Network Embeddings (NEs) map the nodes of a given network into $d$-dimensional Euclidean space $\mathbb{R}^d$. Ideally, this mapping is such that 'similar' nodes are mapped onto nearby points, such that the NE can be used for purposes such as link prediction (if 'similar' means being 'more likely to be connected') or classification (if 'similar' means 'being more likely to have the same label'). In recent years various methods for NE have been introduced, all following a similar strategy: defining a notion of similarity between nodes (typically some distance measure within the network), a distance measure in the embedding space, and a loss function that penalizes large distances for similar nodes and small distances for dissimilar nodes.

 A difficulty faced by existing methods is that certain networks are fundamentally hard to embed due to their structural properties: (approximate) multipartiteness, certain degree distributions, assortativity, etc. To overcome this, we introduce a conceptual innovation to the NE literature and propose to create \emph{Conditional Network Embeddings} (CNEs); embeddings that maximally add information with respect to given structural properties (e.g. node degrees, block densities, etc.). We use a simple Bayesian approach to achieve this, and propose a block stochastic gradient descent algorithm for fitting it efficiently.

 We demonstrate that CNEs are superior for link prediction and multi-label classification when compared to state-of-the-art methods, and this without adding significant mathematical or computational complexity. Finally, we illustrate the potential of CNE for network visualization. Network Embeddings (NEs) map nodes into d-dimensional Euclidean space R d such that an ordinary distance measure allows for meaningful comparisons between nodes. Embeddings directly enable the use of a variety of machine learning methods (classification, clustering, etc.) on networks, explaining their exploding popularity. NE approaches typically have three components (Hamilton et al., 2017) : (1) A measure of similarity between nodes. E.g. nodes can be deemed more similar if they are adjacent, have strongly overlapping neighborhoods, or are otherwise close to each other (link and path-based measures) BID18 BID20 , or if they have similar functional properties (structural measures) BID19 . FORMULA3 A metric in the embedding space. (3) A loss function comparing similarity between node pairs in the network with the proximity of their embeddings. A good NE is then one for which the average loss is small. The literature on NE has so far considered embeddings as tools that are used on their own. Yet, Euclidean embeddings are unable to accurately reflect certain kinds of network topologies, such that this approach is inevitably limited. We proposed the notion of Conditional Network Embeddings (CNEs), which seeks an embedding of a network that maximally adds information with respect to certain given prior knowledge about the network. This prior knowledge can encode information about the network that cannot be represented well by means of an embedding.We implemented this conceptually novel idea in a new algorithm based on a simple probabilistic model for the joint of the data and the network, which scales similarly to state-of-the-art NE approaches. The empirical evaluation of this algorithm confirms our intuition that the combination of structural prior knowledge and a Euclidean embedding is extremely powerful. This is confirmed empirically for both the tasks of link prediction and multi-label classification, where CNE outperforms a range of state-of-the-art baselines on a wide range of networks.In our future work we intend to investigate other models implementing the idea of conditional NEs, alternative and more scalable optimization strategies, as well as the use of other types of structural information as prior knowledge on the network.
We introduce a network embedding method that accounts for prior information about the network, yielding superior empirical performance. The paper proposed to use a prior distribution to constraint the network embedding, for the formulation this paper used very restricted Gaussian distributions. Proposes learning unsupervised node embeddings by considering the structural properties of networks.