently updated. We conduct extensive experiments on 7 real-world large networks, and the results demonstrate the efficiency and effectiveness of the proposed methods.
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paper_title: Finding influential communities in massive networks

paper_content:

Community search is a problem of finding densely connected subgraphs that satisfy the query conditions in a network, which has attracted much attention in recent years. However, all the previous studies on community search do not consider the influence of a community. In this paper, we introduce a novel community model called k-influential community based on the concept of k-core to capture the influence of a community. Based on this community model, we propose a linear time online search algorithm to find the top-r k-influential communities in a network. To further speed up the influential community search algorithm, we devise a linear space data structure which supports efficient search of the top-r k-influential communities in optimal time. We also propose an efficient algorithm to maintain the data structure when the network is frequently updated. Additionally, we propose a novel I/O-efficient algorithm to find the top-r k-influential communities in a disk-resident graph under the assumption of \({{\mathcal {U}}}=O(n)\), where \({{\mathcal {U}}}\) and n denote the size of the main memory and the number of nodes, respectively. Finally, we conduct extensive experiments on six real-world massive networks, and the results demonstrate the efficiency and effectiveness of the proposed methods.
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paper_title: Efficient Computation of Importance Based Communities in Web-Scale Networks Using a Single Machine

paper_content:

Finding decompositions of a graph into a family of communities is crucial to understanding its underlying structure. Algorithms for finding communities in networks often rely only on structural information and search for cohesive subsets of nodes. In practice however, we would like to find communities that are not only cohesive, but also influential or important. In order to capture such communities, Li, Qin, Yu, and Mao introduced a novel community model called "k-influential community" based on the concept of $k$-core, with numerical values representing "influence" assigned to the nodes. They formulate the problem of finding the top-r most important communities as finding r connected k-core subgraphs ordered by the lower-bound of their importance. In this paper, our goal is to scale-up the computation of top-r, k-core communities to web-scale graphs of tens of billions of edges. We feature several fast new algorithms for this problem. With our implementations, we show that we can efficiently handle massive networks using a single consumer-level machine within a reasonable amount of time.
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paper_title: Search of Center-Core Community in Large Graphs

paper_content:

Community search plays an important role in complex network analysis. It aims to find a densely connected subgraph containing the query node in a graph. However, the most existing community search methods do not consider the influence of nodes and can not perfectly support the search in large graphs, making them have limitations in practical applications. In this paper, we introduce a community model called center-core community based on k-core decomposition, which can both capture the influence of nodes and guarantee the cohesiveness of community. Then we devise a center-core community graph index (CCG-Index), and online search algorithms (SingleQuery and MultiQuery) which support efficient search of center-core community in optimal time. Extensive experiments on four real-world large networks demonstrate the efficiency and effectiveness of our methods.
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paper_title: An Optimal and Progressive Approach to Online Search of Top-K Influential Communities

paper_content:

Community search over large graphs is a fundamental problem in graph analysis. Recent studies propose to compute top-k influential communities, where each reported community not only is a cohesive subgraph but also has a high influence value. The existing approaches to the problem of top-k influential community search can be categorized as index-based algorithms and online search algorithms without indexes. The index-based algorithms, although being very efficient in conducting community searches, need to pre-compute a special-purpose index and only work for one built-in vertex weight vector. In this paper, we investigate on-line search approaches and propose an instance-optimal algorithm LocalSearch whose time complexity is linearly proportional to the size of the smallest subgraph that a correct algorithm needs to access without indexes. In addition, we also propose techniques to make LocalSearch progressively compute and report the communities in decreasing influence value order such that k does not need to be specified. Moreover, we extend our framework to the general case of top-k influential community search regarding other cohesiveness measures. Extensive empirical studies on real graphs demonstrate that our algorithms outperform the existing online search algorithms by several orders of magnitude.
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paper_title: Influential Community Search in Large Networks

paper_content:

Community search is a problem of finding densely connected subgraphs that satisfy the query conditions in a network, which has attracted much attention in recent years. However, all the previous studies on community search do not consider the influence of a community. In this paper, we introduce a novel community model called k-influential community based on the concept of k-core, which can capture the influence of a community. Based on the new community model, we propose a linear-time online search algorithm to find the top-r k-influential communities in a network. To further speed up the influential community search algorithm, we devise a linear-space index structure which supports efficient search of the top-r k-influential communities in optimal time. We also propose an efficient algorithm to maintain the index when the network is frequently updated. We conduct extensive experiments on 7 real-world large networks, and the results demonstrate the efficiency and effectiveness of the proposed methods.
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paper_title: Skyline Community Search in Multi-valued Networks

paper_content:

Given a scientific collaboration network, how can we find a group of collaborators with high research indicator (e.g., h-index) and diverse research interests? Given a social network, how can we identify the communities that have high influence (e.g., PageRank) and also have similar interests to a specified user? In such settings, the network can be modeled as a multi-valued network where each node has d ($d \ge 1$) numerical attributes (i.e., h-index, diversity, PageRank, similarity score, etc.). In the multi-valued network, we want to find communities that are not dominated by the other communities in terms of d numerical attributes. Most existing community search algorithms either completely ignore the numerical attributes or only consider one numerical attribute of the nodes. To capture d numerical attributes, we propose a novel community model, called skyline community, based on the concepts of k-core and skyline. A skyline community is a maximal connected k-core that cannot be dominated by the other connected k-cores in the d-dimensional attribute space. We develop an elegant space-partition algorithm to efficiently compute the skyline communities. Two striking advantages of our algorithm are that (1) its time complexity relies mainly on the size of the answer s (i.e., the number of skyline communities), thus it is very efficient if s is small; and (2) it can progressively output the skyline communities, which is very useful for applications that only require part of the skyline communities. Extensive experiments on both synthetic and real-world networks demonstrate the efficiency, scalability, and effectiveness of the proposed algorithm.
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paper_title: Influential Community Search in Large Networks

paper_content:

Community search is a problem of finding densely connected subgraphs that satisfy the query conditions in a network, which has attracted much attention in recent years. However, all the previous studies on community search do not consider the influence of a community. In this paper, we introduce a novel community model called k-influential community based on the concept of k-core, which can capture the influence of a community. Based on the new community model, we propose a linear-time online search algorithm to find the top-r k-influential communities in a network. To further speed up the influential community search algorithm, we devise a linear-space index structure which supports efficient search of the top-r k-influential communities in optimal time. We also propose an efficient algorithm to maintain the index when the network is frequently updated. We conduct extensive experiments on 7 real-world large networks, and the results demonstrate the efficiency and effectiveness of the proposed methods.
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paper_title: Exploring Communities in Large Profiled Graphs

paper_content:

Given a graph $G$G and a vertex $q\in G$q∈G, the community search (CS) problem aims to efficiently find a subgraph of $G$G whose vertices are closely related to $q$q. Communities are prevalent in social and biological networks, and can be used in product advertisement and social event recommendation. In this paper, we study profiled community search (PCS), where CS is performed on a profiled graph. This is a graph in which each vertex has labels arranged in a hierarchical manner. Extensive experiments show that PCS can identify communities with themes that are common to their vertices, and is more effective than existing CS approaches. As a naive solution for PCS is highly expensive, we have also developed a tree index, which facilitates efficient and online solutions for PCS.
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paper_title: A local seed selection algorithm for overlapping community detection

paper_content:

One of the widely studied structural properties of social and information networks is their community structure, and a vast variety of community detection algorithms have been proposed in the literature. Expansion of a seed node into a community is one of the most successful methods for local community detection, especially when the global structure of the network is not accessible. An algorithm for local community detection only requires a partial knowledge of the network and the computations can be done in parallel starting from seed nodes. The parallel nature of local algorithms allow for fast and scalable solutions, however, the coverage of the communities heavily depends on the seed selection. The communities identified by a local algorithm might cover only a subset of the nodes in a network if the seeds are not selected carefully. In this paper, we propose a novel seeding algorithm which is parameter free, utilizes merely the local structure of the network, and identifies good seeds which span over the whole network. In order to find such seeds, our algorithm first computes similarity indices from local link prediction techniques to assign a similarity score to each node, and then a biased graph coloring algorithm is used to enhance the seed selection. Our experiments using large-scale real-world networks show that our algorithm is able to select good seeds which are then expanded into high quality overlapping communities covering the vast majority of the nodes in the network using a personalized PageRank-based community detection algorithm. We also show that using our local seeding algorithm can dramatically reduce the execution time of community detection.
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paper_title: Effective and Efficient Community Search Over Large Directed Graphs

paper_content:

Communities are prevalent in social networks, knowledge graphs, and biological networks. Recently, the topic of community search (CS), extracting a dense subgraph containing a query vertex $q$q from a graph, has received great attention. However, existing CS solutions are designed for undirected graphs, and overlook directions of edges which potentially lose useful information carried on directions. In many applications (e.g., Twitter), users’ relationships are often modeled as directed graphs (e.g., if a user $a$a follows another user $b$b, then there is an edge from $a$a to $b$b). In this paper, we study the problem of CS on directed graph. Given a vertex $q$q of a graph $G$G, we aim to find a densely connected subgraph containing $q$q from $G$G, in which vertices have strong interactions and high similarities, by using the minimum in/out-degrees metric. We first develop a baseline algorithm based on the concept of D-core. We further propose three index structures and corresponding query algorithms. Our experimental results on seven real graphs show that our solutions are very effective and efficient. For example, on a graph with over 1 billion of edges, we only need around 40mins to index it and 1$\sim$∼2sec to answer a query.
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paper_title: Index-based Optimal Algorithms for Computing Steiner Components with Maximum Connectivity

paper_content:

With the proliferation of graph applications, the problem of efficiently computing all $k$-edge connected components of a graph G for a user-given k has been recently investigated. In this paper, we study the problem of efficiently computing the steiner component with the maximum connectivity; that is, given a set q of query vertices in a graph G, we aim to find the maximum induced subgraph g of G such that g contains q and g has the maximum connectivity, where g is denoted as SMCC. To accommodate online query processing, we present an efficient algorithm based on a novel index such that the algorithm runs in linear time regarding the result size; thus, the algorithm is optimal since it needs at least linear time to output the result. Moreover, in this paper we also investigate variations of the above problem. We show that such a problem with the constraint that the size of the SMCC is not smaller than a given size can also be solved in linear time regarding the result size (thus, optimal). We also show that the problem of computing the connectivity (rather than the graph details) of SMCC can be solved in linear time regarding the query size (thus, optimal). To build the index, we extend the techniques in [7] to accommodate batch processing and computation sharing. To efficiently support the applications with graph updates, we also present novel increment techniques. Finally, we conduct extensive performance studies on large real and synthetic graphs, which demonstrate that our index-based algorithms significantly outperform baseline algorithms by several orders of magnitude and our indexing algorithms are efficient.
---
paper_title: The community-search problem and how to plan a successful cocktail party

paper_content:

A lot of research in graph mining has been devoted in the discovery of communities. Most of the work has focused in the scenario where communities need to be discovered with only reference to the input graph. However, for many interesting applications one is interested in finding the community formed by a given set of nodes. In this paper we study a query-dependent variant of the community-detection problem, which we call the community-search problem: given a graph G, and a set of query nodes in the graph, we seek to find a subgraph of G that contains the query nodes and it is densely connected. We motivate a measure of density based on minimum degree and distance constraints, and we develop an optimum greedy algorithm for this measure. We proceed by characterizing a class of monotone constraints and we generalize our algorithm to compute optimum solutions satisfying any set of monotone constraints. Finally we modify the greedy algorithm and we present two heuristic algorithms that find communities of size no greater than a specified upper bound. Our experimental evaluation on real datasets demonstrates the efficiency of the proposed algorithms and the quality of the solutions we obtain.
---
paper_title: Approximate Closest Community Search in Networks

paper_content:

Recently, there has been significant interest in the study of the community search problem in social and information networks: given one or more query nodes, find densely connected communities containing the query nodes. However, most existing studies do not address the "free rider" issue, that is, nodes far away from query nodes and irrelevant to them are included in the detected community. Some state-of-the-art models have attempted to address this issue, but not only are their formulated problems NP-hard, they do not admit any approximations without restrictive assumptions, which may not always hold in practice. ::: ::: In this paper, given an undirected graph G and a set of query nodes Q, we study community search using the k-truss based community model. We formulate our problem of finding a closest truss community (CTC), as finding a connected k-truss subgraph with the largest k that contains Q, and has the minimum diameter among such subgraphs. We prove this problem is NP-hard. Furthermore, it is NP-hard to approximate the problem within a factor (2-e), for any e > 0. However, we develop a greedy algorithmic framework, which first finds a CTC containing Q, and then iteratively removes the furthest nodes from Q, from the graph. The method achieves 2-approximation to the optimal solution. To further improve the efficiency, we make use of a compact truss index and develop efficient algorithms for k-truss identification and maintenance as nodes get eliminated. In addition, using bulk deletion optimization and local exploration strategies, we propose two more efficient algorithms. One of them trades some approximation quality for efficiency while the other is a very efficient heuristic. Extensive experiments on 6 real-world networks show the effectiveness and efficiency of our community model and search algorithms.
---
paper_title: Finding weighted k-truss communities in large networks

paper_content:

Community search is a fundamental problem in social network mining, which has attracted much attention in recent years. However, most previous community models only consider the link structure and ignore the link weights of the community, which may miss some useful properties of the community. In this paper, we propose a novel community model, called weightedk-truss community, based on the concept of k-truss. The proposed model takes the edge weight into consideration, thus can better characterize the properties of a community. Based on the new community model, we design a BFS-based online search algorithm to find the top-rweightedk-truss communities in O(m1.5) time, where m denotes the number of edges in a network. To speed up the online search algorithm, we devise a space-efficient index structure, namely KEP-Index, to support efficient community search. We propose two algorithms to construct the index structure in an offline manner. Based on KEP-Index, the time complexity for finding the top-rweightedk-truss communities is linear to the size of these communities, thus it is optimal. We conduct extensive experiments on six large real-world networks, as well as a case study over a co-authorship network. The results demonstrate the efficiency and effectiveness of the proposed community model and algorithms.
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paper_title: Online search of overlapping communities

paper_content:

A great deal of research has been conducted on modeling and discovering communities in complex networks. In most real life networks, an object often participates in multiple overlapping communities. In view of this, recent research has focused on mining overlapping communities in complex networks. The algorithms essentially materialize a snapshot of the overlapping communities in the network. This approach has three drawbacks, however. First, the mining algorithm uses the same global criterion to decide whether a subgraph qualifies as a community. In other words, the criterion is fixed and predetermined. But in reality, communities for different vertices may have very different characteristics. Second, it is costly, time consuming, and often unnecessary to find communities for an entire network. Third, the approach does not support dynamically evolving networks. In this paper, we focus on online search of overlapping communities, that is, given a query vertex, we find meaningful overlapping communities the vertex belongs to in an online manner. In doing so, each search can use community criterion tailored for the vertex in the search. To support this approach, we introduce a novel model for overlapping communities, and we provide theoretical guidelines for tuning the model. We present several algorithms for online overlapping community search and we conduct comprehensive experiments to demonstrate the effectiveness of the model and the algorithms. We also suggest many potential applications of our model and algorithms.
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paper_title: Local search of communities in large graphs

paper_content:

Community search is important in social network analysis. For a given vertex in a graph, the goal is to find the best community the vertex belongs to. Intuitively, the best community for a given vertex should be in the vicinity of the vertex. However, existing solutions use \emph{global search} to find the best community. These algorithms, although straight-forward, are very costly, as all vertices in the graph may need to be visited. In this paper, we propose a \emph{local search} strategy, which searches in the neighborhood of a vertex to find the best community for the vertex. We show that, because the minimum degree measure used to evaluate the goodness of a community is not \emph{monotonic}, designing efficient local search solutions is a very challenging task. We present theories and algorithms of local search to address this challenge. The efficiency of our local search strategy is verified by extensive experiments on both synthetic networks and a variety of real networks with millions of nodes.
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paper_title: On Minimal Steiner Maximum-Connected Subgraph Queries

paper_content:

Given a graph $G$ and a set $Q$ of query nodes, we examine the Steiner Maximum-Connected Subgraph (SMCS) problem. The SMCS, or $G$ 's induced subgraph that contains $Q$ with the largest connectivity, can be useful for customer prediction, product promotion, and team assembling. Despite its importance, the SMCS problem has only been recently studied. Existing solutions evaluate the maximum SMCS , whose number of nodes is the largest among all the SMCSs of $Q$ . However, the maximum SMCS, which may contain a lot of nodes, can be difficult to interpret. In this paper, we investigate the minimal SMCS , which is the minimal subgraph of $G$ with the maximum connectivity containing $Q$ . The minimal SMCS contains much fewer nodes than its maximum counterpart, and is thus easier to be understood. However, the minimal SMCS can be costly to evaluate. We thus propose efficient Expand-Refine algorithms, as well as their approximate versions with accuracy guarantees. We further develop a cache-based processing model to improve the efficiency for an important case when $Q$ consists of a single node. Extensive experiments on large real and synthetic graph datasets validate the effectiveness and efficiency of our approaches.
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paper_title: Index-Based Densest Clique Percolation Community Search in Networks

paper_content:

Community search is important in graph analysis and can be used in many real applications. In the literature, various community models have been proposed. However, most of them cannot well identify the overlaps between communities which is an essential feature of real graphs. To address this issue, the $k$ -clique percolation community model was proposed and has been proven effective in many applications. Motivated by this, in this paper, we adopt the $k$ -clique percolation community model and study the densest clique percolation community search problem which aims to find the $k$ -clique percolation community with the maximum $k$ value that contains a given set of query nodes. We adopt an index-based approach to solve this problem. Based on the observation that a $k$ -clique percolation community is a union of maximal cliques, we devise a novel compact index, $\mathsf {DCPC}$ - $\mathsf {Index}$ , to preserve the maximal cliques and their connectivity information of the input graph. With $\mathsf {DCPC}$ - $\mathsf {Index}$ , we can answer the densest clique percolation community query efficiently. Besides, we also propose an index construction algorithm based on the definition of $\mathsf {DCPC}$ - $\mathsf {Index}$ and further improve the algorithm in terms of efficiency and memory consumption. We conduct extensive performance studies on real graphs and the experimental results demonstrate the efficiency of our index-based query processing algorithm and index construction algorithm.
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paper_title: On Social-Temporal Group Query with Acquaintance Constraint

paper_content:

Three essential criteria are important for activity planning, including: (1) finding a group of attendees familiar with the initiator, (2) ensuring each attendee in the group to have tight social relations with most of the members in the group, and (3) selecting an activity period available for all attendees. Therefore, this paper proposes Social-Temporal Group Query to find the activity time and attendees with the minimum total social distance to the initiator. Moreover, this query incorporates an acquaintance constraint to avoid finding a group with mutually unfamiliar attendees. Efficient processing of the social-temporal group query is very challenging. We show that the problem is NP-hard via a proof and formulate the problem with Integer Programming. We then propose two efficient algorithms, SGSelect and STGSelect, which include effective pruning techniques and employ the idea of pivot time slots to substantially reduce the running time, for finding the optimal solutions. Experimental results indicate that the proposed algorithms are much more efficient and scalable. In the comparison of solution quality, we show that STGSelect outperforms the algorithm that represents manual coordination by the initiator.
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paper_title: Query Optimal k-Plex Based Community in Graphs

paper_content:

Community search problem, which is to find good communities given a set of query nodes in a graph, has attracted increasing research interest recently. Though various measurement models have been proposed to define and solve community search problem. Few of them could define a community concisely and have good quality of query results. They either involve additional constraints for modeling communities, such as size and diameter, or suffer from the free rider effect, i.e., include irrelevant subgraphs. In this paper, we propose a new k-plex based community model for community search. We show that our model not only is simple and clear, but also meets with basic requirements of defining a community search problem. We formulate the maximum k-plex community query (MCKPQ) problem, that is, given a set of query nodes Q, searching for optimal k-plex containing Q. We prove that MCKPQ is NP-hard, and it is hard to approximate in any constant factor. We first give exact solutions. Then, we propose an efficient branch-and-bound (B&B) method and design an effective upper bound function and a pruning strategy. Furthermore, we optimize the basic B&B by fast candidate generation. We also give a fast heuristic solution, which produces high-quality results in practice. The effectiveness of our model of community and the efficiency of our methods are verified by elaborate experiments.
---
paper_title: Most Influential Community Search over Large Social Networks

paper_content:

Detecting social communities in large social networks provides an effective way to analyze the social media users' behaviors and activities. It has drawn extensive attention from both academia and industry. One essential aspect of communities in social networks is outer influence which is the capability to spread internal information of communities to external users. Detecting the communities of high outer influence has particular interest in a wide range of applications, e.g., Ads trending analytics, social opinion mining and news propagation pattern discovery. However, the existing detection techniques largely ignore the outer influence of the communities. To fill the gap, this work investigates the Most Influential Community Search problem to disclose the communities with the highest outer influences. We firstly propose a new community model, maximal kr-Clique community, which has desirable properties, i.e., society, cohesiveness, connectivity, and maximum. Then, we design a novel tree-based index structure, denoted as C-Tree, to maintain the offline computed r-cliques. To efficiently search the most influential communities, we also develop four advanced index-based algorithms which improve the search performance of non-indexed solution by about 200 times. The efficiency and effectiveness of our solution have been extensively verified using six real datasets and a small case study.
---
paper_title: Approximate Closest Community Search in Networks

paper_content:

Recently, there has been significant interest in the study of the community search problem in social and information networks: given one or more query nodes, find densely connected communities containing the query nodes. However, most existing studies do not address the "free rider" issue, that is, nodes far away from query nodes and irrelevant to them are included in the detected community. Some state-of-the-art models have attempted to address this issue, but not only are their formulated problems NP-hard, they do not admit any approximations without restrictive assumptions, which may not always hold in practice. ::: ::: In this paper, given an undirected graph G and a set of query nodes Q, we study community search using the k-truss based community model. We formulate our problem of finding a closest truss community (CTC), as finding a connected k-truss subgraph with the largest k that contains Q, and has the minimum diameter among such subgraphs. We prove this problem is NP-hard. Furthermore, it is NP-hard to approximate the problem within a factor (2-e), for any e > 0. However, we develop a greedy algorithmic framework, which first finds a CTC containing Q, and then iteratively removes the furthest nodes from Q, from the graph. The method achieves 2-approximation to the optimal solution. To further improve the efficiency, we make use of a compact truss index and develop efficient algorithms for k-truss identification and maintenance as nodes get eliminated. In addition, using bulk deletion optimization and local exploration strategies, we propose two more efficient algorithms. One of them trades some approximation quality for efficiency while the other is a very efficient heuristic. Extensive experiments on 6 real-world networks show the effectiveness and efficiency of our community model and search algorithms.
---
paper_title: Finding weighted k-truss communities in large networks

paper_content:

Community search is a fundamental problem in social network mining, which has attracted much attention in recent years. However, most previous community models only consider the link structure and ignore the link weights of the community, which may miss some useful properties of the community. In this paper, we propose a novel community model, called weightedk-truss community, based on the concept of k-truss. The proposed model takes the edge weight into consideration, thus can better characterize the properties of a community. Based on the new community model, we design a BFS-based online search algorithm to find the top-rweightedk-truss communities in O(m1.5) time, where m denotes the number of edges in a network. To speed up the online search algorithm, we devise a space-efficient index structure, namely KEP-Index, to support efficient community search. We propose two algorithms to construct the index structure in an offline manner. Based on KEP-Index, the time complexity for finding the top-rweightedk-truss communities is linear to the size of these communities, thus it is optimal. We conduct extensive experiments on six large real-world networks, as well as a case study over a co-authorship network. The results demonstrate the efficiency and effectiveness of the proposed community model and algorithms.
---
paper_title: Extracting Analyzing and Visualizing Triangle K-Core Motifs within Networks

paper_content:

Cliques are topological structures that usually provide important information for understanding the structure of a graph or network. However, detecting and extracting cliques efficiently is known to be very hard. In this paper, we define and introduce the notion of a Triangle K-Core, a simpler topological structure and one that is more tractable and can moreover be used as a proxy for extracting clique-like structure from large graphs. Based on this definition we first develop a localized algorithm for extracting Triangle K-Cores from large graphs. Subsequently we extend the simple algorithm to accommodate dynamic graphs (where edges can be dynamically added and deleted). Finally, we extend the basic definition to support various template pattern cliques with applications to network visualization and event detection on graphs and networks. Our empirical results reveal the efficiency and efficacy of the proposed methods on many real world datasets.
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paper_title: A graph‐theoretic generalization of the clique concept*

paper_content:

For at least twenty‐five years, the concept of the clique has had a prominent place in sociometric and other kinds of sociological research. Recently, with the advent of large, fast computers and with the growth of interest in graph‐theoretic social network studies, research on the definition and investigation of the graph theoretic properties of clique‐like structures has grown. In the present paper, several of these formulations are examined, and their mathematical properties analyzed. A family of new clique‐like structures is proposed which captures an aspect of cliques which is seldom treated in the existing literature. The new structures, when used to complement existing concepts, provide a new means of tapping several important properties of social networks.
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paper_title: Finding maximal k-edge-connected subgraphs from a large graph

paper_content:

In this paper, we study how to find maximal k-edge-connected subgraphs from a large graph. k-edge-connected subgraphs can be used to capture closely related vertices, and finding such vertex clusters is interesting in many applications, e. g., social network analysis, bioinformatics, web link research. Compared with other explicit structures for modeling vertex clusters, such as quasi-clique, k-core, which only set the requirement on vertex degrees, k-edge-connected subgraph further requires high connectivity within a subgraph (a stronger requirement), and hence defines a more closely related vertex cluster. To find maximal k-edge-connected subgraphs from a graph, a basic approach is to repeatedly apply minimum cut algorithm to the connected components of the input graph until all connected components are k-connected. However, the basic approach is very expensive if the input graph is large. To tackle the problem, we propose three major techniques: vertex reduction, edge reduction and cut pruning. These speed-up techniques are applied on top of the basic approach. We conduct extensive experiments and show that the speed-up techniques are very effective.
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paper_title: Extracting Communities from Complex Networks by the k-Dense Method

paper_content:

To understand the structural and functional properties of large-scale complex networks, it is crucial to efficiently extract a set of cohesive subnetworks as communities. There have been proposed several such community extraction methods in the literature, including the classical k-core decomposition method and, more recently, the k-clique based community extraction method. The k-core method, although computationally efficient, is often not powerful enough for uncovering a detailed community structure and it produces only coarse-grained and loosely connected communities. The k-clique method, on the other hand, can extract fine-grained and tightly connected communities but requires a substantial amount of computational load for large-scale complex networks. In this paper, we present a new notion of a subnetwork called k-dense, and propose an efficient algorithm for extracting k-dense communities. We applied our method to the three different types of networks assembled from real data, namely, from blog trackbacks, word associations and Wikipedia references, and demonstrated that the k-dense method could extract communities almost as efficiently as the k-core method, while the qualities of the extracted communities are comparable to those obtained by the k-clique method.
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paper_title: Network structure and minimum degree

paper_content:

Abstract Social network researchers have long sought measures of network cohesion, Density has often been used for this purpose, despite its generally admitted deficiencies. An approach to network cohesion is proposed that is based on minimum degree and which produces a sequence of subgraphs of gradually increasing cohesion. The approach also associates with any network measures of local density which promise to be useful both in characterizing network structures and in comparing networks.
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paper_title: Discovering Strong Communities with User Engagement and Tie Strength

paper_content:

In this paper, we propose and study a novel cohesive subgraph model, named (\(k\),\(s\))-core, which requires each user to have at least k familiars or friends (not just acquaintances) in the subgraph. The model considers both user engagement and tie strength to discover strong communities. We compare the (\(k\),\(s\))-core model with \(k\)-core and \(k\)-truss theoretically and experimentally. We propose efficient algorithms to compute the (\(k\),\(s\))-core and decompose the graph by a particular sub-model \(k\)-fami. Extensive experiments show (1) our (\(k\),\(s\))-core and \(k\)-fami are effective cohesive subgraph models and (2) the (\(k\),\(s\))-core computation and \(k\)-fami decomposition are efficient on various real-life social networks.
---
paper_title: Index-based Optimal Algorithms for Computing Steiner Components with Maximum Connectivity

paper_content:

With the proliferation of graph applications, the problem of efficiently computing all $k$-edge connected components of a graph G for a user-given k has been recently investigated. In this paper, we study the problem of efficiently computing the steiner component with the maximum connectivity; that is, given a set q of query vertices in a graph G, we aim to find the maximum induced subgraph g of G such that g contains q and g has the maximum connectivity, where g is denoted as SMCC. To accommodate online query processing, we present an efficient algorithm based on a novel index such that the algorithm runs in linear time regarding the result size; thus, the algorithm is optimal since it needs at least linear time to output the result. Moreover, in this paper we also investigate variations of the above problem. We show that such a problem with the constraint that the size of the SMCC is not smaller than a given size can also be solved in linear time regarding the result size (thus, optimal). We also show that the problem of computing the connectivity (rather than the graph details) of SMCC can be solved in linear time regarding the query size (thus, optimal). To build the index, we extend the techniques in [7] to accommodate batch processing and computation sharing. To efficiently support the applications with graph updates, we also present novel increment techniques. Finally, we conduct extensive performance studies on large real and synthetic graphs, which demonstrate that our index-based algorithms significantly outperform baseline algorithms by several orders of magnitude and our indexing algorithms are efficient.
---
paper_title: Listing $k$-cliques in Sparse Real-World Graphs

paper_content:

Motivated by recent studies in the data mining community, we develop the most efficient parallel algorithm for listing all k-cliques in a graph. Our theoretical analysis shows that our algorithm boasts the best asymptotic upper bound on the running time for the case when the input graph is sparse. Our experimental evaluation on large real-world graphs demonstrates that our parallel algorithm is faster than state-of-the-art algorithms, while boasting an excellent degree of parallelism. In particular, we are able to list all k-cliques (for any value of k) in graphs containing up to tens of millions of edges as well as all 10-cliques in graphs containing billions of edges, within a few minutes and a few hours respectively. We show how it can be employed as an effective subroutine for finding the k-clique core decomposition and an approximate k-clique densest subgraphs in very large real-world graphs.
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paper_title: Locally Densest Subgraph Discovery

paper_content:

Mining dense subgraphs from a large graph is a fundamental graph mining task and can be widely applied in a variety of application domains such as network science, biology, graph database, web mining, graph compression, and micro-blogging systems. Here a dense subgraph is defined as a subgraph with high density (#.edge / #.node). Existing studies of this problem either focus on finding the densest subgraph or identifying an optimal clique-like dense subgraph, and they adopt a simple greedy approach to find the top-k dense subgraphs. However, their identified subgraphs cannot be used to represent the dense regions of the graph. Intuitively, to represent a dense region, the subgraph identified should be the subgraph with highest density in its local region in the graph. However, it is non-trivial to formally model a locally densest subgraph. In this paper, we aim to discover top-k such representative locally densest subgraphs of a graph. We provide an elegant parameter-free definition of a locally densest subgraph. The definition not only fits well with the intuition, but is also associated with several nice structural properties. We show that the set of locally densest subgraphs in a graph can be computed in polynomial time. We further propose three novel pruning strategies to largely reduce the search space of the algorithm. In our experiments, we use several real datasets with various graph properties to evaluate the effectiveness of our model using four quality measures and a case study. We also test our algorithms on several real web-scale graphs, one of which contains 118.14 million nodes and 1.02 billion edges, to demonstrate the high efficiency of the proposed algorithms.
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paper_title: Distributed k-core decomposition

paper_content:

Among the novel metrics used to study the relative importance of nodes in complex networks, k-core decomposition has found a number of applications in areas as diverse as sociology, proteinomics, graph visualization, and distributed system analysis and design. This paper proposes new distributed algorithms for the computation of the k-core decomposition of a network, with the purpose of (i) enabling the run-time computation of k-cores in "live" distributed systems and (ii) allowing the decomposition, over a set of connected machines, of very large graphs, that cannot be hosted in a single machine. Lower bounds on the algorithms complexity are given, and an exhaustive experimental analysis on real-world graphs is provided.
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paper_title: The community-search problem and how to plan a successful cocktail party

paper_content:

A lot of research in graph mining has been devoted in the discovery of communities. Most of the work has focused in the scenario where communities need to be discovered with only reference to the input graph. However, for many interesting applications one is interested in finding the community formed by a given set of nodes. In this paper we study a query-dependent variant of the community-detection problem, which we call the community-search problem: given a graph G, and a set of query nodes in the graph, we seek to find a subgraph of G that contains the query nodes and it is densely connected. We motivate a measure of density based on minimum degree and distance constraints, and we develop an optimum greedy algorithm for this measure. We proceed by characterizing a class of monotone constraints and we generalize our algorithm to compute optimum solutions satisfying any set of monotone constraints. Finally we modify the greedy algorithm and we present two heuristic algorithms that find communities of size no greater than a specified upper bound. Our experimental evaluation on real datasets demonstrates the efficiency of the proposed algorithms and the quality of the solutions we obtain.
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paper_title: Linear-time enumeration of maximal K-edge-connected subgraphs in large networks by random contraction

paper_content:

Capturing sets of closely related vertices from large networks is an essential task in many applications such as social network analysis, bioinformatics, and web link research. Decomposing a graph into k-core components is a standard and efficient method for this task, but obtained clusters might not be well-connected. The idea of using maximal k-edge-connected subgraphs was recently proposed to address this issue. Although we can obtain better clusters with this idea, the state-of-the-art method is not efficient enough to process large networks with millions of vertices. In this paper, we propose a new method to decompose a graph into maximal k-edge-connected components, based on random contraction of edges. Our method is simple to implement but improves performance drastically. We experimentally show that our method can successfully decompose large networks and it is thousands times faster than the previous method. Also, we theoretically explain why our method is efficient in practice. To see the importance of maximal k-edge-connected subgraphs, we also conduct experiments using real-world networks to show that many k-core components have small edge-connectivity and they can be decomposed into a lot of maximal k-edge-connected subgraphs.
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paper_title: Classifying Molecular Sequences Using a Linkage Graph With Their Pairwise Similarities

paper_content:

This paper presents a method for classifying a large and mixed set of uncharacterized sequences provided by genome projects. As the measure of sequence similarity, we use similarity score computed by a method based on the dynamic programming (DP), such as the Smith{Waterman local alignment algorithm. Although comparison by DP based method is very sensitive, when given sequences include a family of sequences that are much diverged in evolutionary process, similarity among some of them may be hidden behind spurious similarity of some unrelated sequences. Also the distance derived from the similarity score may not be metric (i.e., triangle inequality may not hold) when some sequences have multi-domain structure. To cope with these problems, we introduce a new graph structure called p-quasi complete graph for describing a family of sequences with a condence measure. We prove that a restricted version of the pquasi complete graph problem (given a positive integer k, whether a graph contains a 0.5-quasi complete subgraph of which size >k or not) is NP-complete. Thus we present an approximation algorithm for classifying a set of sequences using p-quasi complete subgraphs. The eectiveness of our method is demonstrated by the result of classifying over 4000 protein sequences on the Escherichia coli genome that was completely determined recently. c 1999|Elsevier Science B.V. All rights reserved
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paper_title: Fast Hierarchy Construction for Dense Subgraphs

paper_content:

Discovering dense subgraphs and understanding the relations among them is a fundamental problem in graph mining. We want to not only identify dense subgraphs, but also build a hierarchy among them (e.g., larger but sparser subgraphs formed by two smaller dense subgraphs). Peeling algorithms (k-core, k-truss, and nucleus decomposition) have been effective to locate many dense subgraphs. However, constructing a hierarchical representation of density structure, even correctly computing the connected k-cores and k-trusses, have been mostly overlooked. Keeping track of connected components during peeling requires an additional traversal operation, which is as expensive as the peeling process. In this paper, we start with a thorough survey and point to nuances in problem formulations that lead to significant differences in runtimes. We then propose efficient and generic algorithms to construct the hierarchy of dense subgraphs for k-core, k-truss, or any nucleus decomposition. Our algorithms leverage the disjoint-set forest data structure to efficiently construct the hierarchy during traversal. Furthermore, we introduce a new idea to avoid traversal. We construct the subgraphs while visiting neighborhoods in the peeling process, and build the relations to previously constructed subgraphs. We also consider an existing idea to find the k-core hierarchy and adapt for our objectives efficiently. Experiments on different types of large scale real-world networks show significant speedups over naive algorithms and existing alternatives. Our algorithms also outperform the hypothetical limits of any possible traversal-based solution.
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paper_title: Efficient Algorithms for Densest Subgraph Discovery

paper_content:

Densest subgraph discovery (DSD) is a fundamental problem in graph mining. It has been studied for decades, and is widely used in various areas, including network science, biological analysis, and graph databases. Given a graph G, DSD aims to find a subgraph D of G with the highest density (e.g., the number of edges over the number of vertices in D). Because DSD is difficult to solve, we propose a new solution paradigm in this paper. Our main observation is that a densest subgraph can be accurately found through a k-core (a kind of dense subgraph of G), with theoretical guarantees. Based on this intuition, we develop efficient exact and approximation solutions for DSD. Moreover, our solutions are able to find the densest subgraphs for a wide range of graph density definitions, including clique-based and general pattern-based density. We have performed extensive experimental evaluation on eleven real datasets. Our results show that our algorithms are up to four orders of magnitude faster than existing approaches.
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paper_title: Large Scale Cohesive Subgraphs Discovery for Social Network Visual Analysis

paper_content:

Graphs are widely used in large scale social network analysis nowadays. Not only analysts need to focus on cohesive subgraphs to study patterns among social actors, but also normal users are interested in discovering what happening in their neighborhood. However, effectively storing large scale social network and efficiently identifying cohesive subgraphs is challenging. In this work we introduce a novel subgraph concept to capture the cohesion in social interactions, and propose an I/O efficient approach to discover cohesive subgraphs. ::: ::: Besides, we propose an analytic system which allows users to perform intuitive, visual browsing on large scale social networks. Our system stores the network as a social graph in the graph database, retrieves a local cohesive subgraph based on the input keywords, and then hierarchically visualizes the subgraph out on orbital layout, in which more important social actors are located in the center. By summarizing textual interactions between social actors as tag cloud, we provide a way to quickly locate active social communities and their interactions in a unified view.
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paper_title: Fast algorithms for maximal clique enumeration with limited memory

paper_content:

Maximal clique enumeration (MCE) is a long-standing problem in graph theory and has numerous important applications. Though extensively studied, most existing algorithms become impractical when the input graph is too large and is disk-resident. We first propose an efficient partition-based algorithm for MCE that addresses the problem of processing large graphs with limited memory. We then further reduce the high cost of CPU computation of MCE by a careful nested partition based on a cost model. Finally, we parallelize our algorithm to further reduce the overall running time. We verified the efficiency of our algorithms by experiments in large real-world graphs.
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paper_title: Efficient Truss Maintenance in Evolving Networks

paper_content:

Truss was proposed to study social network data represented by graphs. A k-truss of a graph is a cohesive subgraph, in which each edge is contained in at least k-2 triangles within the subgraph. While truss has been demonstrated as superior to model the close relationship in social networks and efficient algorithms for finding trusses have been extensively studied, very little attention has been paid to truss maintenance. However, most social networks are evolving networks. It may be infeasible to recompute trusses from scratch from time to time in order to find the up-to-date $k$-trusses in the evolving networks. In this paper, we discuss how to maintain trusses in a graph with dynamic updates. We first discuss a set of properties on maintaining trusses, then propose algorithms on maintaining trusses on edge deletions and insertions, finally, we discuss truss index maintenance. We test the proposed techniques on real datasets. The experiment results show the promise of our work.
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paper_title: Algorithmic Graph Theory

paper_content:

Preface 1. Introducing graphs and algorithmic complexity 2. Spanning-trees, branchings and connectivity 3. Planar graphs 4. Networks and flows 5. Matchings 6. Eulerian and Hamiltonian tours 7. Colouring graphs 8. Graph problems and intractability Appendix Author Index Subject Index.
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paper_title: Denser than the densest subgraph: extracting optimal quasi-cliques with quality guarantees

paper_content:

Finding dense subgraphs is an important graph-mining task with many applications. Given that the direct optimization of edge density is not meaningful, as even a single edge achieves maximum density, research has focused on optimizing alternative density functions. A very popular among such functions is the average degree, whose maximization leads to the well-known densest-subgraph notion. Surprisingly enough, however, densest subgraphs are typically large graphs, with small edge density and large diameter. In this paper, we define a novel density function, which gives subgraphs of much higher quality than densest subgraphs: the graphs found by our method are compact, dense, and with smaller diameter. We show that the proposed function can be derived from a general framework, which includes other important density functions as subcases and for which we show interesting general theoretical properties. To optimize the proposed function we provide an additive approximation algorithm and a local-search heuristic. Both algorithms are very efficient and scale well to large graphs. We evaluate our algorithms on real and synthetic datasets, and we also devise several application studies as variants of our original problem. When compared with the method that finds the subgraph of the largest average degree, our algorithms return denser subgraphs with smaller diameter. Finally, we discuss new interesting research directions that our problem leaves open.
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paper_title: Index-Based Densest Clique Percolation Community Search in Networks

paper_content:

Community search is important in graph analysis and can be used in many real applications. In the literature, various community models have been proposed. However, most of them cannot well identify the overlaps between communities which is an essential feature of real graphs. To address this issue, the $k$ -clique percolation community model was proposed and has been proven effective in many applications. Motivated by this, in this paper, we adopt the $k$ -clique percolation community model and study the densest clique percolation community search problem which aims to find the $k$ -clique percolation community with the maximum $k$ value that contains a given set of query nodes. We adopt an index-based approach to solve this problem. Based on the observation that a $k$ -clique percolation community is a union of maximal cliques, we devise a novel compact index, $\mathsf {DCPC}$ - $\mathsf {Index}$ , to preserve the maximal cliques and their connectivity information of the input graph. With $\mathsf {DCPC}$ - $\mathsf {Index}$ , we can answer the densest clique percolation community query efficiently. Besides, we also propose an index construction algorithm based on the definition of $\mathsf {DCPC}$ - $\mathsf {Index}$ and further improve the algorithm in terms of efficiency and memory consumption. We conduct extensive performance studies on real graphs and the experimental results demonstrate the efficiency of our index-based query processing algorithm and index construction algorithm.
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paper_title: Top-k overlapping densest subgraphs

paper_content:

Finding dense subgraphs is an important problem in graph mining and has many practical applications. At the same time, while large real-world networks are known to have many communities that are not well-separated, the majority of the existing work focuses on the problem of finding a single densest subgraph. Hence, it is natural to consider the question of finding the top-kdensest subgraphs. One major challenge in addressing this question is how to handle overlaps: eliminating overlaps completely is one option, but this may lead to extracting subgraphs not as dense as it would be possible by allowing a limited amount of overlap. Furthermore, overlaps are desirable as in most real-world graphs there are vertices that belong to more than one community, and thus, to more than one densest subgraph. In this paper we study the problem of finding top-koverlapping densest subgraphs, and we present a new approach that improves over the existing techniques, both in theory and practice. First, we reformulate the problem definition in a way that we are able to obtain an algorithm with constant-factor approximation guarantee. Our approach relies on using techniques for solving the max-sum diversification problem, which however, we need to extend in order to make them applicable to our setting. Second, we evaluate our algorithm on a collection of benchmark datasets and show that it convincingly outperforms the previous methods, both in terms of quality and efficiency.
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paper_title: An O(m) Algorithm for Cores Decomposition of Networks

paper_content:

The structure of large networks can be revealed by partitioning them to smaller parts, which are easier to handle. One of such decompositions is based on $k$--cores, proposed in 1983 by Seidman. In the paper an efficient, $O(m)$, $m$ is the number of lines, algorithm for determining the cores decomposition of a given network is presented.
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paper_title: Efficient Core Maintenance in Large Dynamic Graphs

paper_content:

The $k$-core decomposition in a graph is a fundamental problem for social network analysis. The problem of $k$-core decomposition is to calculate the core number for every node in a graph. Previous studies mainly focus on $k$-core decomposition in a static graph. There exists a linear time algorithm for $k$-core decomposition in a static graph. However, in many real-world applications such as online social networks and the Internet, the graph typically evolves over time. Under such applications, a key issue is to maintain the core number of nodes given the graph changes over time. A simple implementation is to perform the linear time algorithm to recompute the core number for every node after the graph is updated. Such simple implementation is expensive when the graph is very large. In this paper, we propose a new efficient algorithm to maintain the core number for every node in a dynamic graph. Our main result is that only certain nodes need to update their core number given the graph is changed by inserting/deleting an edge. We devise an efficient algorithm to identify and recompute the core number of such nodes. The complexity of our algorithm is independent of the graph size. In addition, to further accelerate the algorithm, we develop two pruning strategies by exploiting the lower and upper bounds of the core number. Finally, we conduct extensive experiments over both real-world and synthetic datasets, and the results demonstrate the efficiency of the proposed algorithm.
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paper_title: Graph Clustering Using Distance-k Cliques

paper_content:

Identifying the natural clusters of nodes in a graph and treating them as supernodes or metanodes for a higher level graph (or an abstract graph) is a technique used for the reduction of visual complexity of graphs with a large number of nodes. In this paper we report on the implementation of a clustering algorithm based on the idea of distance-k cliques, a generalization of the idea of the cliques in graphs. The performance of the clustering algorithm on some large graphs obtained from the archives of Bell Laboratories is presented.
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paper_title: K-Core Decomposition of Large Networks on a Single PC

paper_content:

Studying the topology of a network is critical to inferring underlying dynamics such as tolerance to failure, group behavior and spreading patterns. k-core decomposition is a well-established metric which partitions a graph into layers from external to more central vertices. In this paper we aim to explore whether k-core decomposition of large networks can be computed using a consumer-grade PC. We feature implementations of the "vertex-centric" distributed protocol introduced by Montresor, De Pellegrini and Miorandi on GraphChi and Webgraph. Also, we present an accurate implementation of the Batagelj and Zaversnik algorithm for k-core decomposition in Webgraph. With our implementations, we show that we can efficiently handle networks of billions of edges using a single consumer-level machine within reasonable time and can produce excellent approximations in only a fraction of the execution time. To the best of our knowledge, our biggest graphs are considerably larger than the graphs considered in the literature. Next, we present an optimized implementation of an external-memory algorithm (EMcore) by Cheng, Ke, Chu, and Ozsu. We show that this algorithm also performs well for large datasets, however, it cannot predict whether a given memory budget is sufficient for a new dataset. We present a thorough analysis of all algorithms concluding that it is viable to compute k-core decomposition for large networks in a consumer-grade PC.
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paper_title: Efficiently computing k-edge connected components via graph decomposition

paper_content:

Efficiently computing k-edge connected components in a large graph, G = (V, E), where V is the vertex set and E is the edge set, is a long standing research problem. It is not only fundamental in graph analysis but also crucial in graph search optimization algorithms. Consider existing techniques for computing k-edge connected components are quite time consuming and are unlikely to be scalable for large scale graphs, in this paper we firstly propose a novel graph decomposition paradigm to iteratively decompose a graph G for computing its k-edge connected components such that the number of drilling-down iterations h is bounded by the "depth" of the k-edge connected components nested together to form G, where h usually is a small integer in practice. Secondly, we devise a novel, efficient threshold-based graph decomposition algorithm, with time complexity O(l × |E|), to decompose a graph G at each iteration, where l usually is a small integer with l « |V|. As a result, our algorithm for computing k-edge connected components significantly improves the time complexity of an existing state-of-the-art technique from O(|V|2|E| + |V|3 log |V|) to O(h × l × |E|). Finally, we conduct extensive performance studies on large real and synthetic graphs. The performance studies demonstrate that our techniques significantly outperform the state-of-the-art solution by several orders of magnitude.
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paper_title: A Fast Order-Based Approach for Core Maintenance

paper_content:

Graphs have been widely used in many applications such as social networks, collaboration networks, and biological networks. One important graph analytics is to explore cohesive subgraphs in a large graph. Among several cohesive subgraphs studied, k-core is one that can be computed in linear time for a static graph. Since graphs are evolving in real applications, in this paper, we study core maintenance which is to reduce the computational cost to compute k-cores for a graph when graphs are updated from time to time dynamically. We identify drawbacks of the existing efficient algorithm, which needs a large search space to find the vertices that need to be updated, and has high overhead to maintain the index built, when a graph is updated. We propose a new order-based approach to maintain an order, called k-order, among vertices, while a graph is updated. Our new algorithm can significantly outperform the state-of-theart algorithm up to 3 orders of magnitude for the 11 large real graphs tested. We report our findings in this paper.
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paper_title: Distributed algorithms for k-truss decomposition

paper_content:

k-truss, a type of cohesive subgraphs of a network, is an important measure for a social network graph. However, with the emergence of large online social networks, the running time of the traditional batch algorithms for k-truss decomposition is usually prohibitively long on such a graph with billions of edges and millions of vertices. Moreover, the size of a graph becomes too large to load into the main memory of a single machine. Currently, cloud computing has become an imperative way to process the big data. Thus, our aim is to design a scalable algorithm of k-truss decomposition in the scenario of cloud computing. In this paper, we first improve the existing distributed k-truss decomposition in the MapReduce framework. We then propose a theoretical basis for k-truss and use it to design an algorithm based on graph-parallel abstractions. Our experiment results show that our method in the graph-parallel abstraction significantly outperforms the methods based on MapReduce in terms of running time and disk usage.
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paper_title: Efficient core decomposition in massive networks

paper_content:

The k-core of a graph is the largest subgraph in which every vertex is connected to at least k other vertices within the subgraph. Core decomposition finds the k-core of the graph for every possible k. Past studies have shown important applications of core decomposition such as in the study of the properties of large networks (e.g., sustainability, connectivity, centrality, etc.), for solving NP-hard problems efficiently in real networks (e.g., maximum clique finding, densest subgraph approximation, etc.), and for large-scale network fingerprinting and visualization. The k-core is a well accepted concept partly because there exists a simple and efficient algorithm for core decomposition, by recursively removing the lowest degree vertices and their incident edges. However, this algorithm requires random access to the graph and hence assumes the entire graph can be kept in main memory. Nevertheless, real-world networks such as online social networks have become exceedingly large in recent years and still keep growing at a steady rate. In this paper, we propose the first external-memory algorithm for core decomposition in massive graphs. When the memory is large enough to hold the graph, our algorithm achieves comparable performance as the in-memory algorithm. When the graph is too large to be kept in the memory, our algorithm requires only O(k max ) scans of the graph, where k max is the largest core number of the graph. We demonstrate the efficiency of our algorithm on real networks with up to 52.9 million vertices and 1.65 billion edges.
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paper_title: Incremental k-core decomposition: algorithms and evaluation

paper_content:

A k-core of a graph is a maximal connected subgraph in which every vertex is connected to at least k vertices in the subgraph. k-core decomposition is often used in large-scale network analysis, such as community detection, protein function prediction, visualization, and solving NP-hard problems on real networks efficiently, like maximal clique finding. In many real-world applications, networks change over time. As a result, it is essential to develop efficient incremental algorithms for dynamic graph data. In this paper, we propose a suite of incremental k-core decomposition algorithms for dynamic graph data. These algorithms locate a small subgraph that is guaranteed to contain the list of vertices whose maximum k-core values have changed and efficiently process this subgraph to update the k-core decomposition. We present incremental algorithms for both insertion and deletion operations, and propose auxiliary vertex state maintenance techniques that can further accelerate these operations. Our results show a significant reduction in runtime compared to non-incremental alternatives. We illustrate the efficiency of our algorithms on different types of real and synthetic graphs, at varying scales. For a graph of 16 million vertices, we observe relative throughputs reaching a million times, relative to the non-incremental algorithms.
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paper_title: I/O Efficient Core Graph Decomposition: Application to Degeneracy Ordering

paper_content:

Core decomposition is a fundamental graph problem with a large number of applications. Most existing approaches for core decomposition assume that the graph is kept in memory of a machine. Nevertheless, many real-world graphs are too big to reside in memory. In this paper, we study I/O efficient core decomposition following a semi-external model, which only allows node information to be loaded in memory. We propose a semi-external algorithm and an optimized algorithm for I/O efficient core decomposition. To handle dynamic graph updates, we firstly show that our algorithm can be naturally extended to handle edge deletion. Then, we propose an I/O efficient core maintenance algorithm to handle edge insertion, and an improved algorithm to further reduce I/O and CPU cost. In addition, based on our core decomposition algorithms, we further propose an I/O efficient semi-external algorithm for degeneracy ordering, which is an important graph problem that is highly related to core decomposition. We also consider how to maintain the degeneracy order. We conduct extensive experiments on 12 real large graphs. Our optimal core decomposition algorithm significantly outperforms the existing I/O efficient algorithm in terms of both processing time and memory consumption. They are very scalable to handle web-scale graphs. As an example, we are the first to handle a web graph with 978.5 million nodes and 42.6 billion edges using less than 4.2 GB memory. We also show that our proposed algorithms for degeneracy order computation and maintenance can handle big graphs efficiently with small memory overhead.
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paper_title: Uncovering the overlapping community structure of complex networks in nature and society

paper_content:

Many complex systems in nature and society can be described in terms of networks capturing the intricate web of connections among the units they are made of. A key question is how to interpret the global organization of such networks as the coexistence of their structural subunits (communities) associated with more highly interconnected parts. Identifying these a priori unknown building blocks (such as functionally related proteins, industrial sectors and groups of people) is crucial to the understanding of the structural and functional properties of networks. The existing deterministic methods used for large networks find separated communities, whereas most of the actual networks are made of highly overlapping cohesive groups of nodes. Here we introduce an approach to analysing the main statistical features of the interwoven sets of overlapping communities that makes a step towards uncovering the modular structure of complex systems. After defining a set of new characteristic quantities for the statistics of communities, we apply an efficient technique for exploring overlapping communities on a large scale. We find that overlaps are significant, and the distributions we introduce reveal universal features of networks. Our studies of collaboration, word-association and protein interaction graphs show that the web of communities has non-trivial correlations and specific scaling properties.
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paper_title: Truss Decomposition in Massive Networks

paper_content:

The k-truss is a type of cohesive subgraphs proposed recently for the study of networks. While the problem of computing most cohesive subgraphs is NP-hard, there exists a polynomial time algorithm for computing k-truss. Compared with k-core which is also efficient to compute, k-truss represents the "core" of a k-core that keeps the key information of, while filtering out less important information from, the k-core. However, existing algorithms for computing k-truss are inefficient for handling today's massive networks. We first improve the existing in-memory algorithm for computing k-truss in networks of moderate size. Then, we propose two I/O-efficient algorithms to handle massive networks that cannot fit in main memory. Our experiments on real datasets verify the efficiency of our algorithms and the value of k-truss.
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paper_title: Approximate Closest Community Search in Networks

paper_content:

Recently, there has been significant interest in the study of the community search problem in social and information networks: given one or more query nodes, find densely connected communities containing the query nodes. However, most existing studies do not address the "free rider" issue, that is, nodes far away from query nodes and irrelevant to them are included in the detected community. Some state-of-the-art models have attempted to address this issue, but not only are their formulated problems NP-hard, they do not admit any approximations without restrictive assumptions, which may not always hold in practice. ::: ::: In this paper, given an undirected graph G and a set of query nodes Q, we study community search using the k-truss based community model. We formulate our problem of finding a closest truss community (CTC), as finding a connected k-truss subgraph with the largest k that contains Q, and has the minimum diameter among such subgraphs. We prove this problem is NP-hard. Furthermore, it is NP-hard to approximate the problem within a factor (2-e), for any e > 0. However, we develop a greedy algorithmic framework, which first finds a CTC containing Q, and then iteratively removes the furthest nodes from Q, from the graph. The method achieves 2-approximation to the optimal solution. To further improve the efficiency, we make use of a compact truss index and develop efficient algorithms for k-truss identification and maintenance as nodes get eliminated. In addition, using bulk deletion optimization and local exploration strategies, we propose two more efficient algorithms. One of them trades some approximation quality for efficiency while the other is a very efficient heuristic. Extensive experiments on 6 real-world networks show the effectiveness and efficiency of our community model and search algorithms.
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paper_title: Finding weighted k-truss communities in large networks

paper_content:

Community search is a fundamental problem in social network mining, which has attracted much attention in recent years. However, most previous community models only consider the link structure and ignore the link weights of the community, which may miss some useful properties of the community. In this paper, we propose a novel community model, called weightedk-truss community, based on the concept of k-truss. The proposed model takes the edge weight into consideration, thus can better characterize the properties of a community. Based on the new community model, we design a BFS-based online search algorithm to find the top-rweightedk-truss communities in O(m1.5) time, where m denotes the number of edges in a network. To speed up the online search algorithm, we devise a space-efficient index structure, namely KEP-Index, to support efficient community search. We propose two algorithms to construct the index structure in an offline manner. Based on KEP-Index, the time complexity for finding the top-rweightedk-truss communities is linear to the size of these communities, thus it is optimal. We conduct extensive experiments on six large real-world networks, as well as a case study over a co-authorship network. The results demonstrate the efficiency and effectiveness of the proposed community model and algorithms.
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paper_title: Online search of overlapping communities

paper_content:

A great deal of research has been conducted on modeling and discovering communities in complex networks. In most real life networks, an object often participates in multiple overlapping communities. In view of this, recent research has focused on mining overlapping communities in complex networks. The algorithms essentially materialize a snapshot of the overlapping communities in the network. This approach has three drawbacks, however. First, the mining algorithm uses the same global criterion to decide whether a subgraph qualifies as a community. In other words, the criterion is fixed and predetermined. But in reality, communities for different vertices may have very different characteristics. Second, it is costly, time consuming, and often unnecessary to find communities for an entire network. Third, the approach does not support dynamically evolving networks. In this paper, we focus on online search of overlapping communities, that is, given a query vertex, we find meaningful overlapping communities the vertex belongs to in an online manner. In doing so, each search can use community criterion tailored for the vertex in the search. To support this approach, we introduce a novel model for overlapping communities, and we provide theoretical guidelines for tuning the model. We present several algorithms for online overlapping community search and we conduct comprehensive experiments to demonstrate the effectiveness of the model and the algorithms. We also suggest many potential applications of our model and algorithms.
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paper_title: A graph‐theoretic generalization of the clique concept*

paper_content:

For at least twenty‐five years, the concept of the clique has had a prominent place in sociometric and other kinds of sociological research. Recently, with the advent of large, fast computers and with the growth of interest in graph‐theoretic social network studies, research on the definition and investigation of the graph theoretic properties of clique‐like structures has grown. In the present paper, several of these formulations are examined, and their mathematical properties analyzed. A family of new clique‐like structures is proposed which captures an aspect of cliques which is seldom treated in the existing literature. The new structures, when used to complement existing concepts, provide a new means of tapping several important properties of social networks.
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paper_title: R.: On effectively finding maximal quasi-cliques in graphs

paper_content:

The problem of finding a maximum clique in a graph is prototypical for many clustering and similarity problems; however, in many real-world scenarios, the classical problem of finding a complete subgraph needs to be relaxed to finding an almost complete subgraph, a so-called quasi-clique . In this work, we demonstrate how two previously existing definitions of quasi-cliques can be unified and how the resulting, more general quasi-clique finding problem can be solved by extending two state-of-the-art stochastic local search algorithms for the classical maximum clique problem. Preliminary results for these algorithms applied to both, artificial and real-world problem instances demonstrate the usefulness of the new quasi-clique definition and the effectiveness of our algorithms.
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paper_title: Online search of overlapping communities

paper_content:

A great deal of research has been conducted on modeling and discovering communities in complex networks. In most real life networks, an object often participates in multiple overlapping communities. In view of this, recent research has focused on mining overlapping communities in complex networks. The algorithms essentially materialize a snapshot of the overlapping communities in the network. This approach has three drawbacks, however. First, the mining algorithm uses the same global criterion to decide whether a subgraph qualifies as a community. In other words, the criterion is fixed and predetermined. But in reality, communities for different vertices may have very different characteristics. Second, it is costly, time consuming, and often unnecessary to find communities for an entire network. Third, the approach does not support dynamically evolving networks. In this paper, we focus on online search of overlapping communities, that is, given a query vertex, we find meaningful overlapping communities the vertex belongs to in an online manner. In doing so, each search can use community criterion tailored for the vertex in the search. To support this approach, we introduce a novel model for overlapping communities, and we provide theoretical guidelines for tuning the model. We present several algorithms for online overlapping community search and we conduct comprehensive experiments to demonstrate the effectiveness of the model and the algorithms. We also suggest many potential applications of our model and algorithms.
---
paper_title: Most Influential Community Search over Large Social Networks

paper_content:

Detecting social communities in large social networks provides an effective way to analyze the social media users' behaviors and activities. It has drawn extensive attention from both academia and industry. One essential aspect of communities in social networks is outer influence which is the capability to spread internal information of communities to external users. Detecting the communities of high outer influence has particular interest in a wide range of applications, e.g., Ads trending analytics, social opinion mining and news propagation pattern discovery. However, the existing detection techniques largely ignore the outer influence of the communities. To fill the gap, this work investigates the Most Influential Community Search problem to disclose the communities with the highest outer influences. We firstly propose a new community model, maximal kr-Clique community, which has desirable properties, i.e., society, cohesiveness, connectivity, and maximum. Then, we design a novel tree-based index structure, denoted as C-Tree, to maintain the offline computed r-cliques. To efficiently search the most influential communities, we also develop four advanced index-based algorithms which improve the search performance of non-indexed solution by about 200 times. The efficiency and effectiveness of our solution have been extensively verified using six real datasets and a small case study.
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paper_title: Uncovering the overlapping community structure of complex networks in nature and society

paper_content:

Many complex systems in nature and society can be described in terms of networks capturing the intricate web of connections among the units they are made of. A key question is how to interpret the global organization of such networks as the coexistence of their structural subunits (communities) associated with more highly interconnected parts. Identifying these a priori unknown building blocks (such as functionally related proteins, industrial sectors and groups of people) is crucial to the understanding of the structural and functional properties of networks. The existing deterministic methods used for large networks find separated communities, whereas most of the actual networks are made of highly overlapping cohesive groups of nodes. Here we introduce an approach to analysing the main statistical features of the interwoven sets of overlapping communities that makes a step towards uncovering the modular structure of complex systems. After defining a set of new characteristic quantities for the statistics of communities, we apply an efficient technique for exploring overlapping communities on a large scale. We find that overlaps are significant, and the distributions we introduce reveal universal features of networks. Our studies of collaboration, word-association and protein interaction graphs show that the web of communities has non-trivial correlations and specific scaling properties.
---
paper_title: Uncovering the overlapping community structure of complex networks in nature and society

paper_content:

Many complex systems in nature and society can be described in terms of networks capturing the intricate web of connections among the units they are made of. A key question is how to interpret the global organization of such networks as the coexistence of their structural subunits (communities) associated with more highly interconnected parts. Identifying these a priori unknown building blocks (such as functionally related proteins, industrial sectors and groups of people) is crucial to the understanding of the structural and functional properties of networks. The existing deterministic methods used for large networks find separated communities, whereas most of the actual networks are made of highly overlapping cohesive groups of nodes. Here we introduce an approach to analysing the main statistical features of the interwoven sets of overlapping communities that makes a step towards uncovering the modular structure of complex systems. After defining a set of new characteristic quantities for the statistics of communities, we apply an efficient technique for exploring overlapping communities on a large scale. We find that overlaps are significant, and the distributions we introduce reveal universal features of networks. Our studies of collaboration, word-association and protein interaction graphs show that the web of communities has non-trivial correlations and specific scaling properties.
---
paper_title: Uncovering the overlapping community structure of complex networks in nature and society

paper_content:

Many complex systems in nature and society can be described in terms of networks capturing the intricate web of connections among the units they are made of. A key question is how to interpret the global organization of such networks as the coexistence of their structural subunits (communities) associated with more highly interconnected parts. Identifying these a priori unknown building blocks (such as functionally related proteins, industrial sectors and groups of people) is crucial to the understanding of the structural and functional properties of networks. The existing deterministic methods used for large networks find separated communities, whereas most of the actual networks are made of highly overlapping cohesive groups of nodes. Here we introduce an approach to analysing the main statistical features of the interwoven sets of overlapping communities that makes a step towards uncovering the modular structure of complex systems. After defining a set of new characteristic quantities for the statistics of communities, we apply an efficient technique for exploring overlapping communities on a large scale. We find that overlaps are significant, and the distributions we introduce reveal universal features of networks. Our studies of collaboration, word-association and protein interaction graphs show that the web of communities has non-trivial correlations and specific scaling properties.
---
paper_title: Online search of overlapping communities

paper_content:

A great deal of research has been conducted on modeling and discovering communities in complex networks. In most real life networks, an object often participates in multiple overlapping communities. In view of this, recent research has focused on mining overlapping communities in complex networks. The algorithms essentially materialize a snapshot of the overlapping communities in the network. This approach has three drawbacks, however. First, the mining algorithm uses the same global criterion to decide whether a subgraph qualifies as a community. In other words, the criterion is fixed and predetermined. But in reality, communities for different vertices may have very different characteristics. Second, it is costly, time consuming, and often unnecessary to find communities for an entire network. Third, the approach does not support dynamically evolving networks. In this paper, we focus on online search of overlapping communities, that is, given a query vertex, we find meaningful overlapping communities the vertex belongs to in an online manner. In doing so, each search can use community criterion tailored for the vertex in the search. To support this approach, we introduce a novel model for overlapping communities, and we provide theoretical guidelines for tuning the model. We present several algorithms for online overlapping community search and we conduct comprehensive experiments to demonstrate the effectiveness of the model and the algorithms. We also suggest many potential applications of our model and algorithms.
---
paper_title: Index-Based Densest Clique Percolation Community Search in Networks

paper_content:

Community search is important in graph analysis and can be used in many real applications. In the literature, various community models have been proposed. However, most of them cannot well identify the overlaps between communities which is an essential feature of real graphs. To address this issue, the $k$ -clique percolation community model was proposed and has been proven effective in many applications. Motivated by this, in this paper, we adopt the $k$ -clique percolation community model and study the densest clique percolation community search problem which aims to find the $k$ -clique percolation community with the maximum $k$ value that contains a given set of query nodes. We adopt an index-based approach to solve this problem. Based on the observation that a $k$ -clique percolation community is a union of maximal cliques, we devise a novel compact index, $\mathsf {DCPC}$ - $\mathsf {Index}$ , to preserve the maximal cliques and their connectivity information of the input graph. With $\mathsf {DCPC}$ - $\mathsf {Index}$ , we can answer the densest clique percolation community query efficiently. Besides, we also propose an index construction algorithm based on the definition of $\mathsf {DCPC}$ - $\mathsf {Index}$ and further improve the algorithm in terms of efficiency and memory consumption. We conduct extensive performance studies on real graphs and the experimental results demonstrate the efficiency of our index-based query processing algorithm and index construction algorithm.
---
paper_title: Uncovering the overlapping community structure of complex networks in nature and society

paper_content:

Many complex systems in nature and society can be described in terms of networks capturing the intricate web of connections among the units they are made of. A key question is how to interpret the global organization of such networks as the coexistence of their structural subunits (communities) associated with more highly interconnected parts. Identifying these a priori unknown building blocks (such as functionally related proteins, industrial sectors and groups of people) is crucial to the understanding of the structural and functional properties of networks. The existing deterministic methods used for large networks find separated communities, whereas most of the actual networks are made of highly overlapping cohesive groups of nodes. Here we introduce an approach to analysing the main statistical features of the interwoven sets of overlapping communities that makes a step towards uncovering the modular structure of complex systems. After defining a set of new characteristic quantities for the statistics of communities, we apply an efficient technique for exploring overlapping communities on a large scale. We find that overlaps are significant, and the distributions we introduce reveal universal features of networks. Our studies of collaboration, word-association and protein interaction graphs show that the web of communities has non-trivial correlations and specific scaling properties.
---
paper_title: Index-Based Densest Clique Percolation Community Search in Networks

paper_content:

Community search is important in graph analysis and can be used in many real applications. In the literature, various community models have been proposed. However, most of them cannot well identify the overlaps between communities which is an essential feature of real graphs. To address this issue, the $k$ -clique percolation community model was proposed and has been proven effective in many applications. Motivated by this, in this paper, we adopt the $k$ -clique percolation community model and study the densest clique percolation community search problem which aims to find the $k$ -clique percolation community with the maximum $k$ value that contains a given set of query nodes. We adopt an index-based approach to solve this problem. Based on the observation that a $k$ -clique percolation community is a union of maximal cliques, we devise a novel compact index, $\mathsf {DCPC}$ - $\mathsf {Index}$ , to preserve the maximal cliques and their connectivity information of the input graph. With $\mathsf {DCPC}$ - $\mathsf {Index}$ , we can answer the densest clique percolation community query efficiently. Besides, we also propose an index construction algorithm based on the definition of $\mathsf {DCPC}$ - $\mathsf {Index}$ and further improve the algorithm in terms of efficiency and memory consumption. We conduct extensive performance studies on real graphs and the experimental results demonstrate the efficiency of our index-based query processing algorithm and index construction algorithm.
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paper_title: An O(m) Algorithm for Cores Decomposition of Networks

paper_content:

The structure of large networks can be revealed by partitioning them to smaller parts, which are easier to handle. One of such decompositions is based on $k$--cores, proposed in 1983 by Seidman. In the paper an efficient, $O(m)$, $m$ is the number of lines, algorithm for determining the cores decomposition of a given network is presented.
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paper_title: Uncovering the overlapping community structure of complex networks in nature and society

paper_content:

Many complex systems in nature and society can be described in terms of networks capturing the intricate web of connections among the units they are made of. A key question is how to interpret the global organization of such networks as the coexistence of their structural subunits (communities) associated with more highly interconnected parts. Identifying these a priori unknown building blocks (such as functionally related proteins, industrial sectors and groups of people) is crucial to the understanding of the structural and functional properties of networks. The existing deterministic methods used for large networks find separated communities, whereas most of the actual networks are made of highly overlapping cohesive groups of nodes. Here we introduce an approach to analysing the main statistical features of the interwoven sets of overlapping communities that makes a step towards uncovering the modular structure of complex systems. After defining a set of new characteristic quantities for the statistics of communities, we apply an efficient technique for exploring overlapping communities on a large scale. We find that overlaps are significant, and the distributions we introduce reveal universal features of networks. Our studies of collaboration, word-association and protein interaction graphs show that the web of communities has non-trivial correlations and specific scaling properties.
---
paper_title: Spatial Networks

paper_content:

Complex systems are very often organized under the form of networks where nodes and edges are embedded in space. Transportation and mobility networks, Internet, mobile phone networks, power grids, social and contact networks, neural networks, are all examples where space is relevant and where topology alone does not contain all the information. Characterizing and understanding the structure and the evolution of spatial networks is thus crucial for many different fields ranging from urbanism to epidemiology. An important consequence of space on networks is that there is a cost associated to the length of edges which in turn has dramatic effects on the topological structure of these networks. We will expose thoroughly the current state of our understanding of how the spatial constraints affect the structure and properties of these networks. We will review the most recent empirical observations and the most important models of spatial networks. We will also discuss various processes which take place on these spatial networks, such as phase transitions, random walks, synchronization, navigation, resilience, and disease spread.
---
paper_title: On Social-Temporal Group Query with Acquaintance Constraint

paper_content:

Three essential criteria are important for activity planning, including: (1) finding a group of attendees familiar with the initiator, (2) ensuring each attendee in the group to have tight social relations with most of the members in the group, and (3) selecting an activity period available for all attendees. Therefore, this paper proposes Social-Temporal Group Query to find the activity time and attendees with the minimum total social distance to the initiator. Moreover, this query incorporates an acquaintance constraint to avoid finding a group with mutually unfamiliar attendees. Efficient processing of the social-temporal group query is very challenging. We show that the problem is NP-hard via a proof and formulate the problem with Integer Programming. We then propose two efficient algorithms, SGSelect and STGSelect, which include effective pruning techniques and employ the idea of pivot time slots to substantially reduce the running time, for finding the optimal solutions. Experimental results indicate that the proposed algorithms are much more efficient and scalable. In the comparison of solution quality, we show that STGSelect outperforms the algorithm that represents manual coordination by the initiator.
---
paper_title: On Social-Temporal Group Query with Acquaintance Constraint

paper_content:

Three essential criteria are important for activity planning, including: (1) finding a group of attendees familiar with the initiator, (2) ensuring each attendee in the group to have tight social relations with most of the members in the group, and (3) selecting an activity period available for all attendees. Therefore, this paper proposes Social-Temporal Group Query to find the activity time and attendees with the minimum total social distance to the initiator. Moreover, this query incorporates an acquaintance constraint to avoid finding a group with mutually unfamiliar attendees. Efficient processing of the social-temporal group query is very challenging. We show that the problem is NP-hard via a proof and formulate the problem with Integer Programming. We then propose two efficient algorithms, SGSelect and STGSelect, which include effective pruning techniques and employ the idea of pivot time slots to substantially reduce the running time, for finding the optimal solutions. Experimental results indicate that the proposed algorithms are much more efficient and scalable. In the comparison of solution quality, we show that STGSelect outperforms the algorithm that represents manual coordination by the initiator.
---
paper_title: Query Optimal k-Plex Based Community in Graphs

paper_content:

Community search problem, which is to find good communities given a set of query nodes in a graph, has attracted increasing research interest recently. Though various measurement models have been proposed to define and solve community search problem. Few of them could define a community concisely and have good quality of query results. They either involve additional constraints for modeling communities, such as size and diameter, or suffer from the free rider effect, i.e., include irrelevant subgraphs. In this paper, we propose a new k-plex based community model for community search. We show that our model not only is simple and clear, but also meets with basic requirements of defining a community search problem. We formulate the maximum k-plex community query (MCKPQ) problem, that is, given a set of query nodes Q, searching for optimal k-plex containing Q. We prove that MCKPQ is NP-hard, and it is hard to approximate in any constant factor. We first give exact solutions. Then, we propose an efficient branch-and-bound (B&B) method and design an effective upper bound function and a pruning strategy. Furthermore, we optimize the basic B&B by fast candidate generation. We also give a fast heuristic solution, which produces high-quality results in practice. The effectiveness of our model of community and the efficiency of our methods are verified by elaborate experiments.
---
paper_title: A Query Approach for Influence Maximization on Specific Users in Social Networks

paper_content:

Influence maximization is introduced to maximize the profit of viral marketing in social networks. The weakness of influence maximization is that it does not distinguish specific users from others, even if some items can be only useful for the specific users. For such items, it is a better strategy to focus on maximizing the influence on the specific users. In this paper, we formulate an influence maximization problem as query processing to distinguish specific users from others. We show that the query processing problem is NP-hard and its objective function is submodular. We propose an expectation model for the value of the objective function and a fast greedy-based approximation method using the expectation model. For the expectation model, we investigate a relationship of paths between users. For the greedy method, we work out an efficient incremental updating of the marginal gain to our objective function. We conduct experiments to evaluate the proposed method with real-life datasets, and compare the results with those of existing methods that are adapted to the problem. From our experimental results, the proposed method is at least an order of magnitude faster than the existing methods in most cases while achieving high accuracy.
---
paper_title: Most Influential Community Search over Large Social Networks

paper_content:

Detecting social communities in large social networks provides an effective way to analyze the social media users' behaviors and activities. It has drawn extensive attention from both academia and industry. One essential aspect of communities in social networks is outer influence which is the capability to spread internal information of communities to external users. Detecting the communities of high outer influence has particular interest in a wide range of applications, e.g., Ads trending analytics, social opinion mining and news propagation pattern discovery. However, the existing detection techniques largely ignore the outer influence of the communities. To fill the gap, this work investigates the Most Influential Community Search problem to disclose the communities with the highest outer influences. We firstly propose a new community model, maximal kr-Clique community, which has desirable properties, i.e., society, cohesiveness, connectivity, and maximum. Then, we design a novel tree-based index structure, denoted as C-Tree, to maintain the offline computed r-cliques. To efficiently search the most influential communities, we also develop four advanced index-based algorithms which improve the search performance of non-indexed solution by about 200 times. The efficiency and effectiveness of our solution have been extensively verified using six real datasets and a small case study.
---
paper_title: Most Influential Community Search over Large Social Networks

paper_content:

Detecting social communities in large social networks provides an effective way to analyze the social media users' behaviors and activities. It has drawn extensive attention from both academia and industry. One essential aspect of communities in social networks is outer influence which is the capability to spread internal information of communities to external users. Detecting the communities of high outer influence has particular interest in a wide range of applications, e.g., Ads trending analytics, social opinion mining and news propagation pattern discovery. However, the existing detection techniques largely ignore the outer influence of the communities. To fill the gap, this work investigates the Most Influential Community Search problem to disclose the communities with the highest outer influences. We firstly propose a new community model, maximal kr-Clique community, which has desirable properties, i.e., society, cohesiveness, connectivity, and maximum. Then, we design a novel tree-based index structure, denoted as C-Tree, to maintain the offline computed r-cliques. To efficiently search the most influential communities, we also develop four advanced index-based algorithms which improve the search performance of non-indexed solution by about 200 times. The efficiency and effectiveness of our solution have been extensively verified using six real datasets and a small case study.
---
paper_title: Index-based Optimal Algorithms for Computing Steiner Components with Maximum Connectivity

paper_content:

With the proliferation of graph applications, the problem of efficiently computing all $k$-edge connected components of a graph G for a user-given k has been recently investigated. In this paper, we study the problem of efficiently computing the steiner component with the maximum connectivity; that is, given a set q of query vertices in a graph G, we aim to find the maximum induced subgraph g of G such that g contains q and g has the maximum connectivity, where g is denoted as SMCC. To accommodate online query processing, we present an efficient algorithm based on a novel index such that the algorithm runs in linear time regarding the result size; thus, the algorithm is optimal since it needs at least linear time to output the result. Moreover, in this paper we also investigate variations of the above problem. We show that such a problem with the constraint that the size of the SMCC is not smaller than a given size can also be solved in linear time regarding the result size (thus, optimal). We also show that the problem of computing the connectivity (rather than the graph details) of SMCC can be solved in linear time regarding the query size (thus, optimal). To build the index, we extend the techniques in [7] to accommodate batch processing and computation sharing. To efficiently support the applications with graph updates, we also present novel increment techniques. Finally, we conduct extensive performance studies on large real and synthetic graphs, which demonstrate that our index-based algorithms significantly outperform baseline algorithms by several orders of magnitude and our indexing algorithms are efficient.
---
paper_title: Index-based Optimal Algorithms for Computing Steiner Components with Maximum Connectivity

paper_content:

With the proliferation of graph applications, the problem of efficiently computing all $k$-edge connected components of a graph G for a user-given k has been recently investigated. In this paper, we study the problem of efficiently computing the steiner component with the maximum connectivity; that is, given a set q of query vertices in a graph G, we aim to find the maximum induced subgraph g of G such that g contains q and g has the maximum connectivity, where g is denoted as SMCC. To accommodate online query processing, we present an efficient algorithm based on a novel index such that the algorithm runs in linear time regarding the result size; thus, the algorithm is optimal since it needs at least linear time to output the result. Moreover, in this paper we also investigate variations of the above problem. We show that such a problem with the constraint that the size of the SMCC is not smaller than a given size can also be solved in linear time regarding the result size (thus, optimal). We also show that the problem of computing the connectivity (rather than the graph details) of SMCC can be solved in linear time regarding the query size (thus, optimal). To build the index, we extend the techniques in [7] to accommodate batch processing and computation sharing. To efficiently support the applications with graph updates, we also present novel increment techniques. Finally, we conduct extensive performance studies on large real and synthetic graphs, which demonstrate that our index-based algorithms significantly outperform baseline algorithms by several orders of magnitude and our indexing algorithms are efficient.
---
paper_title: Index-based Optimal Algorithms for Computing Steiner Components with Maximum Connectivity

paper_content:

With the proliferation of graph applications, the problem of efficiently computing all $k$-edge connected components of a graph G for a user-given k has been recently investigated. In this paper, we study the problem of efficiently computing the steiner component with the maximum connectivity; that is, given a set q of query vertices in a graph G, we aim to find the maximum induced subgraph g of G such that g contains q and g has the maximum connectivity, where g is denoted as SMCC. To accommodate online query processing, we present an efficient algorithm based on a novel index such that the algorithm runs in linear time regarding the result size; thus, the algorithm is optimal since it needs at least linear time to output the result. Moreover, in this paper we also investigate variations of the above problem. We show that such a problem with the constraint that the size of the SMCC is not smaller than a given size can also be solved in linear time regarding the result size (thus, optimal). We also show that the problem of computing the connectivity (rather than the graph details) of SMCC can be solved in linear time regarding the query size (thus, optimal). To build the index, we extend the techniques in [7] to accommodate batch processing and computation sharing. To efficiently support the applications with graph updates, we also present novel increment techniques. Finally, we conduct extensive performance studies on large real and synthetic graphs, which demonstrate that our index-based algorithms significantly outperform baseline algorithms by several orders of magnitude and our indexing algorithms are efficient.
---
paper_title: Linear-time enumeration of maximal K-edge-connected subgraphs in large networks by random contraction

paper_content:

Capturing sets of closely related vertices from large networks is an essential task in many applications such as social network analysis, bioinformatics, and web link research. Decomposing a graph into k-core components is a standard and efficient method for this task, but obtained clusters might not be well-connected. The idea of using maximal k-edge-connected subgraphs was recently proposed to address this issue. Although we can obtain better clusters with this idea, the state-of-the-art method is not efficient enough to process large networks with millions of vertices. In this paper, we propose a new method to decompose a graph into maximal k-edge-connected components, based on random contraction of edges. Our method is simple to implement but improves performance drastically. We experimentally show that our method can successfully decompose large networks and it is thousands times faster than the previous method. Also, we theoretically explain why our method is efficient in practice. To see the importance of maximal k-edge-connected subgraphs, we also conduct experiments using real-world networks to show that many k-core components have small edge-connectivity and they can be decomposed into a lot of maximal k-edge-connected subgraphs.
---
paper_title: Efficiently computing k-edge connected components via graph decomposition

paper_content:

Efficiently computing k-edge connected components in a large graph, G = (V, E), where V is the vertex set and E is the edge set, is a long standing research problem. It is not only fundamental in graph analysis but also crucial in graph search optimization algorithms. Consider existing techniques for computing k-edge connected components are quite time consuming and are unlikely to be scalable for large scale graphs, in this paper we firstly propose a novel graph decomposition paradigm to iteratively decompose a graph G for computing its k-edge connected components such that the number of drilling-down iterations h is bounded by the "depth" of the k-edge connected components nested together to form G, where h usually is a small integer in practice. Secondly, we devise a novel, efficient threshold-based graph decomposition algorithm, with time complexity O(l × |E|), to decompose a graph G at each iteration, where l usually is a small integer with l « |V|. As a result, our algorithm for computing k-edge connected components significantly improves the time complexity of an existing state-of-the-art technique from O(|V|2|E| + |V|3 log |V|) to O(h × l × |E|). Finally, we conduct extensive performance studies on large real and synthetic graphs. The performance studies demonstrate that our techniques significantly outperform the state-of-the-art solution by several orders of magnitude.
---
paper_title: Index-based Optimal Algorithms for Computing Steiner Components with Maximum Connectivity

paper_content:

With the proliferation of graph applications, the problem of efficiently computing all $k$-edge connected components of a graph G for a user-given k has been recently investigated. In this paper, we study the problem of efficiently computing the steiner component with the maximum connectivity; that is, given a set q of query vertices in a graph G, we aim to find the maximum induced subgraph g of G such that g contains q and g has the maximum connectivity, where g is denoted as SMCC. To accommodate online query processing, we present an efficient algorithm based on a novel index such that the algorithm runs in linear time regarding the result size; thus, the algorithm is optimal since it needs at least linear time to output the result. Moreover, in this paper we also investigate variations of the above problem. We show that such a problem with the constraint that the size of the SMCC is not smaller than a given size can also be solved in linear time regarding the result size (thus, optimal). We also show that the problem of computing the connectivity (rather than the graph details) of SMCC can be solved in linear time regarding the query size (thus, optimal). To build the index, we extend the techniques in [7] to accommodate batch processing and computation sharing. To efficiently support the applications with graph updates, we also present novel increment techniques. Finally, we conduct extensive performance studies on large real and synthetic graphs, which demonstrate that our index-based algorithms significantly outperform baseline algorithms by several orders of magnitude and our indexing algorithms are efficient.
---
paper_title: On Minimal Steiner Maximum-Connected Subgraph Queries

paper_content:

Given a graph $G$ and a set $Q$ of query nodes, we examine the Steiner Maximum-Connected Subgraph (SMCS) problem. The SMCS, or $G$ 's induced subgraph that contains $Q$ with the largest connectivity, can be useful for customer prediction, product promotion, and team assembling. Despite its importance, the SMCS problem has only been recently studied. Existing solutions evaluate the maximum SMCS , whose number of nodes is the largest among all the SMCSs of $Q$ . However, the maximum SMCS, which may contain a lot of nodes, can be difficult to interpret. In this paper, we investigate the minimal SMCS , which is the minimal subgraph of $G$ with the maximum connectivity containing $Q$ . The minimal SMCS contains much fewer nodes than its maximum counterpart, and is thus easier to be understood. However, the minimal SMCS can be costly to evaluate. We thus propose efficient Expand-Refine algorithms, as well as their approximate versions with accuracy guarantees. We further develop a cache-based processing model to improve the efficiency for an important case when $Q$ consists of a single node. Extensive experiments on large real and synthetic graph datasets validate the effectiveness and efficiency of our approaches.
---
paper_title: Expanding network communities from representative examples

paper_content:

We present an approach to leverage a small subset of a coherent community within a social network into a much larger, more representative sample. Our problem becomes identifying a small conductance subgraph containing many (but not necessarily all) members of the given seed set. Starting with an initial seed set representing a sample of a community, we seek to discover as much of the full community as possible. We present a general method for network community expansion, demonstrating that our methods work well in expanding communities in real world networks starting from small given seed groups (20 to 400 members). Our approach is marked by incremental expansion from the seeds with retrospective analysis to determine the ultimate boundaries of our community. We demonstrate how to increase the robustness of the general approach through bootstrapping multiple random partitions of the input set into seed and evaluation groups. We go beyond statistical comparisons against gold standards to careful subjective evaluations of our expanded communities. This process explains the causes of most disagreement between our expanded communities and our gold-standards—arguing that our expansion methods provide more reliable communities than can be extracted from reference sources/gazetteers such as Wikipedia.
---
paper_title: Community membership identification from small seed sets

paper_content:

In many applications we have a social network of people and would like to identify the members of an interesting but unlabeled group or community. We start with a small number of exemplar group members -- they may be followers of a political ideology or fans of a music genre -- and need to use those examples to discover the additional members. This problem gives rise to the seed expansion problem in community detection: given example community members, how can the social graph be used to predict the identities of remaining, hidden community members? In contrast with global community detection (graph partitioning or covering), seed expansion is best suited for identifying communities locally concentrated around nodes of interest. A growing body of work has used seed expansion as a scalable means of detecting overlapping communities. Yet despite growing interest in seed expansion, there are divergent approaches in the literature and there still isn't a systematic understanding of which approaches work best in different domains. Here we evaluate several variants and uncover subtle trade-offs between different approaches. We explore which properties of the seed set can improve performance, focusing on heuristics that one can control in practice. As a consequence of this systematic understanding we have found several opportunities for performance gains. We also consider an adaptive version in which requests are made for additional membership labels of particular nodes, such as one finds in field studies of social communities. This leads to interesting connections and contrasts with active learning and the trade-offs of exploration and exploitation. Finally, we explore topological properties of communities and seed sets that correlate with algorithm performance, and explain these empirical observations with theoretical ones. We evaluate our methods across multiple domains, using publicly available datasets with labeled, ground-truth communities.
---
paper_title: Robust Local Community Detection: On Free Rider Effect and Its Elimination

paper_content:

Given a large network, local community detection aims at finding the community that contains a set of query nodes and also maximizes (minimizes) a goodness metric. This problem has recently drawn intense research interest. Various goodness metrics have been proposed. However, most existing metrics tend to include irrelevant subgraphs in the detected local community. We refer to such irrelevant subgraphs as free riders. We systematically study the existing goodness metrics and provide theoretical explanations on why they may cause the free rider effect. We further develop a query biased node weighting scheme to reduce the free rider effect. In particular, each node is weighted by its proximity to the query node. We define a query biased density metric to integrate the edge and node weights. The query biased densest subgraph, which has the largest query biased density, will shift to the neighborhood of the query nodes after node weighting. We then formulate the query biased densest connected subgraph (QDC) problem, study its complexity, and provide efficient algorithms to solve it. We perform extensive experiments on a variety of real and synthetic networks to evaluate the effectiveness and efficiency of the proposed methods.
---
paper_title: Finding local community structure in networks.

paper_content:

Although the inference of global community structure in networks has recently become a topic of great interest in the physics community, all such algorithms require that the graph be completely known. Here, we define both a measure of local community structure and an algorithm that infers the hierarchy of communities that enclose a given vertex by exploring the graph one vertex at a time. This algorithm runs in time O(k2d) for general graphs when d is the mean degree and k is the number of vertices to be explored. For graphs where exploring a new vertex is time consuming, the running time is linear, O(k). We show that on computer-generated graphs the average behavior of this technique approximates that of algorithms that require global knowledge. As an application, we use this algorithm to extract meaningful local clustering information in the large recommender network of an online retailer.
---
paper_title: Exploring Local Community Structures in Large Networks

paper_content:

In this paper, we extend the concept of degree from single vertex to sub-graph, and present a formal definition of module/community in a network based on this extension. A new locally optimized algorithm is designed to find the module for a given source vertex in a network. Our analysis shows that the complexity of this algorithm is O(K2d), where K is the number of vertices to be explored in the sub-graph and d is the average degree of the vertices in the sub-graph. Based on this algorithm, we implement a JAVA tool, MoNet, for exploring local community structures in large networks. Using this tool to analyze a co-purchase network from Amazon shows that there are local community structures in this network. Further analyses on these local community structures demonstrate that media items are much easier to form compact local modules than book items do, indicating that recommending digital media items to customers based on co-purchasing information in the online store will be more efficient than recommending books.
---
paper_title: Finding local community structure in networks.

paper_content:

Although the inference of global community structure in networks has recently become a topic of great interest in the physics community, all such algorithms require that the graph be completely known. Here, we define both a measure of local community structure and an algorithm that infers the hierarchy of communities that enclose a given vertex by exploring the graph one vertex at a time. This algorithm runs in time O(k2d) for general graphs when d is the mean degree and k is the number of vertices to be explored. For graphs where exploring a new vertex is time consuming, the running time is linear, O(k). We show that on computer-generated graphs the average behavior of this technique approximates that of algorithms that require global knowledge. As an application, we use this algorithm to extract meaningful local clustering information in the large recommender network of an online retailer.
---
paper_title: Exploring Local Community Structures in Large Networks

paper_content:

In this paper, we extend the concept of degree from single vertex to sub-graph, and present a formal definition of module/community in a network based on this extension. A new locally optimized algorithm is designed to find the module for a given source vertex in a network. Our analysis shows that the complexity of this algorithm is O(K2d), where K is the number of vertices to be explored in the sub-graph and d is the average degree of the vertices in the sub-graph. Based on this algorithm, we implement a JAVA tool, MoNet, for exploring local community structures in large networks. Using this tool to analyze a co-purchase network from Amazon shows that there are local community structures in this network. Further analyses on these local community structures demonstrate that media items are much easier to form compact local modules than book items do, indicating that recommending digital media items to customers based on co-purchasing information in the online store will be more efficient than recommending books.
---
paper_title: Robust Local Community Detection: On Free Rider Effect and Its Elimination

paper_content:

Given a large network, local community detection aims at finding the community that contains a set of query nodes and also maximizes (minimizes) a goodness metric. This problem has recently drawn intense research interest. Various goodness metrics have been proposed. However, most existing metrics tend to include irrelevant subgraphs in the detected local community. We refer to such irrelevant subgraphs as free riders. We systematically study the existing goodness metrics and provide theoretical explanations on why they may cause the free rider effect. We further develop a query biased node weighting scheme to reduce the free rider effect. In particular, each node is weighted by its proximity to the query node. We define a query biased density metric to integrate the edge and node weights. The query biased densest subgraph, which has the largest query biased density, will shift to the neighborhood of the query nodes after node weighting. We then formulate the query biased densest connected subgraph (QDC) problem, study its complexity, and provide efficient algorithms to solve it. We perform extensive experiments on a variety of real and synthetic networks to evaluate the effectiveness and efficiency of the proposed methods.
---
paper_title: Community membership identification from small seed sets

paper_content:

In many applications we have a social network of people and would like to identify the members of an interesting but unlabeled group or community. We start with a small number of exemplar group members -- they may be followers of a political ideology or fans of a music genre -- and need to use those examples to discover the additional members. This problem gives rise to the seed expansion problem in community detection: given example community members, how can the social graph be used to predict the identities of remaining, hidden community members? In contrast with global community detection (graph partitioning or covering), seed expansion is best suited for identifying communities locally concentrated around nodes of interest. A growing body of work has used seed expansion as a scalable means of detecting overlapping communities. Yet despite growing interest in seed expansion, there are divergent approaches in the literature and there still isn't a systematic understanding of which approaches work best in different domains. Here we evaluate several variants and uncover subtle trade-offs between different approaches. We explore which properties of the seed set can improve performance, focusing on heuristics that one can control in practice. As a consequence of this systematic understanding we have found several opportunities for performance gains. We also consider an adaptive version in which requests are made for additional membership labels of particular nodes, such as one finds in field studies of social communities. This leads to interesting connections and contrasts with active learning and the trade-offs of exploration and exploitation. Finally, we explore topological properties of communities and seed sets that correlate with algorithm performance, and explain these empirical observations with theoretical ones. We evaluate our methods across multiple domains, using publicly available datasets with labeled, ground-truth communities.
---
paper_title: Community membership identification from small seed sets

paper_content:

In many applications we have a social network of people and would like to identify the members of an interesting but unlabeled group or community. We start with a small number of exemplar group members -- they may be followers of a political ideology or fans of a music genre -- and need to use those examples to discover the additional members. This problem gives rise to the seed expansion problem in community detection: given example community members, how can the social graph be used to predict the identities of remaining, hidden community members? In contrast with global community detection (graph partitioning or covering), seed expansion is best suited for identifying communities locally concentrated around nodes of interest. A growing body of work has used seed expansion as a scalable means of detecting overlapping communities. Yet despite growing interest in seed expansion, there are divergent approaches in the literature and there still isn't a systematic understanding of which approaches work best in different domains. Here we evaluate several variants and uncover subtle trade-offs between different approaches. We explore which properties of the seed set can improve performance, focusing on heuristics that one can control in practice. As a consequence of this systematic understanding we have found several opportunities for performance gains. We also consider an adaptive version in which requests are made for additional membership labels of particular nodes, such as one finds in field studies of social communities. This leads to interesting connections and contrasts with active learning and the trade-offs of exploration and exploitation. Finally, we explore topological properties of communities and seed sets that correlate with algorithm performance, and explain these empirical observations with theoretical ones. We evaluate our methods across multiple domains, using publicly available datasets with labeled, ground-truth communities.
---
paper_title: Expanding network communities from representative examples

paper_content:

We present an approach to leverage a small subset of a coherent community within a social network into a much larger, more representative sample. Our problem becomes identifying a small conductance subgraph containing many (but not necessarily all) members of the given seed set. Starting with an initial seed set representing a sample of a community, we seek to discover as much of the full community as possible. We present a general method for network community expansion, demonstrating that our methods work well in expanding communities in real world networks starting from small given seed groups (20 to 400 members). Our approach is marked by incremental expansion from the seeds with retrospective analysis to determine the ultimate boundaries of our community. We demonstrate how to increase the robustness of the general approach through bootstrapping multiple random partitions of the input set into seed and evaluation groups. We go beyond statistical comparisons against gold standards to careful subjective evaluations of our expanded communities. This process explains the causes of most disagreement between our expanded communities and our gold-standards—arguing that our expansion methods provide more reliable communities than can be extracted from reference sources/gazetteers such as Wikipedia.
---
paper_title: AutoG: a visual query autocompletion framework for graph databases

paper_content:

Composing queries is evidently a tedious task. This is particularly true of graph queries as they are typically complex and prone to errors, compounded by the fact that graph schemas can be missing or too loose to be helpful for query formulation. Despite the great success of query formulation aids, in particular, automatic query completion, graph query autocompletion has received much less research attention. In this paper, we propose a novel framework for subgraph query autocompletion (called AutoG). Given an initial query q and a user’s preference as input, AutoG returns ranked query suggestions $$Q'$$Q′ as output. Users may choose a query from $$Q'$$Q′ and iteratively apply AutoG to compose their queries. The novelties of AutoG are as follows: First, we formalize query composition. Second, we propose to increment a query with the logical units called c-prime features that are (i) frequent subgraphs and (ii) constructed from smaller c-prime features in no more than c ways. Third, we propose algorithms to rank candidate suggestions. Fourth, we propose a novel index called feature Dag (FDag) to optimize the ranking. We study the query suggestion quality with simulations and real users and conduct an extensive performance evaluation. The results show that the query suggestions are useful (saved roughly 40% of users’ mouse clicks), and AutoG returns suggestions shortly under a large variety of parameter settings.
---
paper_title: Large scale graph processing systems: survey and an experimental evaluation

paper_content:

Graph is a fundamental data structure that captures relationships between different data entities. In practice, graphs are widely used for modeling complicated data in different application domains such as social networks, protein networks, transportation networks, bibliographical networks, knowledge bases and many more. Currently, graphs with millions and billions of nodes and edges have become very common. In principle, graph analytics is an important big data discovery technique. Therefore, with the increasing abundance of large graphs, designing scalable systems for processing and analyzing large scale graphs has become one of the most timely problems facing the big data research community. In general, scalable processing of big graphs is a challenging task due to their size and the inherent irregular structure of graph computations. Thus, in recent years, we have witnessed an unprecedented interest in building big graph processing systems that attempted to tackle these challenges. In this article, we provide a comprehensive survey over the state-of-the-art of large scale graph processing platforms. In addition, we present an extensive experimental study of five popular systems in this domain, namely, GraphChi, Apache Giraph, GPS, GraphLab and GraphX. In particular, we report and analyze the performance characteristics of these systems using five common graph processing algorithms and seven large graph datasets. Finally, we identify a set of the current open research challenges and discuss some promising directions for future research in the domain of large scale graph processing.
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paper_title: ExpFinder: Finding experts by graph pattern matching

paper_content:

We present ExpFinder, a system for finding experts in social networks based on graph pattern matching. We demonstrate (1) how ExpFinder identifies top-K experts in a social network by supporting bounded simulation of graph patterns, and by ranking the matches based on a metric for social impact; (2) how it copes with the sheer size of real-life social graphs by supporting incremental query evaluation and query preserving graph compression, and (3) how the GUI of ExpFinder interacts with users to help them construct queries and inspect matches.
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paper_title: VizCS: Online Searching and Visualizing Communities in Dynamic Graphs

paper_content:

Given a query vertex in a graph, the task of community search is to find all meaningful communities containing the query vertex in an online manner. In this demonstration, we propose a novel query processing system for searching and visualizing communities in graphs, called VizCS. It exhibits three key innovative features. First, VizCS adopts several community models and supports community search on dynamic graphs where nodes/edges undergo frequently insertions/deletions. Second, VizCS offers a user-friendly visual interface to formulate queries and a real-time response query processing engine. Last but not least, VizCS generates a community exploration wall by offering interactive community visualization, which facilitates users to in-depth understanding of the data. Furthermore, VizCS becomes a community search platform that can visualize and compare different community results by various state-of-the-art algorithms and user-uploaded approaches.
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paper_title: VIIQ: Auto-Suggestion Enabled Visual Interface for Interactive Graph Query Formulation

paper_content:

We present VIIQ (pronounced as wick), an interactive and iterative visual query formulation interface that helps users construct query graphs specifying their exact query intent. Heterogeneous graphs are increasingly used to represent complex relationships in schemaless data, which are usually queried using query graphs. Existing graph query systems offer little help to users in easily choosing the exact labels of the edges and vertices in the query graph. VIIQ helps users easily specify their exact query intent by providing a visual interface that lets them graphically add various query graph components, backed by an edge suggestion mechanism that suggests edges relevant to the user's query intent. In this demo we present: 1) a detailed description of the various features and user-friendly graphical interface of VIIQ, 2) a brief description of the edge suggestion algorithm, and 3) a demonstration scenario that we intend to show the audience.
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paper_title: GraphX: graph processing in a distributed dataflow framework

paper_content:

In pursuit of graph processing performance, the systems community has largely abandoned general-purpose distributed dataflow frameworks in favor of specialized graph processing systems that provide tailored programming abstractions and accelerate the execution of iterative graph algorithms. In this paper we argue that many of the advantages of specialized graph processing systems can be recovered in a modern general-purpose distributed dataflow system. We introduce GraphX, an embedded graph processing framework built on top of Apache Spark, a widely used distributed dataflow system. GraphX presents a familiar composable graph abstraction that is sufficient to express existing graph APIs, yet can be implemented using only a few basic dataflow operators (e.g., join, map, group-by). To achieve performance parity with specialized graph systems, GraphX recasts graph-specific optimizations as distributed join optimizations and materialized view maintenance. By leveraging advances in distributed dataflow frameworks, GraphX brings low-cost fault tolerance to graph processing. We evaluate GraphX on real workloads and demonstrate that GraphX achieves an order of magnitude performance gain over the base dataflow framework and matches the performance of specialized graph processing systems while enabling a wider range of computation.
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paper_title: Pregel: a system for large-scale graph processing

paper_content:

Many practical computing problems concern large graphs. Standard examples include the Web graph and various social networks. The scale of these graphs - in some cases billions of vertices, trillions of edges - poses challenges to their efficient processing. In this paper we present a computational model suitable for this task. Programs are expressed as a sequence of iterations, in each of which a vertex can receive messages sent in the previous iteration, send messages to other vertices, and modify its own state and that of its outgoing edges or mutate graph topology. This vertex-centric approach is flexible enough to express a broad set of algorithms. The model has been designed for efficient, scalable and fault-tolerant implementation on clusters of thousands of commodity computers, and its implied synchronicity makes reasoning about programs easier. Distribution-related details are hidden behind an abstract API. The result is a framework for processing large graphs that is expressive and easy to program.
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paper_title: C-Explorer: Browsing Communities in Large Graphs

paper_content:

Community retrieval (CR) algorithms, which enable the extraction of subgraphs from large social networks (e.g., Facebook and Twitter), have attracted tremendous interest. Various CR solutions, such as k-core and codicil, have been proposed to obtain graphs whose vertices are closely related. In this paper, we propose the C-Explorer system to assist users in extracting, visualizing, and analyzing communities. C-Explorer provides online and interactive CR facilities, allowing a user to view her interesting graphs, indicate her required vertex q, and display the communities to which q belongs. A seminal feature of C-Explorer is that it uses an attributed graph, whose vertices are associated with labels and keywords, and looks for an attributed community (or AC), whose vertices are structurally and semantically related. Moreover, C-Explorer implements several state-of-the-art CR algorithms, as well as functions for analyzing their effectiveness. We plan to make C-Explorer an open-source web-based platform, and design API functions for software developers to test their CR algorithms in our system.
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paper_title: Efficient community detection in large networks using content and links

paper_content:

In this paper we discuss a very simple approach of combining content and link information in graph structures for the purpose of community discovery, a fundamental task in network analysis. Our approach hinges on the basic intuition that many networks contain noise in the link structure and that content information can help strengthen the community signal. This enables ones to eliminate the impact of noise (false positives and false negatives), which is particularly prevalent in online social networks and Web-scale information networks. Specifically we introduce a measure of signal strength between two nodes in the network by fusing their link strength with content similarity. Link strength is estimated based on whether the link is likely (with high probability) to reside within a community. Content similarity is estimated through cosine similarity or Jaccard coefficient. We discuss a simple mechanism for fusing content and link similarity. We then present a biased edge sampling procedure which retains edges that are locally relevant for each graph node. The resulting backbone graph can be clustered using standard community discovery algorithms such as Metis and Markov clustering. Through extensive experiments on multiple real-world datasets (Flickr, Wikipedia and CiteSeer) with varying sizes and characteristics, we demonstrate the effectiveness and efficiency of our methods over state-of-the-art learning and mining approaches several of which also attempt to combine link and content analysis for the purposes of community discovery. Specifically we always find a qualitative benefit when combining content with link analysis. Additionally our biased graph sampling approach realizes a quantitative benefit in that it is typically several orders of magnitude faster than competing approaches.
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paper_title: VizCS: Online Searching and Visualizing Communities in Dynamic Graphs

paper_content:

Given a query vertex in a graph, the task of community search is to find all meaningful communities containing the query vertex in an online manner. In this demonstration, we propose a novel query processing system for searching and visualizing communities in graphs, called VizCS. It exhibits three key innovative features. First, VizCS adopts several community models and supports community search on dynamic graphs where nodes/edges undergo frequently insertions/deletions. Second, VizCS offers a user-friendly visual interface to formulate queries and a real-time response query processing engine. Last but not least, VizCS generates a community exploration wall by offering interactive community visualization, which facilitates users to in-depth understanding of the data. Furthermore, VizCS becomes a community search platform that can visualize and compare different community results by various state-of-the-art algorithms and user-uploaded approaches.
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paper_title: Finding influential communities in massive networks

paper_content:

Community search is a problem of finding densely connected subgraphs that satisfy the query conditions in a network, which has attracted much attention in recent years. However, all the previous studies on community search do not consider the influence of a community. In this paper, we introduce a novel community model called k-influential community based on the concept of k-core to capture the influence of a community. Based on this community model, we propose a linear time online search algorithm to find the top-r k-influential communities in a network. To further speed up the influential community search algorithm, we devise a linear space data structure which supports efficient search of the top-r k-influential communities in optimal time. We also propose an efficient algorithm to maintain the data structure when the network is frequently updated. Additionally, we propose a novel I/O-efficient algorithm to find the top-r k-influential communities in a disk-resident graph under the assumption of \({{\mathcal {U}}}=O(n)\), where \({{\mathcal {U}}}\) and n denote the size of the main memory and the number of nodes, respectively. Finally, we conduct extensive experiments on six real-world massive networks, and the results demonstrate the efficiency and effectiveness of the proposed methods.
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paper_title: Skyline Community Search in Multi-valued Networks

paper_content:

Given a scientific collaboration network, how can we find a group of collaborators with high research indicator (e.g., h-index) and diverse research interests? Given a social network, how can we identify the communities that have high influence (e.g., PageRank) and also have similar interests to a specified user? In such settings, the network can be modeled as a multi-valued network where each node has d ($d \ge 1$) numerical attributes (i.e., h-index, diversity, PageRank, similarity score, etc.). In the multi-valued network, we want to find communities that are not dominated by the other communities in terms of d numerical attributes. Most existing community search algorithms either completely ignore the numerical attributes or only consider one numerical attribute of the nodes. To capture d numerical attributes, we propose a novel community model, called skyline community, based on the concepts of k-core and skyline. A skyline community is a maximal connected k-core that cannot be dominated by the other connected k-cores in the d-dimensional attribute space. We develop an elegant space-partition algorithm to efficiently compute the skyline communities. Two striking advantages of our algorithm are that (1) its time complexity relies mainly on the size of the answer s (i.e., the number of skyline communities), thus it is very efficient if s is small; and (2) it can progressively output the skyline communities, which is very useful for applications that only require part of the skyline communities. Extensive experiments on both synthetic and real-world networks demonstrate the efficiency, scalability, and effectiveness of the proposed algorithm.
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paper_title: Finding dense and connected subgraphs in dual networks

paper_content:

Finding dense subgraphs is an important problem that has recently attracted a lot of interests. Most of the existing work focuses on a single graph (or network1). In many real-life applications, however, there exist dual networks, in which one network represents the physical world and another network represents the conceptual world. In this paper, we investigate the problem of finding the densest connected subgraph (DCS) which has the largest density in the conceptual network and is also connected in the physical network. Such pattern cannot be identified using the existing algorithms for a single network. We show that even though finding the densest subgraph in a single network is polynomial time solvable, the DCS problem is NP-hard. We develop a two-step approach to solve the DCS problem. In the first step, we effectively prune the dual networks while guarantee that the optimal solution is contained in the remaining networks. For the second step, we develop two efficient greedy methods based on different search strategies to find the DCS. Different variations of the DCS problem are also studied. We perform extensive experiments on a variety of real and synthetic dual networks to evaluate the effectiveness and efficiency of the developed methods.
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paper_title: Efficient Computation of Importance Based Communities in Web-Scale Networks Using a Single Machine

paper_content:

Finding decompositions of a graph into a family of communities is crucial to understanding its underlying structure. Algorithms for finding communities in networks often rely only on structural information and search for cohesive subsets of nodes. In practice however, we would like to find communities that are not only cohesive, but also influential or important. In order to capture such communities, Li, Qin, Yu, and Mao introduced a novel community model called "k-influential community" based on the concept of $k$-core, with numerical values representing "influence" assigned to the nodes. They formulate the problem of finding the top-r most important communities as finding r connected k-core subgraphs ordered by the lower-bound of their importance. In this paper, our goal is to scale-up the computation of top-r, k-core communities to web-scale graphs of tens of billions of edges. We feature several fast new algorithms for this problem. With our implementations, we show that we can efficiently handle massive networks using a single consumer-level machine within a reasonable amount of time.
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paper_title: Search of Center-Core Community in Large Graphs

paper_content:

Community search plays an important role in complex network analysis. It aims to find a densely connected subgraph containing the query node in a graph. However, the most existing community search methods do not consider the influence of nodes and can not perfectly support the search in large graphs, making them have limitations in practical applications. In this paper, we introduce a community model called center-core community based on k-core decomposition, which can both capture the influence of nodes and guarantee the cohesiveness of community. Then we devise a center-core community graph index (CCG-Index), and online search algorithms (SingleQuery and MultiQuery) which support efficient search of center-core community in optimal time. Extensive experiments on four real-world large networks demonstrate the efficiency and effectiveness of our methods.
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paper_title: Finding weighted k-truss communities in large networks

paper_content:

Community search is a fundamental problem in social network mining, which has attracted much attention in recent years. However, most previous community models only consider the link structure and ignore the link weights of the community, which may miss some useful properties of the community. In this paper, we propose a novel community model, called weightedk-truss community, based on the concept of k-truss. The proposed model takes the edge weight into consideration, thus can better characterize the properties of a community. Based on the new community model, we design a BFS-based online search algorithm to find the top-rweightedk-truss communities in O(m1.5) time, where m denotes the number of edges in a network. To speed up the online search algorithm, we devise a space-efficient index structure, namely KEP-Index, to support efficient community search. We propose two algorithms to construct the index structure in an offline manner. Based on KEP-Index, the time complexity for finding the top-rweightedk-truss communities is linear to the size of these communities, thus it is optimal. We conduct extensive experiments on six large real-world networks, as well as a case study over a co-authorship network. The results demonstrate the efficiency and effectiveness of the proposed community model and algorithms.
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paper_title: An Optimal and Progressive Approach to Online Search of Top-K Influential Communities

paper_content:

Community search over large graphs is a fundamental problem in graph analysis. Recent studies propose to compute top-k influential communities, where each reported community not only is a cohesive subgraph but also has a high influence value. The existing approaches to the problem of top-k influential community search can be categorized as index-based algorithms and online search algorithms without indexes. The index-based algorithms, although being very efficient in conducting community searches, need to pre-compute a special-purpose index and only work for one built-in vertex weight vector. In this paper, we investigate on-line search approaches and propose an instance-optimal algorithm LocalSearch whose time complexity is linearly proportional to the size of the smallest subgraph that a correct algorithm needs to access without indexes. In addition, we also propose techniques to make LocalSearch progressively compute and report the communities in decreasing influence value order such that k does not need to be specified. Moreover, we extend our framework to the general case of top-k influential community search regarding other cohesiveness measures. Extensive empirical studies on real graphs demonstrate that our algorithms outperform the existing online search algorithms by several orders of magnitude.
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paper_title: Influential Community Search in Large Networks

paper_content:

Community search is a problem of finding densely connected subgraphs that satisfy the query conditions in a network, which has attracted much attention in recent years. However, all the previous studies on community search do not consider the influence of a community. In this paper, we introduce a novel community model called k-influential community based on the concept of k-core, which can capture the influence of a community. Based on the new community model, we propose a linear-time online search algorithm to find the top-r k-influential communities in a network. To further speed up the influential community search algorithm, we devise a linear-space index structure which supports efficient search of the top-r k-influential communities in optimal time. We also propose an efficient algorithm to maintain the index when the network is frequently updated. We conduct extensive experiments on 7 real-world large networks, and the results demonstrate the efficiency and effectiveness of the proposed methods.
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paper_title: Most Influential Community Search over Large Social Networks

paper_content:

Detecting social communities in large social networks provides an effective way to analyze the social media users' behaviors and activities. It has drawn extensive attention from both academia and industry. One essential aspect of communities in social networks is outer influence which is the capability to spread internal information of communities to external users. Detecting the communities of high outer influence has particular interest in a wide range of applications, e.g., Ads trending analytics, social opinion mining and news propagation pattern discovery. However, the existing detection techniques largely ignore the outer influence of the communities. To fill the gap, this work investigates the Most Influential Community Search problem to disclose the communities with the highest outer influences. We firstly propose a new community model, maximal kr-Clique community, which has desirable properties, i.e., society, cohesiveness, connectivity, and maximum. Then, we design a novel tree-based index structure, denoted as C-Tree, to maintain the offline computed r-cliques. To efficiently search the most influential communities, we also develop four advanced index-based algorithms which improve the search performance of non-indexed solution by about 200 times. The efficiency and effectiveness of our solution have been extensively verified using six real datasets and a small case study.
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paper_title: Effective and efficient attributed community search

paper_content:

Given a graph G and a vertex $$q \in G$$ ::: , the community search query returns a subgraph of G that contains vertices related to q. Communities, which are prevalent in attributed graphs such as social networks and knowledge bases, can be used in emerging applications such as product advertisement and setting up of social events. In this paper, we investigate the attributed community query (or ACQ), which returns an attributed community (AC) for an attributed graph. The AC is a subgraph of G, which satisfies both structure cohesiveness (i.e., its vertices are tightly connected) and keyword cohesiveness (i.e., its vertices share common keywords). The AC enables a better understanding of how and why a community is formed (e.g., members of an AC have a common interest in music, because they all have the same keyword “music”). An AC can be “personalized”; for example, an ACQ user may specify that an AC returned should be related to some specific keywords like “research” and “sports”. To enable efficient AC search, we develop the CL-tree index structure and three algorithms based on it. We further propose efficient algorithms for maintaining the index on dynamic graphs. Moreover, we study two problems that are related to the ACQ problem. We evaluate our solutions on six large graphs. Our results show that ACQ is more effective and efficient than existing community retrieval approaches. Moreover, an AC contains more precise and personalized information than that of existing community search and detection methods.
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paper_title: Effective Community Search for Large Attributed Graphs

paper_content:

Given a graph G and a vertex q ∈ G, the community search query returns a subgraph of G that contains vertices related to q. Communities, which are prevalent in attributed graphs such as social networks and knowledge bases, can be used in emerging applications such as product advertisement and setting up of social events. In this paper, we investigate the attributed community query (or ACQ), which returns an attributed community (AC) for an attributed graph. The AC is a subgraph of G, which satisfies both structure cohesiveness (i.e., its vertices are tightly connected) and keyword cohesiveness (i.e., its vertices share common keywords). The AC enables a better understanding of how and why a community is formed (e.g., members of an AC have a common interest in music, because they all have the same keyword "music"). An AC can be "personalized"; for example, an ACQ user may specify that an AC returned should be related to some specific keywords like "research" and "sports". ::: ::: To enable efficient AC search, we develop the CL-tree index structure and three algorithms based on it. We evaluate our solutions on four large graphs, namely Flickr, DBLP, Tencent, and DBpedia. Our results show that ACs are more effective and efficient than existing community retrieval approaches. Moreover, an AC contains more precise and personalized information than that of existing community search and detection methods.
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paper_title: When Engagement Meets Similarity: Efficient (k, r)-Core Computation on Social Networks

paper_content:

In this paper, we investigate the problem of (k,r)-core which intends to find cohesive subgraphs on social networks considering both user engagement and similarity perspectives. In particular, we adopt the popular concept of k-core to guarantee the engagement of the users (vertices) in a group (subgraph) where each vertex in a (k,r)-core connects to at least k other vertices. Meanwhile, we also consider the pairwise similarity between users based on their profiles. For a given similarity metric and a similarity threshold r, the similarity between any two vertices in a (k,r)-core is ensured not less than r. Efficient algorithms are proposed to enumerate all maximal (k,r)-cores and find the maximum (k,r)-core, where both problems are shown to be NP-hard. Effective pruning techniques significantly reduce the search space of two algorithms and a novel (k,k')-core based (k,r)-core size upper bound enhances performance of the maximum (k,r)-core computation. We also devise effective search orders to accommodate the different nature of two mining algorithms. Comprehensive experiments on real-life data demonstrate that the maximal/maximum (k,r)-cores enable us to find interesting cohesive subgraphs, and performance of two mining algorithms is significantly improved by proposed techniques.
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paper_title: A SURVEY OF ALGORITHMS FOR KEYWORD SEARCH ON GRAPH DATA

paper_content:

In this chapter, we survey methods that perform keyword search on graph data. Keyword search provides a simple but user-friendly interface to retrieve information from complicated data structures. Since many real life datasets are represented by trees and graphs, keyword search has become an attractive mechanism for data of a variety of types. In this survey, we discuss methods of keyword search on schema graphs, which are abstract representation for XML data and relational data, and methods of keyword search on schema-free graphs. In our discussion, we focus on three major challenges of keyword search on graphs. First, what is the semantics of keyword search on graphs, or, what qualifies as an answer to a keyword search; second, what constitutes a good answer, or, how to rank the answers; third, how to perform keyword search efficiently. We also discuss some unresolved challenges and propose some new research directions on this topic.
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paper_title: Querying Communities in Relational Databases

paper_content:

Keyword search on relational databases provides users with insights that they can not easily observe using the traditional RDBMS techniques. Here, an l-keyword query is specified by a set of l keywords, {k1, k2, · · · , kl}. It finds how the tuples that contain the keywords are connected in a relational database via the possible foreign key references. Conceptually, it is to find some structural information in a database graph, where nodes are tuples and edges are foreign key references. The existing work studied how to find connected trees for an l-keyword query. However, a tree may only show partial information about how those tuples that contain the keywords are connected. In this paper, we focus on finding communities for an l-keyword query. A community is an induced subgraph that contains all the l-keywords within a given distance. We propose new efficient algorithms to find all/top-k communities which consume small memory, for an l-keyword query. For top kl-keyword queries, our algorithm allows users to interactively enlarge k at run time. We conducted extensive performance studies using two large real datasets to confirm the efficiency of our algorithms.
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paper_title: Keyword Search in Databases

paper_content:

It has become highly desirable to provide users with flexible ways to query/search information over databases as simple as keyword search like Google search. This book surveys the recent developments on keyword search over databases, and focuses on finding structural information among objects in a database using a set of keywords. Such structural information to be returned can be either trees or subgraphs representing how the objects, that contain the required keywords, are interconnected in a relational database or in an XML database. The structural keyword search is completely different from finding documents that contain all the user-given keywords. The former focuses on the interconnected object structures, whereas the latter focuses on the object content. The book is organized as follows. In Chapter 1, we highlight the main research issues on the structural keyword search in different contexts. In Chapter 2, we focus on supporting structural keyword search in a relational database management system using the SQL query language. We concentrate on how to generate a set of SQL queries that can find all the structural information among records in a relational database completely, and how to evaluate the generated set of SQL queries efficiently. In Chapter 3, we discuss graph algorithms for structural keyword search by treating an entire relational database as a large data graph. In Chapter 4, we discuss structural keyword search in a large tree-structured XML database. In Chapter 5, we highlight several interesting research issues regarding keyword search on databases. The book can be used as either an extended survey for people who are interested in the structural keyword search or a reference book for a postgraduate course on the related topics. Table of Contents: Introduction / Schema-Based Keyword Search on Relational Databases / Graph-Based Keyword Search / Keyword Search in XML Databases / Other Topics for Keyword Search on Databases
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paper_title: Keyword Search over Distributed Graphs with Compressed Signature

paper_content:

Graph keyword search has drawn many research interests, since graph models can generally represent both structured and unstructured databases and keyword searches can extract valuable information for users without the knowledge of the underlying schema and query language. In practice, data graphs can be extremely large, e.g., a Web-scale graph containing billions of vertices. The state-of-the-art approaches employ centralized algorithms to process graph keyword searches, and thus they are infeasible for such large graphs, due to the limited computational power and storage space of a centralized server. To address this problem, we investigate keyword search for Web-scale graphs deployed in a distributed environment. We first give a naive search algorithm to answer the query efficiently. However, the naive search algorithm uses a flooding search strategy that incurs large time and network overhead. To remedy this shortcoming, we then propose a signature-based search algorithm. Specifically, we design a vertex signature that encodes the shortest-path distance from a vertex to any given keyword in the graph. As a result, we can find query answers by exploring fewer paths, so that the time and communication costs are low. Moreover, we reorganize the graph data in the cluster after its initial random partitioning so that the signature-based techniques are more effective. Finally, our experimental results demonstrate the feasibility of our proposed approach in performing keyword searches over Web-scale graph data.
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paper_title: EASE: an effective 3-in-1 keyword search method for unstructured, semi-structured and structured data

paper_content:

Conventional keyword search engines are restricted to a given data model and cannot easily adapt to unstructured, semi-structured or structured data. In this paper, we propose an efficient and adaptive keyword search method, called EASE, for indexing and querying large collections of heterogenous data. To achieve high efficiency in processing keyword queries, we first model unstructured, semi-structured and structured data as graphs, and then summarize the graphs and construct graph indices instead of using traditional inverted indices. We propose an extended inverted index to facilitate keyword-based search, and present a novel ranking mechanism for enhancing search effectiveness. We have conducted an extensive experimental study using real datasets, and the results show that EASE achieves both high search efficiency and high accuracy, and outperforms the existing approaches significantly.
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paper_title: Keyword searching and browsing in databases using BANKS

paper_content:

With the growth of the Web, there has been a rapid increase in the number of users who need to access online databases without having a detailed knowledge of the schema or of query languages; even relatively simple query languages designed for non-experts are too complicated for them. We describe BANKS, a system which enables keyword-based search on relational databases, together with data and schema browsing. BANKS enables users to extract information in a simple manner without any knowledge of the schema or any need for writing complex queries. A user can get information by typing a few keywords, following hyperlinks, and interacting with controls on the displayed results. BANKS models tuples as nodes in a graph, connected by links induced by foreign key and other relationships. Answers to a query are modeled as rooted trees connecting tuples that match individual keywords in the query. Answers are ranked using a notion of proximity coupled with a notion of prestige of nodes based on inlinks, similar to techniques developed for Web search. We present an efficient heuristic algorithm for finding and ranking query results.
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paper_title: Efficient Keyword Search on Uncertain Graph Data

paper_content:

As a popular search mechanism, keyword search has been applied to retrieve useful data in documents, texts, graphs, and even relational databases. However, so far, there is no work on keyword search over uncertain graph data even though the uncertain graphs have been widely used in many real applications, such as modeling road networks, influential detection in social networks, and data analysis on PPI networks. Therefore, in this paper, we study the problem of top-k keyword search over uncertain graph data. Following the similar answer definition for keyword search over deterministic graphs, we consider a subtree in the uncertain graph as an answer to a keyword query if 1) it contains all the keywords; 2) it has a high score (defined by users or applications) based on keyword matching; and 3) it has low uncertainty. Keyword search over deterministic graphs is already a hard problem as stated in [1], [2], [3]. Due to the existence of uncertainty, keyword search over uncertain graphs is much harder. Therefore, to improve the search efficiency, we employ a filtering-and-verification strategy based on a probabilistic keyword index, PKIndex. For each keyword, we offline compute path-based top-k probabilities, and attach these values to PKIndex in an optimal, compressed way. In the filtering phase, we perform existence, path-based and tree-based probabilistic pruning phases, which filter out most false subtrees. In the verification, we propose a sampling algorithm to verify the candidates. Extensive experimental results demonstrate the effectiveness of the proposed algorithms.
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paper_title: Keyword Search in Graphs: Finding r-cliques

paper_content:

Keyword search over a graph finds a substructure of the graph containing all or some of the input keywords. Most of previous methods in this area find connected minimal trees that cover all the query keywords. Recently, it has been shown that finding subgraphs rather than trees can be more useful and informative for the users. However, the current tree or graph based methods may produce answers in which some content nodes (i.e., nodes that contain input keywords) are not very close to each other. In addition, when searching for answers, these methods may explore the whole graph rather than only the content nodes. This may lead to poor performance in execution time. To address the above problems, we propose the problem of finding r-cliques in graphs. An r-clique is a group of content nodes that cover all the input keywords and the distance between each two nodes is less than or equal to r. An exact algorithm is proposed that finds all r-cliques in the input graph. In addition, an approximation algorithm that produces r-cliques with 2-approximation in polynomial delay is proposed. Extensive performance studies using two large real data sets confirm the efficiency and accuracy of finding r-cliques in graphs.
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paper_title: An Algorithm for Subgraph Isomorphism

paper_content:

Subgraph isomorphism can be determined by means of a brute-force tree-search enumeration procedure. In this paper a new algorithm is introduced that attains efficiency by inferentially eliminating successor nodes in the tree search. To assess the time actually taken by the new algorithm, subgraph isomorphism, clique detection, graph isomorphism, and directed graph isomorphism experiments have been carried out with random and with various nonrandom graphs. A parallel asynchronous logic-in-memory implementation of a vital part of the algorithm is also described, although this hardware has not actually been built. The hardware implementation would allow very rapid determination of isomorphism.
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paper_title: PTE: Enumerating Trillion Triangles On Distributed Systems

paper_content:

How can we enumerate triangles from an enormous graph with billions of vertices and edges? Triangle enumeration is an important task for graph data analysis with many applications including identifying suspicious users in social networks, detecting web spams, finding communities, etc. However, recent networks are so large that most of the previous algorithms fail to process them. Recently, several MapReduce algorithms have been proposed to address such large networks; however, they suffer from the massive shuffled data resulting in a very long processing time. In this paper, we propose PTE (Pre-partitioned Triangle Enumeration), a new distributed algorithm for enumerating triangles in enormous graphs by resolving the structural inefficiency of the previous MapReduce algorithms. PTE enumerates trillions of triangles in a billion scale graph by decreasing three factors: the amount of shuffled data, total work, and network read. Experimental results show that PTE provides up to 47 times faster performance than recent distributed algorithms on real world graphs, and succeeds in enumerating more than 3 trillion triangles on the ClueWeb12 graph with 6.3 billion vertices and 72 billion edges, which any previous triangle computation algorithm fail to process.
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paper_title: I/O-Efficient Algorithms on Triangle Listing and Counting

paper_content:

This article studies I/O-efficient algorithms for the triangle listing problem and the triangle counting problem, whose solutions are basic operators in dealing with many other graph problems. In the former problem, given an undirected graph G, the objective is to find all the cliques involving 3 vertices in G. In the latter problem, the objective is to report just the number of such cliques without having to enumerate them. Both problems have been well studied in internal memory, but still remain as difficult challenges when G does not fit in memory, thus making it crucial to minimize the number of disk I/Os performed. Although previous research has attempted to tackle these challenges, the state-of-the-art solutions rely on a set of crippling assumptions to guarantee good performance. Motivated by this, we develop a new algorithm that is provably I/O and CPU efficient at the same time, without making any assumption on the input G at all. The algorithm uses ideas drastically different from all the previous approaches, and outperforms the existing competitors by a factor of over an order of magnitude in our extensive experimentation.
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paper_title: SpaceKey: Exploring Patterns in Spatial Databases

paper_content:

Spatial objects associated with keywords are prevalent in applications such as Google Maps and Twitter. Recently, the topic of spatial keyword queries has received plenty of attention. Spatial Group Keyword (SGK) search is a popular class of queries; their goal is to find a set of objects which are close to each other and are associated to a set of input keywords. In this paper, we propose SpaceKey, a system for retrieving and visualizing spatial objects returned by SGK queries. In addition to existing SGK query types, SpaceKey supports a novel query, called SPM query. An SPM query is defined by a spatial pattern, a graph whose vertices contain keywords and its edges are associated with distance constraints. The results are sets of objects that match the pattern. SpaceKey allows users to perform comparison analysis between different SGK query types. We plan to make SpaceKey an open-source web-based platform, and design API functions for software developers to plug other SGK query algorithms into our system.
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paper_title: On Spatial Pattern Matching

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In this paper, we study the spatial pattern matching (SPM) query. Given a set D of spatial objects (e.g., houses and shops), each with a textual description, we aim at finding all combinations of objects from D that match a user-defined spatial pattern P. A pattern P is a graph where vertices represent spatial objects, and edges denote distance relationships between them. The SPM query returns the instances that satisfy P. An example of P can be "a house within 10-minute walk from a school, which is at least 2km away from a hospital". The SPM query can benefit users such as house buyers, urban planners, and archaeologists. We prove that answering such queries is computationally intractable, and propose two efficient algorithms for their evaluation. Extensive experimental evaluation and cases studies on four real datasets show that our proposed solutions are highly effective and efficient.
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paper_title: Subgraph Matching: on Compression and Computation

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Subgraph matching finds a set I of all occurrences of a pattern graph in a target graph. It has a wide range of applications while suffers an expensive computation. This efficiency issue has been studied extensively. All existing approaches, however, turn a blind eye to the output crisis, that is, when the system has to materialize I as a preprocessing/intermediate/final result or an index, the cost of the export of I dominates the overall cost, which could be prohibitive even for a small pattern graph. ::: ::: This paper studies subgraph matching via two problems. 1) Is there an ideal compression of I? 2) Will the compression of I reversely boost the computation of I? For the problem 1), we propose a technique called VCBC to compress I to code (I) which serves effectively the same as I. For problem 2), we propose a subgraph matching computation framework CBF which computes code(I)instead of I to bring down the output cost. CBF further reduces the overall cost by reducing the intermediate results. Extensive experiments show that the compression ratio of VCBC can be up to 105 which also significantly lowers the output cost of CBF. Extensive experiments show the superior performance of CBF over existing approaches.
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paper_title: Graph Pattern Matching: From Intractable to Polynomial Time

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Graph pattern matching is typically defined in terms of subgraph isomorphism, which makes it an np-complete problem. Moreover, it requires bijective functions, which are often too restrictive to characterize patterns in emerging applications. We propose a class of graph patterns, in which an edge denotes the connectivity in a data graph within a predefined number of hops. In addition, we define matching based on a notion of bounded simulation, an extension of graph simulation. We show that with this revision, graph pattern matching can be performed in cubic-time, by providing such an algorithm. We also develop algorithms for incrementally finding matches when data graphs are updated, with performance guarantees for dag patterns. We experimentally verify that these algorithms scale well, and that the revised notion of graph pattern matching allows us to identify communities commonly found in real-world networks.
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paper_title: Triangle listing in massive networks and its applications

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Triangle listing is one of the fundamental algorithmic problems whose solution has numerous applications especially in the analysis of complex networks, such as the computation of clustering coefficient, transitivity, triangular connectivity, etc. Existing algorithms for triangle listing are mainly in-memory algorithms, whose performance cannot scale with the massive volume of today's fast growing networks. When the input graph cannot fit into main memory, triangle listing requires random disk accesses that can incur prohibitively large I/O cost. Some streaming and sampling algorithms have been proposed but these are approximation algorithms. We propose an I/O-efficient algorithm for triangle listing. Our algorithm is exact and avoids random disk access. Our results show that our algorithm is scalable and outperforms the state-of-the-art local triangle estimation algorithm.
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paper_title: Top-k graph pattern matching over large graphs

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There exist many graph-based applications including bioinformatics, social science, link analysis, citation analysis, and collaborative work. All need to deal with a large data graph. Given a large data graph, in this paper, we study finding top-k answers for a graph pattern query (kGPM), and in particular, we focus on top-k cyclic graph queries where a graph query is cyclic and can be complex. The capability of supporting kGPM provides much more flexibility for a user to search graphs. And the problem itself is challenging. In this paper, we propose a new framework of processing kGPM with on-the-fly ranked lists based on spanning trees of the cyclic graph query. We observe a multidimensional representation for using multiple ranked lists to answer a given kGPM query. Under this representation, we propose a cost model to estimate the least number of tree answers to be consumed in each ranked list for a given kGPM query. This leads to a query optimization approach for kGPM processing, and a top-k algorithm to process kGPM with the optimal query plan. We conducted extensive performance studies using a synthetic dataset and a real dataset, and we confirm the efficiency of our proposed approach.
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paper_title: Fast best-effort pattern matching in large attributed graphs

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We focus on large graphs where nodes have attributes, such as a social network where the nodes are labelled with each person's job title. In such a setting, we want to find subgraphs that match a user query pattern. For example, a "star" query would be, "find a CEO who has strong interactions with a Manager, a Lawyer,and an Accountant, or another structure as close to that as possible". Similarly, a "loop" query could help spot a money laundering ring. Traditional SQL-based methods, as well as more recent graph indexing methods, will return no answer when an exact match does not exist. This is the first main feature of our method. It can find exact-, as well as near-matches, and it will present them to the user in our proposed "goodness" order. For example, our method tolerates indirect paths between, say, the "CEO" and the "Accountant" of the above sample query, when direct paths don't exist. Its second feature is scalability. In general, if the query has nq nodes and the data graph has n nodes, the problem needs polynomial time complexity O(n n q), which is prohibitive. Our G-Ray ("Graph X-Ray") method finds high-quality subgraphs in time linear on the size of the data graph. Experimental results on the DLBP author-publication graph (with 356K nodes and 1.9M edges) illustrate both the effectiveness and scalability of our approach. The results agree with our intuition, and the speed is excellent. It takes 4 seconds on average fora 4-node query on the DBLP graph.
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paper_title: Arboricity and subgraph listing algorithms

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In this paper we introduce a new simple strategy into edge-searching of a graph, which is useful to the various subgraph listing problems. Applying the strategy, we obtain the following four algorithms. The first one lists all the triangles in a graph G in $O(a(G)m)$ time, where m is the number of edges of G and $a(G)$ the arboricity of G. The second finds all the quadrangles in $O(a(G)m)$ time. Since $a(G)$ is at most three for a planar graph G, both run in linear time for a planar graph. The third lists all the complete subgraphs $K_l $ of order l in $O(la(G)^{l - 2} m)$ time. The fourth lists all the cliques in $O(a(G)m)$ time per clique. All the algorithms require linear space. We also establish an upper bound on $a(G)$ for a graph $G:a(G) \leqq \lceil (2m + n)^{1/2} \rceil $, where n is the number of vertices in G.
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paper_title: Scalable Distributed Subgraph Enumeration

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Subgraph enumeration aims to find all the subgraphs of a large data graph that are isomorphic to a given pattern graph. As the subgraph isomorphism operation is computationally intensive, researchers have recently focused on solving this problem in distributed environments, such as MapReduce and Pregel. Among them, the state-of-the-art algorithm, Twin TwigJoin, is proven to be instance optimal based on a left-deep join framework. However, it is still not scalable to large graphs because of the constraints in the left-deep join framework and that each decomposed component (join unit) must be a star. In this paper, we propose SEED - a scalable sub-graph enumeration approach in the distributed environment. Compared to Twin TwigJoin, SEED returns optimal solution in a generalized join framework without the constraints in Twin TwigJoin. We use both star and clique as the join units, and design an effective distributed graph storage mechanism to support such an extension. We develop a comprehensive cost model, that estimates the number of matches of any given pattern graph by considering power-law degree distribution in the data graph. We then generalize the left-deep join framework and develop a dynamic-programming algorithm to compute an optimal bushy join plan. We also consider overlaps among the join units. Finally, we propose clique compression to further improve the algorithm by reducing the number of the intermediate results. Extensive performance studies are conducted on several real graphs, one containing billions of edges. The results demonstrate that our algorithm outperforms all other state-of-the-art algorithms by more than one order of magnitude.
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paper_title: Association Rules with Graph Patterns

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We propose graph-pattern association rules (GPARs) for social media marketing. Extending association rules for item-sets, GPARs help us discover regularities between entities in social graphs, and identify potential customers by exploring social influence. We study the problem of discovering top-k diversified GPARs. While this problem is NP-hard, we develop a parallel algorithm with accuracy bound. We also study the problem of identifying potential customers with GPARs. While it is also NP-hard, we provide a parallel scalable algorithm that guarantees a polynomial speedup over sequential algorithms with the increase of processors. Using real-life and synthetic graphs, we experimentally verify the scalability and effectiveness of the algorithms.
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paper_title: Efficient Subgraph Search over Large Uncertain Graphs

paper_content:

Retrieving graphs containing a query graph from a large graph database is a key task in many graph-based applications, including chemical compounds discovery, protein complex prediction, and structural pattern recognition. However, graph data handled by these applications is often noisy, incomplete, and inaccurate because of the way the data is produced. In this paper,we study subgraph queries over uncertain graphs. Specifically, we consider the problem of answering threshold-based probabilistic queries over a large uncertain graph database with the possible world semantics. We prove that problem is #P-complete, therefore, we adopt a filtering-and-verification strategy to speed up the search. In the filtering phase, we use a probabilistic inverted index, PIndex, based on subgraph features obtained by an optimal feature selection process. During the verification phase, we develop exact and bound algorithms to validate the remaining candidates. Extensive experimental results demonstrate the effectiveness of the proposed algorithms.
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paper_title: Scalable Subgraph Enumeration in MapReduce

paper_content:

Subgraph enumeration, which aims to find all the subgraphs of a large data graph that are isomorphic to a given pattern graph, is a fundamental graph problem with a wide range of applications. However, existing sequential algorithms for subgraph enumeration fall short in handling large graphs due to the involvement of computationally intensive subgraph isomorphism operations. Thus, some recent researches focus on solving the problem using MapReduce. Nevertheless, exiting MapReduce approaches are not scalable to handle very large graphs since they either produce a huge number of partial results or consume a large amount of memory. Motivated by this, in this paper, we propose a new algorithm TwinTwigJoin based on a left-deep-join framework in MapReduce, in which the basic join unit is a TwinTwig (an edge or two incident edges of a node). We show that in the Erdos-Renyi random-graph model, TwinTwigJoin is instance optimal in the left-deep-join framework under reasonable assumptions, and we devise an algorithm to compute the optimal join plan. Three optimization strategies are explored to improve our algorithm. Furthermore, we discuss how our approach can be adapted in the power-law random-graph model. We conduct extensive performance studies in several real graphs, one of which contains billions of edges. Our approach significantly outperforms existing solutions in all tests.
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paper_title: Efficient Subgraph Similarity Search on Large Probabilistic Graph Databases

paper_content:

Many studies have been conducted on seeking the efficient solution for subgraph similarity search over certain (deterministic) graphs due to its wide application in many fields, including bioinformatics, social network analysis, and Resource Description Framework (RDF) data management. All these works assume that the underlying data are certain. However, in reality, graphs are often noisy and uncertain due to various factors, such as errors in data extraction, inconsistencies in data integration, and privacy preserving purposes. Therefore, in this paper, we study subgraph similarity search on large probabilistic graph databases. Different from previous works assuming that edges in an uncertain graph are independent of each other, we study the uncertain graphs where edges' occurrences are correlated. We formally prove that subgraph similarity search over probabilistic graphs is #P-complete, thus, we employ a filter-and-verify framework to speed up the search. In the filtering phase, we develop tight lower and upper bounds of subgraph similarity probability based on a probabilistic matrix index, PMI. PMI is composed of discriminative subgraph features associated with tight lower and upper bounds of subgraph isomorphism probability. Based on PMI, we can sort out a large number of probabilistic graphs and maximize the pruning capability. During the verification phase, we develop an efficient sampling algorithm to validate the remaining candidates. The efficiency of our proposed solutions has been verified through extensive experiments.
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paper_title: The community-search problem and how to plan a successful cocktail party

paper_content:

A lot of research in graph mining has been devoted in the discovery of communities. Most of the work has focused in the scenario where communities need to be discovered with only reference to the input graph. However, for many interesting applications one is interested in finding the community formed by a given set of nodes. In this paper we study a query-dependent variant of the community-detection problem, which we call the community-search problem: given a graph G, and a set of query nodes in the graph, we seek to find a subgraph of G that contains the query nodes and it is densely connected. We motivate a measure of density based on minimum degree and distance constraints, and we develop an optimum greedy algorithm for this measure. We proceed by characterizing a class of monotone constraints and we generalize our algorithm to compute optimum solutions satisfying any set of monotone constraints. Finally we modify the greedy algorithm and we present two heuristic algorithms that find communities of size no greater than a specified upper bound. Our experimental evaluation on real datasets demonstrates the efficiency of the proposed algorithms and the quality of the solutions we obtain.
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paper_title: Local search of communities in large graphs

paper_content:

Community search is important in social network analysis. For a given vertex in a graph, the goal is to find the best community the vertex belongs to. Intuitively, the best community for a given vertex should be in the vicinity of the vertex. However, existing solutions use \emph{global search} to find the best community. These algorithms, although straight-forward, are very costly, as all vertices in the graph may need to be visited. In this paper, we propose a \emph{local search} strategy, which searches in the neighborhood of a vertex to find the best community for the vertex. We show that, because the minimum degree measure used to evaluate the goodness of a community is not \emph{monotonic}, designing efficient local search solutions is a very challenging task. We present theories and algorithms of local search to address this challenge. The efficiency of our local search strategy is verified by extensive experiments on both synthetic networks and a variety of real networks with millions of nodes.
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paper_title: Effective Community Search for Large Attributed Graphs

paper_content:

Given a graph G and a vertex q ∈ G, the community search query returns a subgraph of G that contains vertices related to q. Communities, which are prevalent in attributed graphs such as social networks and knowledge bases, can be used in emerging applications such as product advertisement and setting up of social events. In this paper, we investigate the attributed community query (or ACQ), which returns an attributed community (AC) for an attributed graph. The AC is a subgraph of G, which satisfies both structure cohesiveness (i.e., its vertices are tightly connected) and keyword cohesiveness (i.e., its vertices share common keywords). The AC enables a better understanding of how and why a community is formed (e.g., members of an AC have a common interest in music, because they all have the same keyword "music"). An AC can be "personalized"; for example, an ACQ user may specify that an AC returned should be related to some specific keywords like "research" and "sports". ::: ::: To enable efficient AC search, we develop the CL-tree index structure and three algorithms based on it. We evaluate our solutions on four large graphs, namely Flickr, DBLP, Tencent, and DBpedia. Our results show that ACs are more effective and efficient than existing community retrieval approaches. Moreover, an AC contains more precise and personalized information than that of existing community search and detection methods.
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paper_title: A survey of query recommendation techniques for data warehouse exploration.

paper_content:

Lots of data are gathered in datawarehouses that are navigated andexplored for analytical purposes. Only recently has the problem of recommending a datawarehouse query to a datawarehouse user attracted attention. In this paper, we propose a simple formal framework for expressing datawarehouse queryrecommendations. We propose to see the problem of recommending a datawarehouse query for exploration purposes as a function computing a set of queriesand associated ratings given a query log, a session, a user profile, a datawarehouse instance, and an expectation function. The rating computed indicates theusefulness of each query for a session. With this viewpoint, we review and categorize the few techniques that, to the best of our knowledge, have been proposedand we illustrate them with a case study.
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paper_title: Finding the Hierarchy of Dense Subgraphs using Nucleus Decompositions

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Finding dense substructures in a graph is a fundamental graph mining operation, with applications in bioinformatics, social networks, and visualization to name a few. Yet most standard formulations of this problem (like clique, quasiclique, k-densest subgraph) are NP-hard. Furthermore, the goal is rarely to find the "true optimum", but to identify many (if not all) dense substructures, understand their distribution in the graph, and ideally determine relationships among them. Current dense subgraph finding algorithms usually optimize some objective, and only find a few such subgraphs without providing any structural relations. We define the nucleus decomposition of a graph, which represents the graph as a forest of nuclei. Each nucleus is a subgraph where smaller cliques are present in many larger cliques. The forest of nuclei is a hierarchy by containment, where the edge density increases as we proceed towards leaf nuclei. Sibling nuclei can have limited intersections, which enables discovering overlapping dense subgraphs. With the right parameters, the nucleus decomposition generalizes the classic notions of k-cores and k-truss decompositions. We give provably efficient algorithms for nucleus decompositions, and empirically evaluate their behavior in a variety of real graphs. The tree of nuclei consistently gives a global, hierarchical snapshot of dense substructures, and outputs dense subgraphs of higher quality than other state-of-the-art solutions. Our algorithm can process graphs with tens of millions of edges in less than an hour.
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paper_title: Fast Hierarchy Construction for Dense Subgraphs

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Discovering dense subgraphs and understanding the relations among them is a fundamental problem in graph mining. We want to not only identify dense subgraphs, but also build a hierarchy among them (e.g., larger but sparser subgraphs formed by two smaller dense subgraphs). Peeling algorithms (k-core, k-truss, and nucleus decomposition) have been effective to locate many dense subgraphs. However, constructing a hierarchical representation of density structure, even correctly computing the connected k-cores and k-trusses, have been mostly overlooked. Keeping track of connected components during peeling requires an additional traversal operation, which is as expensive as the peeling process. In this paper, we start with a thorough survey and point to nuances in problem formulations that lead to significant differences in runtimes. We then propose efficient and generic algorithms to construct the hierarchy of dense subgraphs for k-core, k-truss, or any nucleus decomposition. Our algorithms leverage the disjoint-set forest data structure to efficiently construct the hierarchy during traversal. Furthermore, we introduce a new idea to avoid traversal. We construct the subgraphs while visiting neighborhoods in the peeling process, and build the relations to previously constructed subgraphs. We also consider an existing idea to find the k-core hierarchy and adapt for our objectives efficiently. Experiments on different types of large scale real-world networks show significant speedups over naive algorithms and existing alternatives. Our algorithms also outperform the hypothetical limits of any possible traversal-based solution.
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paper_title: Indexing Public-Private Graphs

paper_content:

We consider the reachability indexing problem for private-public directed graphs. In these graphs nodes come in three flavors: public--nodes visible to all users, private--nodes visible to a specific set of users, and protected--nodes visible to any user who can see at least one of the node's parents. We are interested in computing the set of nodes visible to a specific user online. There are two obvious algorithms: precompute the result for every user, or run a reachability algorithm at query time. This paper explores the trade-off between these two strategies. ::: ::: Our approach is to identify a set of additional visible seed nodes for each user. The online reachability algorithm explores the graph starting at these nodes. We first formulate the problem as asymmetric k-center with outliers, and then give an efficient and practical algorithm. We prove new theoretical guarantees for this problem and show empirically that it performs very well in practice.
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paper_title: A Survey of Heterogeneous Information Network Analysis

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Most real systems consist of a large number of interacting, multi-typed components, while most contemporary researches model them as homogeneous networks, without distinguishing different types of objects and links in the networks. Recently, more and more researchers begin to consider these interconnected, multi-typed data as heterogeneous information networks, and develop structural analysis approaches by leveraging the rich semantic meaning of structural types of objects and links in the networks. Compared to widely studied homogeneous network, the heterogeneous information network contains richer structure and semantic information, which provides plenty of opportunities as well as a lot of challenges for data mining. In this paper, we provide a survey of heterogeneous information network analysis. We will introduce basic concepts of heterogeneous information network analysis, examine its developments on different data mining tasks, discuss some advanced topics, and point out some future research directions.
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paper_title: On Embedding Uncertain Graphs

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Graph data are prevalent in communication networks, social media, and biological networks. These data, which are often noisy or inexact, can be represented by uncertain graphs, whose edges are associated with probabilities to indicate the chances that they exist. Recently, researchers have studied various algorithms (e.g., clustering, classification, and k-NN) for uncertain graphs. These solutions face two problems: (1) high dimensionality: uncertain graphs are often highly complex, which can affect the mining quality; and (2) low reusability, where an existing mining algorithm has to be redesigned to deal with uncertain graphs. To tackle these problems, we propose a solution called URGE, or UnceRtain Graph Embedding. Given an uncertain graph G, URGE generates G's embedding, or a set of low-dimensional vectors, which carry the proximity information of nodes in G. This embedding enables the dimensionality of G to be reduced, without destroying node proximity information. Due to its simplicity, existing mining solutions can be used on the embedding. We investigate two low- and high-order node proximity measures in the embedding generation process, and develop novel algorithms to enable fast evaluation. To our best knowledge, there is no prior study on the use of embedding for uncertain graphs. We have further performed extensive experiments for clustering, classification, and k-NN on several uncertain graph datasets. Our results show that URGE attains better effectiveness than current uncertain data mining algorithms, as well as state-of-the-art embedding solutions. The embedding and mining performance is also highly efficient in our experiments.
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paper_title: PP-DBLP: Modeling and Generating Attributed Public-Private Networks with DBLP

paper_content:

In many online social networks (e.g., Facebook, Google+, Twitter, and Instagram), users prefer to hide her/his partial or all relationships, which makes such private relationships not visible to public users or even friends. This leads to a new graph model called public-private networks, where each user has her/his own perspective of the network including the private connections. Recently, public-private network analysis has attracted significant research interest in the literature. A great deal of important graph computing problems (e.g., shortest paths, centrality, PageRank, and reachability tree) has been studied. However, due to the limited data sources and privacy concerns, proposed approaches are not tested on real-world datasets, but on synthetic datasets by randomly selecting vertices as private ones. Therefore, real-world datasets of public-private networks are essential and urgently needed for such algorithms in the evaluation of efficiency and effectiveness. In this paper, we generate four public-private networks from real-world DBLP records, called PPDBLP. We take published articles as public information and regard ongoing collaborations as the hidden information, which is only known by the authors. Our released datasets of PPDBLP offer the prospects for verifying various kinds of efficient public-private analysis algorithms in a fair way. In addition, motivated by widely existing attributed graphs, we propose an advanced model of attributed public-private graphs where vertices have not only private edges but also private attributes. We also discuss open problems on attributed public-private graphs. Preliminary experimental results on our generated real-world datasets verify the effectiveness and efficiency of public-private models and state-of-the-art algorithms.
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paper_title: Community Mining from Signed Social Networks

paper_content:

Many complex systems in the real world can be modeled as signed social networks that contain both positive and negative relations. Algorithms for mining social networks have been developed in the past; however, most of them were designed primarily for networks containing only positive relations and, thus, are not suitable for signed networks. In this work, we propose a new algorithm, called FEC, to mine signed social networks where both positive within-group relations and negative between-group relations are dense. FEC considers both the sign and the density of relations as the clustering attributes, making it effective for not only signed networks but also conventional social networks including only positive relations. Also, FEC adopts an agent-based heuristic that makes the algorithm efficient (in linear time with respect to the size of a network) and capable of giving nearly optimal solutions. FEC depends on only one parameter whose value can easily be set and requires no prior knowledge on hidden community structures. The effectiveness and efficacy of FEC have been demonstrated through a set of rigorous experiments involving both benchmark and randomly generated signed networks.
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paper_title: Detecting hot topics from Twitter: A multiview approach

paper_content:

Twitter is widely used all over the world, and a huge number of hot topics are generated by Twitter users in real time. These topics are able to reflect almost every aspect of people's daily lives. Therefore, the detection of topics in Twitter can be used in many real applications, such as monitoring public opinion, hot product recommendation and incidence detection. However, the performance of traditional topic detection methods is still far from perfect largely owing to the tweets' features, such as their limited length and arbitrary abbreviations. To address these problems, we propose a novel framework (MVTD) for Twitter topic detection using multiview clustering, which can integrate multirelations among tweets, such as semantic relations, social tag relations and temporal relations. We also propose some methods for measuring relations among tweets. In particular, to better measure the semantic similarity of tweets, we propose a new document similarity measure based on a suffix tree (STVSM). In addition, a new keyword extraction method based on a suffix tree is proposed. Experiments on real datasets show that the performance of MVTD is much better than that of a single view, and it is useful for detecting topics from Twitter.
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paper_title: Discovering Maximal Motif Cliques in Large Heterogeneous Information Networks

paper_content:

We study the discovery of cliques (or "complete" subgraphs) in heterogeneous information networks (HINs). Existing clique-finding solutions often ignore the rich semantics of HINs. We propose motif clique, or m-clique, which redefines subgraph completeness with respect to a given motif. A motif, essentially a small subgraph pattern, is a fundamental building block of an HIN. The m-clique concept is general and allows us to analyse "complete" subgraphs in an HIN with respect to desired high-order connection patterns. We further investigate the maximal m-clique enumeration problem (MMCE), which finds all maximal m-cliques not contained in any other m-cliques. Because MMCE is NP-hard, developing an accurate and efficient solution for MMCE is not straightforward. We thus present the META algorithm, which employs advanced pruning strategies to effectively reduce the search space. We also design fast techniques to avoid generating duplicated maximal m-clique instances. Our extensive experiments on large real and synthetic HINs show that META is highly effective and efficient.
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paper_title: Efficient Algorithms for Public-Private Social Networks

paper_content:

We introduce the public-private model of graphs. In this model, we have a public graph and each node in the public graph has an associated private graph. The motivation for studying this model stems from social networks, where the nodes are the users, the public graph is visible to everyone, and the private graph at each node is visible only to the user at the node. From each node's viewpoint, the graph is just a union of its private graph and the public graph. We consider the problem of efficiently computing various properties of the graphs from each node's point of view, with minimal amount of recomputation on the public graph. To illustrate the richness of our model, we explore two powerful computational paradigms for studying large graphs, namely, sketching and sampling, and focus on some key problems in social networks and show efficient algorithms in the public-private graph model. In the sketching model, we show how to efficiently approximate the neighborhood function, which in turn can be used to approximate various notions of centrality. In the sampling model, we focus on all-pair shortest path distances, node similarities, and correlation clustering.
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paper_title: Truss Decomposition of Probabilistic Graphs: Semantics and Algorithms

paper_content:

A key operation in network analysis is the discovery of cohesive subgraphs. The notion of $k$-truss has gained considerable popularity in this regard, based on its rich structure and efficient computability. However, many complex networks such as social, biological and communication networks feature uncertainty, best modeled using probabilities. Unfortunately the problem of discovering k-trusses in probabilistic graphs has received little attention to date. In this paper, given a probabilistic graph G, number k and parameter γ --(0,1], we define a (k,γ)-truss as a maximal connected subgraph H ⊆ G, in which for each edge, the probability that it is contained in at least (k-2) triangles is at least γ. We develop an efficient dynamic programming algorithm for decomposing a probabilistic graph into such maximal (k,γ)-trusses. The above definition of a (k,γ)-truss is local in that the "witness" graphs that has the (k-2) triangles containing an edge in H may be quite different for distinct edges. Hence, we also propose: a global (k,γ)-truss, which in addition to being a local (k,γ)-truss, has to satisfy the condition that the probability that H contains a k-truss is at least γ. We show that unlike local (k,γ)-trusses, the global (k,γ)-truss decomposition on a probabilistic graph is intractable. We propose a novel sampling technique which enables approximate discovery of global (k,γ)-trusses with high probability. Our extensive experiments on real datasets demonstrate the efficacy of our proposed approach and the usefulness of local and global (k,γ)-truss.
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paper_title: Effective and Efficient Community Search Over Large Directed Graphs

paper_content:

Communities are prevalent in social networks, knowledge graphs, and biological networks. Recently, the topic of community search (CS), extracting a dense subgraph containing a query vertex $q$q from a graph, has received great attention. However, existing CS solutions are designed for undirected graphs, and overlook directions of edges which potentially lose useful information carried on directions. In many applications (e.g., Twitter), users’ relationships are often modeled as directed graphs (e.g., if a user $a$a follows another user $b$b, then there is an edge from $a$a to $b$b). In this paper, we study the problem of CS on directed graph. Given a vertex $q$q of a graph $G$G, we aim to find a densely connected subgraph containing $q$q from $G$G, in which vertices have strong interactions and high similarities, by using the minimum in/out-degrees metric. We first develop a baseline algorithm based on the concept of D-core. We further propose three index structures and corresponding query algorithms. Our experimental results on seven real graphs show that our solutions are very effective and efficient. For example, on a graph with over 1 billion of edges, we only need around 40mins to index it and 1$\sim$∼2sec to answer a query.
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paper_title: Walking in the Cloud: Parallel SimRank at Scale

paper_content:

Despite its popularity, SimRank is computationally costly, in both time and space. In particular, its recursive nature poses a great challenge in using modern distributed computing power, and also prevents querying similarities individually. Existing solutions suffer greatly from these practical issues. In this paper, we break such dependency for maximum efficiency possible. Our method consists of offline and online phases. In offline phase, a length-n indexing vector is derived by solving a linear system in parallel. At online query time, the similarities are computed instantly from the index vector. Throughout, the Monte Carlo method is used to maximally reduce time and space. Our algorithm, called CloudWalker, is highly parallelizable, with only linear time and space. Remarkably, it responses to both single-pair and single-source queries in constant time. CloudWalker is orders of magnitude more efficient and scalable than existing solutions for large-scale problems. Implemented on Spark with 10 machines and tested on the web-scale clue-web graph with 1 billion nodes and 43 billion edges, it takes 110 hours for offline indexing, 64 seconds for a single-pair query, and 188 seconds for a single-source query. To the best of our knowledge, our work is the first to report results on clue-web, which is 10x larger than the largest graph ever reported for SimRank computation.
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paper_title: Influential Community Search in Large Networks

paper_content:

Community search is a problem of finding densely connected subgraphs that satisfy the query conditions in a network, which has attracted much attention in recent years. However, all the previous studies on community search do not consider the influence of a community. In this paper, we introduce a novel community model called k-influential community based on the concept of k-core, which can capture the influence of a community. Based on the new community model, we propose a linear-time online search algorithm to find the top-r k-influential communities in a network. To further speed up the influential community search algorithm, we devise a linear-space index structure which supports efficient search of the top-r k-influential communities in optimal time. We also propose an efficient algorithm to maintain the index when the network is frequently updated. We conduct extensive experiments on 7 real-world large networks, and the results demonstrate the efficiency and effectiveness of the proposed methods.
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paper_title: GraphX: graph processing in a distributed dataflow framework

paper_content:

In pursuit of graph processing performance, the systems community has largely abandoned general-purpose distributed dataflow frameworks in favor of specialized graph processing systems that provide tailored programming abstractions and accelerate the execution of iterative graph algorithms. In this paper we argue that many of the advantages of specialized graph processing systems can be recovered in a modern general-purpose distributed dataflow system. We introduce GraphX, an embedded graph processing framework built on top of Apache Spark, a widely used distributed dataflow system. GraphX presents a familiar composable graph abstraction that is sufficient to express existing graph APIs, yet can be implemented using only a few basic dataflow operators (e.g., join, map, group-by). To achieve performance parity with specialized graph systems, GraphX recasts graph-specific optimizations as distributed join optimizations and materialized view maintenance. By leveraging advances in distributed dataflow frameworks, GraphX brings low-cost fault tolerance to graph processing. We evaluate GraphX on real workloads and demonstrate that GraphX achieves an order of magnitude performance gain over the base dataflow framework and matches the performance of specialized graph processing systems while enabling a wider range of computation.
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Title: A Survey of Community Search Over Big Graphs

Section 1: Introduction
Description 1: Introduce the importance and applications of community search over big graphs, along with key factors and challenges addressed by existing solutions.

Section 2: Community Cohesiveness Metrics
Description 2: Introduce and discuss various metrics used to measure the cohesiveness of communities in big graphs.

Section 3: K-core-based Community Search
Description 3: Review community search works that use k-core as the structure cohesiveness metric, classified by graph types such as undirected, directed, and attributed graphs.

Section 4: K-truss-based Community Search
Description 4: Review community search works that use k-truss as the structure cohesiveness metric, including both simple and attributed graphs, and discuss their algorithms and applications. 

Section 5: K-clique-based Community Search
Description 5: Survey community search solutions using k-clique or its variants, and their applications in finding dense and cohesive subgraphs.

Section 6: K-ECC-based Community Search
Description 6: Review community search using k-edge connected component (k-ECC) model, including maximum and minimal SMCS problems.

Section 7: Other Metrics-based Community Search
Description 7: Discuss community search models using other metrics such as local modularity, query biased density, personalized PageRank, and neighbor expansion.

Section 8: Community Search Systems
Description 8: Examine systems developed for supporting, visualizing, and analyzing community searches in large graphs.

Section 9: Comparison Analysis
Description 9: Compare and analyze the various community search solutions across different cohesiveness metrics and graph types.

Section 10: Related Work
Description 10: Review related studies on community detection, cohesive subgraph discovery, graph keyword search, and graph pattern matching.

Section 11: Future Work
Description 11: Outline potential future research directions and challenges for community search over large graphs.

Section 12: Conclusion
Description 12: Summarize the survey by highlighting key research achievements and providing insights into community search over big graphs.