A Brief Overview of the Complex Biological and Engineering Networks
---
paper_title: Exploring complex networks

paper_content:

The study of networks pervades all of science, from neurobiology to statistical physics. The most basic issues are structural: how does one characterize the wiring diagram of a food web or the Internet or the metabolic network of the bacterium Escherichia coli? Are there any unifying principles underlying their topology? From the perspective of nonlinear dynamics, we would also like to understand how an enormous network of interacting dynamical systems — be they neurons, power stations or lasers — will behave collectively, given their individual dynamics and coupling architecture. Researchers are only now beginning to unravel the structure and dynamics of complex networks.
---
paper_title: Structural asymmetry and the stability of diverse food webs

paper_content:

Untangling the influence of human activities on food-web stability and persistence is complex given the large numbers of species and overwhelming number of interactions within ecosystems. Although biodiversity has been associated with stability, the actual structures and processes that confer stability to diverse food webs remain largely unknown. Here we show that real food webs are structured such that top predators act as couplers of distinct energy channels that differ in both productivity and turnover rate. Our theoretical analysis shows that coupled fast and slow channels convey both local and non-local stability to food webs. Alarmingly, the same human actions that have been implicated in the loss of biodiversity also directly erode the very structures and processes that we show to confer stability on food webs.
---
paper_title: Flocks, herds and schools: A distributed behavioral model

paper_content:

The aggregate motion of a flock of birds, a herd of land animals, or a school of fish is a beautiful and familiar part of the natural world. But this type of complex motion is rarely seen in computer animation. This paper explores an approach based on simulation as an alternative to scripting the paths of each bird individually. The simulated flock is an elaboration of a particle systems, with the simulated birds being the particles. The aggregate motion of the simulated flock is created by a distributed behavioral model much like that at work in a natural flock; the birds choose their own course. Each simulated bird is implemented as an independent actor that navigates according to its local perception of the dynamic environment, the laws of simulated physics that rule its motion, and a set of behaviors programmed into it by the "animator." The aggregate motion of the simulated flock is the result of the dense interaction of the relatively simple behaviors of the individual simulated birds.
---
paper_title: Ecological networks and their fragility

paper_content:

Darwin used the metaphor of a 'tangled bank' to describe the complex interactions between species. Those interactions are varied: they can be antagonistic ones involving predation, herbivory and parasitism, or mutualistic ones, such as those involving the pollination of flowers by insects. Moreover, the metaphor hints that the interactions may be complex to the point of being impossible to understand. All interactions can be visualized as ecological networks, in which species are linked together, either directly or indirectly through intermediate species. Ecological networks, although complex, have well defined patterns that both illuminate the ecological mechanisms underlying them and promise a better understanding of the relationship between complexity and ecological stability.
---
paper_title: A time-varying complex dynamical network model and its controlled synchronization criteria

paper_content:

Today, complex networks have attracted increasing attention from various fields of science and engineering. It has been demonstrated that many complex networks display various synchronization phenomena. In this note, we introduce a time-varying complex dynamical network model. We then further investigate its synchronization phenomenon and prove several network synchronization theorems. Especially, we show that synchronization of such a time-varying dynamical network is completely determined by the inner-coupling matrix, and by the eigenvalues and the corresponding eigenvectors of the coupling configuration matrix of the network.
---
paper_title: Are network motifs the spandrels of cellular complexity ?

paper_content:

Cellular networks display modular organization at different levels, from small sets of genes exchanging signals in morphogenesis to large groups of proteins involved in cell death. At the smallest scale, minute groups of interacting proteins or genes, so-called 'network motifs', have been suggested to be the functional building blocks of network biology. In this context, the relative abundance of a network motif would reflect its adaptive value. However, although the overabundance of motifs is non-random, recent studies by Mazurie et al. and by Kuo et al. show that motif abundance does not reflect their true adaptive value. Just as some architectural components emerge as a byproduct of a prior decision, common motifs might be a side effect of inevitable rules of genome growth and change.
---
paper_title: Chaos synchronization of general complex dynamical networks

paper_content:

Recently, it has been demonstrated that many large-scale complex dynamical networks display a collective synchronization motion. Here, we introduce a time-varying complex dynamical network model and further investigate its synchronization phenomenon. Based on this new complex network model, two network chaos synchronization theorems are proved. We show that the chaos synchronization of a time-varying complex network is determined by means of the inner coupled link matrix, the eigenvalues and the corresponding eigenvectors of the coupled configuration matrix, rather than the conventional eigenvalues of the coupled configuration matrix for a uniform network. Especially, we do not assume that the coupled configuration matrix is symmetric and its off-diagonal elements are nonnegative, which in a way generalizes the related results existing in the literature.
---
paper_title: Characterizing the synchronizability of small-world dynamical networks

paper_content:

Many real-world complex networks display a small-world feature-a high degree of clustering and a small average distance. We show that the maximum synchronizability of a network is completely determined by its associated feedback system, which has a precise meaning in terms of synchronous communication. We introduce a new concept of synchronizability matrix to characterize the maximum synchronizability of a network. Several new concepts, such as sensitive edge and robust edge, are proposed for analyzing the robustness and fragility of synchronization of a network. Using the knowledge of synchronizability, we can purposefully increase the robustness of the network synchronization and prevent it from attacks. Some applications in small-world networks are also discussed briefly.
---
paper_title: Effective leadership and decision-making in animal groups on the move

paper_content:

Moving groups of animals, including fish, ungulates, birds and honeybee swarms seem able to take complex decisions in the absence of signalling mechanisms, and when group members cannot establish who has or has not got information. A numerical simulation shows how such groups can make accurate consensus decisions, and that the larger the group, the smaller the proportion of informed individuals needed to guide the group. A very small proportion of informed individuals is sufficient for near maximal accuracy. This has implications for our understanding of the evolution of information transfer in groups, and also suggests a new design protocol for the guidance of grouping robots. Cover photo, by Phillip Colla Natural History Photography ( ::: http://www.OceanLight.com ::: ), shows schooling jack mackerel. For animals that forage or travel in groups, making movement decisions often depends on social interactions among group members1,2. However, in many cases, few individuals have pertinent information, such as knowledge about the location of a food source3,4, or of a migration route5,6,7,8,9. Using a simple model we show how information can be transferred within groups both without signalling and when group members do not know which individuals, if any, have information. We reveal that the larger the group the smaller the proportion of informed individuals needed to guide the group, and that only a very small proportion of informed individuals is required to achieve great accuracy. We also demonstrate how groups can make consensus decisions, even though informed individuals do not know whether they are in a majority or minority, how the quality of their information compares with that of others, or even whether there are any other informed individuals. Our model provides new insights into the mechanisms of effective leadership and decision-making in biological systems.
---
paper_title: Computational Ecology: From the Complex to the Simple and Back

paper_content:

In 1958, when ecology was a young science and mathematical models for ecological systems were in their infancy, Elton [1] wrote of the “neolithic days of animal ecology, that is to say about twenty-five years ago.” Acknowledging the influence of Lotka and Volterra, he noted, “Being mathematicians, they did not attempt to contemplate a whole food-chain with all the complications of five stages. They took two: a predator and its prey.” ::: ::: Today, in the era of computational ecological modeling, deterministic systems for two variables—and even a whole food chain—appear like simple idealizations well removed from the complexity of nature. We now consider predator–prey interactions as “consumer–resource” interactions embedded within the large ecological networks that underlie biodiversity (Figure 1) [2]. Consequently, the scale of the problems we model has grown to reflect the world as we now need to observe it. For example, the interplay between ecosystem dynamics and the physical environment that influences global change occurs over a tremendous range of spatial and organizational scales (e.g., [3]). Similarly, the population dynamics of the transmission of infectious diseases often involve spatial or social networks with large numbers of individuals, but the interactions of each individual involve only a subset of the network and can span from local to global distances (e.g., [4–6]). ::: ::: ::: ::: Figure 1 ::: ::: The Network of Trophic Interactions for Little Rock Lake, Wisconsin ::: ::: ::: ::: These examples illustrate the current view of ecological systems as complex adaptive systems [7,8]. Complex adaptive systems are distinguished not only by the multiplicity of components within them, but also by interactions that can be local or distributed among these components and whose rates vary as nonlinear functions of the state of the system itself. One obvious role of computation in the science of complex systems is simply one of synthesis: to reconstruct the whole from the parts as we learn more and more about the components and their interactions. There are obvious limitations to this approach, evident in the famous image of those imperial cartographers who produced a map of the empire of the same size as the empire itself [9]. I argue here that an alternative and more useful role of computation is to address questions on the relationship between dynamics at different temporal, spatial, and organizational scales, that is, to address the importance of variability at small, local scales to the dynamics of aggregated quantities measured at large, global scales. If small-scale “details” matter, we need to ask how much complexity we need to incorporate into large-scale models if we seek to both understand and predict the dynamics of global quantities. ::: ::: Is it possible to incorporate the effect of small-scale variability without resorting to the “brute force” approach of using higher and higher resolution? I start with examples from theoretical ecology that illustrate problems and approaches related to these scaling questions; I then present more specific examples related to global change and ecosystem dynamics, and end with a series of related problems on the dynamics of large food webs, the ultimate networks of ecological interactions.
---
paper_title: NOVEL TYPE OF PHASE TRANSITION IN A SYSTEM OF SELF-DRIVEN PARTICLES

paper_content:

A simple model with a novel type of dynamics is introduced in order to investigate the emergence of self-ordered motion in systems of particles with biologically motivated interaction. In our model particles are driven with a constant absolute velocity and at each time step assume the average direction of motion of the particles in their neighborhood with some random perturbation $(\ensuremath{\eta})$ added. We present numerical evidence that this model results in a kinetic phase transition from no transport (zero average velocity, $|{\mathbf{v}}_{a}|\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$) to finite net transport through spontaneous symmetry breaking of the rotational symmetry. The transition is continuous, since $|{\mathbf{v}}_{a}|$ is found to scale as $({\ensuremath{\eta}}_{c}\ensuremath{-}\ensuremath{\eta}{)}^{\ensuremath{\beta}}$ with $\ensuremath{\beta}\ensuremath{\simeq}0.45$.
---
paper_title: Adaptive synchronization of an uncertain complex dynamical network

paper_content:

This note further investigates the locally and globally adaptive synchronization of an uncertain complex dynamical network. Several network synchronization criteria are deduced. Especially, our hypotheses and designed adaptive controllers for network synchronization are rather simple in form. It is very useful for future practical engineering design. Moreover, numerical simulations are also given to show the effectiveness of our synchronization approaches.
---
paper_title: Flocks, herds and schools: A distributed behavioral model

paper_content:

The aggregate motion of a flock of birds, a herd of land animals, or a school of fish is a beautiful and familiar part of the natural world. But this type of complex motion is rarely seen in computer animation. This paper explores an approach based on simulation as an alternative to scripting the paths of each bird individually. The simulated flock is an elaboration of a particle systems, with the simulated birds being the particles. The aggregate motion of the simulated flock is created by a distributed behavioral model much like that at work in a natural flock; the birds choose their own course. Each simulated bird is implemented as an independent actor that navigates according to its local perception of the dynamic environment, the laws of simulated physics that rule its motion, and a set of behaviors programmed into it by the "animator." The aggregate motion of the simulated flock is the result of the dense interaction of the relatively simple behaviors of the individual simulated birds.
---
paper_title: Effective leadership and decision-making in animal groups on the move

paper_content:

Moving groups of animals, including fish, ungulates, birds and honeybee swarms seem able to take complex decisions in the absence of signalling mechanisms, and when group members cannot establish who has or has not got information. A numerical simulation shows how such groups can make accurate consensus decisions, and that the larger the group, the smaller the proportion of informed individuals needed to guide the group. A very small proportion of informed individuals is sufficient for near maximal accuracy. This has implications for our understanding of the evolution of information transfer in groups, and also suggests a new design protocol for the guidance of grouping robots. Cover photo, by Phillip Colla Natural History Photography ( ::: http://www.OceanLight.com ::: ), shows schooling jack mackerel. For animals that forage or travel in groups, making movement decisions often depends on social interactions among group members1,2. However, in many cases, few individuals have pertinent information, such as knowledge about the location of a food source3,4, or of a migration route5,6,7,8,9. Using a simple model we show how information can be transferred within groups both without signalling and when group members do not know which individuals, if any, have information. We reveal that the larger the group the smaller the proportion of informed individuals needed to guide the group, and that only a very small proportion of informed individuals is required to achieve great accuracy. We also demonstrate how groups can make consensus decisions, even though informed individuals do not know whether they are in a majority or minority, how the quality of their information compares with that of others, or even whether there are any other informed individuals. Our model provides new insights into the mechanisms of effective leadership and decision-making in biological systems.
---
paper_title: Computational Ecology: From the Complex to the Simple and Back

paper_content:

In 1958, when ecology was a young science and mathematical models for ecological systems were in their infancy, Elton [1] wrote of the “neolithic days of animal ecology, that is to say about twenty-five years ago.” Acknowledging the influence of Lotka and Volterra, he noted, “Being mathematicians, they did not attempt to contemplate a whole food-chain with all the complications of five stages. They took two: a predator and its prey.” ::: ::: Today, in the era of computational ecological modeling, deterministic systems for two variables—and even a whole food chain—appear like simple idealizations well removed from the complexity of nature. We now consider predator–prey interactions as “consumer–resource” interactions embedded within the large ecological networks that underlie biodiversity (Figure 1) [2]. Consequently, the scale of the problems we model has grown to reflect the world as we now need to observe it. For example, the interplay between ecosystem dynamics and the physical environment that influences global change occurs over a tremendous range of spatial and organizational scales (e.g., [3]). Similarly, the population dynamics of the transmission of infectious diseases often involve spatial or social networks with large numbers of individuals, but the interactions of each individual involve only a subset of the network and can span from local to global distances (e.g., [4–6]). ::: ::: ::: ::: Figure 1 ::: ::: The Network of Trophic Interactions for Little Rock Lake, Wisconsin ::: ::: ::: ::: These examples illustrate the current view of ecological systems as complex adaptive systems [7,8]. Complex adaptive systems are distinguished not only by the multiplicity of components within them, but also by interactions that can be local or distributed among these components and whose rates vary as nonlinear functions of the state of the system itself. One obvious role of computation in the science of complex systems is simply one of synthesis: to reconstruct the whole from the parts as we learn more and more about the components and their interactions. There are obvious limitations to this approach, evident in the famous image of those imperial cartographers who produced a map of the empire of the same size as the empire itself [9]. I argue here that an alternative and more useful role of computation is to address questions on the relationship between dynamics at different temporal, spatial, and organizational scales, that is, to address the importance of variability at small, local scales to the dynamics of aggregated quantities measured at large, global scales. If small-scale “details” matter, we need to ask how much complexity we need to incorporate into large-scale models if we seek to both understand and predict the dynamics of global quantities. ::: ::: Is it possible to incorporate the effect of small-scale variability without resorting to the “brute force” approach of using higher and higher resolution? I start with examples from theoretical ecology that illustrate problems and approaches related to these scaling questions; I then present more specific examples related to global change and ecosystem dynamics, and end with a series of related problems on the dynamics of large food webs, the ultimate networks of ecological interactions.
---
paper_title: NOVEL TYPE OF PHASE TRANSITION IN A SYSTEM OF SELF-DRIVEN PARTICLES

paper_content:

A simple model with a novel type of dynamics is introduced in order to investigate the emergence of self-ordered motion in systems of particles with biologically motivated interaction. In our model particles are driven with a constant absolute velocity and at each time step assume the average direction of motion of the particles in their neighborhood with some random perturbation $(\ensuremath{\eta})$ added. We present numerical evidence that this model results in a kinetic phase transition from no transport (zero average velocity, $|{\mathbf{v}}_{a}|\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$) to finite net transport through spontaneous symmetry breaking of the rotational symmetry. The transition is continuous, since $|{\mathbf{v}}_{a}|$ is found to scale as $({\ensuremath{\eta}}_{c}\ensuremath{-}\ensuremath{\eta}{)}^{\ensuremath{\beta}}$ with $\ensuremath{\beta}\ensuremath{\simeq}0.45$.
---
paper_title: Flocks, herds and schools: A distributed behavioral model

paper_content:

The aggregate motion of a flock of birds, a herd of land animals, or a school of fish is a beautiful and familiar part of the natural world. But this type of complex motion is rarely seen in computer animation. This paper explores an approach based on simulation as an alternative to scripting the paths of each bird individually. The simulated flock is an elaboration of a particle systems, with the simulated birds being the particles. The aggregate motion of the simulated flock is created by a distributed behavioral model much like that at work in a natural flock; the birds choose their own course. Each simulated bird is implemented as an independent actor that navigates according to its local perception of the dynamic environment, the laws of simulated physics that rule its motion, and a set of behaviors programmed into it by the "animator." The aggregate motion of the simulated flock is the result of the dense interaction of the relatively simple behaviors of the individual simulated birds.
---
paper_title: NOVEL TYPE OF PHASE TRANSITION IN A SYSTEM OF SELF-DRIVEN PARTICLES

paper_content:

A simple model with a novel type of dynamics is introduced in order to investigate the emergence of self-ordered motion in systems of particles with biologically motivated interaction. In our model particles are driven with a constant absolute velocity and at each time step assume the average direction of motion of the particles in their neighborhood with some random perturbation $(\ensuremath{\eta})$ added. We present numerical evidence that this model results in a kinetic phase transition from no transport (zero average velocity, $|{\mathbf{v}}_{a}|\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$) to finite net transport through spontaneous symmetry breaking of the rotational symmetry. The transition is continuous, since $|{\mathbf{v}}_{a}|$ is found to scale as $({\ensuremath{\eta}}_{c}\ensuremath{-}\ensuremath{\eta}{)}^{\ensuremath{\beta}}$ with $\ensuremath{\beta}\ensuremath{\simeq}0.45$.
---
paper_title: Exploring complex networks

paper_content:

The study of networks pervades all of science, from neurobiology to statistical physics. The most basic issues are structural: how does one characterize the wiring diagram of a food web or the Internet or the metabolic network of the bacterium Escherichia coli? Are there any unifying principles underlying their topology? From the perspective of nonlinear dynamics, we would also like to understand how an enormous network of interacting dynamical systems — be they neurons, power stations or lasers — will behave collectively, given their individual dynamics and coupling architecture. Researchers are only now beginning to unravel the structure and dynamics of complex networks.
---
paper_title: Chaos synchronization of general complex dynamical networks

paper_content:

Recently, it has been demonstrated that many large-scale complex dynamical networks display a collective synchronization motion. Here, we introduce a time-varying complex dynamical network model and further investigate its synchronization phenomenon. Based on this new complex network model, two network chaos synchronization theorems are proved. We show that the chaos synchronization of a time-varying complex network is determined by means of the inner coupled link matrix, the eigenvalues and the corresponding eigenvectors of the coupled configuration matrix, rather than the conventional eigenvalues of the coupled configuration matrix for a uniform network. Especially, we do not assume that the coupled configuration matrix is symmetric and its off-diagonal elements are nonnegative, which in a way generalizes the related results existing in the literature.
---
paper_title: Exploring complex networks

paper_content:

The study of networks pervades all of science, from neurobiology to statistical physics. The most basic issues are structural: how does one characterize the wiring diagram of a food web or the Internet or the metabolic network of the bacterium Escherichia coli? Are there any unifying principles underlying their topology? From the perspective of nonlinear dynamics, we would also like to understand how an enormous network of interacting dynamical systems — be they neurons, power stations or lasers — will behave collectively, given their individual dynamics and coupling architecture. Researchers are only now beginning to unravel the structure and dynamics of complex networks.
---
paper_title: Chaos synchronization of general complex dynamical networks

paper_content:

Recently, it has been demonstrated that many large-scale complex dynamical networks display a collective synchronization motion. Here, we introduce a time-varying complex dynamical network model and further investigate its synchronization phenomenon. Based on this new complex network model, two network chaos synchronization theorems are proved. We show that the chaos synchronization of a time-varying complex network is determined by means of the inner coupled link matrix, the eigenvalues and the corresponding eigenvectors of the coupled configuration matrix, rather than the conventional eigenvalues of the coupled configuration matrix for a uniform network. Especially, we do not assume that the coupled configuration matrix is symmetric and its off-diagonal elements are nonnegative, which in a way generalizes the related results existing in the literature.
---
paper_title: A time-varying complex dynamical network model and its controlled synchronization criteria

paper_content:

Today, complex networks have attracted increasing attention from various fields of science and engineering. It has been demonstrated that many complex networks display various synchronization phenomena. In this note, we introduce a time-varying complex dynamical network model. We then further investigate its synchronization phenomenon and prove several network synchronization theorems. Especially, we show that synchronization of such a time-varying dynamical network is completely determined by the inner-coupling matrix, and by the eigenvalues and the corresponding eigenvectors of the coupling configuration matrix of the network.
---
paper_title: Chaos synchronization of general complex dynamical networks

paper_content:

Recently, it has been demonstrated that many large-scale complex dynamical networks display a collective synchronization motion. Here, we introduce a time-varying complex dynamical network model and further investigate its synchronization phenomenon. Based on this new complex network model, two network chaos synchronization theorems are proved. We show that the chaos synchronization of a time-varying complex network is determined by means of the inner coupled link matrix, the eigenvalues and the corresponding eigenvectors of the coupled configuration matrix, rather than the conventional eigenvalues of the coupled configuration matrix for a uniform network. Especially, we do not assume that the coupled configuration matrix is symmetric and its off-diagonal elements are nonnegative, which in a way generalizes the related results existing in the literature.
---
paper_title: Characterizing the synchronizability of small-world dynamical networks

paper_content:

Many real-world complex networks display a small-world feature-a high degree of clustering and a small average distance. We show that the maximum synchronizability of a network is completely determined by its associated feedback system, which has a precise meaning in terms of synchronous communication. We introduce a new concept of synchronizability matrix to characterize the maximum synchronizability of a network. Several new concepts, such as sensitive edge and robust edge, are proposed for analyzing the robustness and fragility of synchronization of a network. Using the knowledge of synchronizability, we can purposefully increase the robustness of the network synchronization and prevent it from attacks. Some applications in small-world networks are also discussed briefly.
---

Title: A Brief Overview of the Complex Biological and Engineering Networks

Section 1: INTRODUCTION
Description 1: This section introduces the ubiquity and significance of complex networks in biological and engineering sciences, laying out the challenges and recent advances in understanding these networks.

Section 2: MODELING COMPLEX BIOLOGICAL NETWORKS
Description 2: This section explains various models used to represent complex biological networks, including examples such as the boid model, Vicsek model, Couzin-Levin model, and time-varying discrete biological network.

Section 3: MODELING COMPLEX ENGINEERING NETWORKS
Description 3: This section outlines models for complex engineering networks, including the Small-world network, Scale-free network, time-varying discrete engineering network, and time-varying continuous engineering network.

Section 4: CONCLUSIONS AND REMARKS
Description 4: This section provides a summary of the main advances in the study of complex biological and engineering networks and emphasizes the need for collaborative efforts between biologists and engineers to further understand these networks.