Information Processing and Structure of Dynamical Networks

Complex dynamical networks are ubiquitous in many fields of science from engineering to biology, physics, and sociology. Collective behavior, and in particular synchronization,) is one of the most interesting consequences of interaction of dynamical systems over complex networks. In this thesis we study some aspects of synchronization in dynamical networks. The first section of the study discuses the problem of synchronizability in dynamical networks. Although synchronizability, i.e. the ease by which interacting dynamical systems can synchronize their activity, has been frequently used in research studies, there is no single interpretation for that. Here we give some possible interpretations of synchronizability and investigate to what extent they coincide. We show that in unweighted dynamical networks different interpretations of synchronizability do not lie in the same line, in general. However, in networks with high degrees of synchronization properties, the networks with properly assigned weights for the links or the ones with well-performed link rewirings, the different interpretations of synchronizability go hand in hand. We also show that networks with nonidentical diffusive connections whose weights are assigned using the connection-graph-stability method are better synchronizable compared to networks with identical diffusive couplings. Furthermore, we give an algorithm based on node and edge betweenness centrality measures to enhance the synchronizability of dynamical networks. The algorithm is tested on some artificially constructed dynamical networks as well as on some real-world networks from different disciplines. In the second section we study the synchronization phenomenon in networks of Hindmarsh-Rose neurons. First, the complete synchronization of Hindmarsh-Rose neurons over Newman-Watts networks is investigated. By numerically solving the differential equations of the dynamical network as well as using the master-stability-function method we determine the synchronizing coupling strength for diffusively coupled Hindmarsh-Rose neurons. We also consider clustered networks with dense intra-cluster connections and sparse inter-cluster links. In such networks, the synchronizability is more influenced by the inter-cluster links than intra-cluster connections. We also consider the case where the neurons are coupled through both electrical and chemical connections and obtain the synchronizing coupling strength using numerical calculations. We investigate the behavior of interacting locally synchronized gamma oscillations. We construct a network of minimal number of neurons producing synchronized gamma oscillations. By simulating giant networks of this minimal module we study the dependence of the spike synchrony on some parameters of the network such as the probability and strength of excitatory/inhibitory couplings, parameter mismatch, correlation of thalamic input and transmission time-delay. In the third section of the thesis we study the interdependencies within the time series obtained through electroencephalography (EEG) and give the EEG specific maps for patients suffering from schizophrenia or Alzheimer's disease. Capturing the collective coherent spatiotemporal activity of neuronal populations measured by high density EEG is addressed using measures estimating the synchronization within multivariate time series. Our EEG power analysis on schizophrenic patients, which is based on a new parametrization of the multichannel EEG, shows a relative increase of power in alpha rhythm over the anterior brain regions against its reduction over posterior regions. The correlations of these patterns with the clinical picture of schizophrenia as well as discriminating of the schizophrenia patients from normal control subjects supports the concept of hypofrontality in schizophrenia and renders the alpha rhythm as a sensitive marker of it. By applying a multivariate synchronization estimator, called S-estimator, we reveal the whole-head synchronization topography in schizophrenia. Our finding shows bilaterally increased synchronization over temporal brain regions and decreased synchronization over the postcentral/parietal brain regions. The topography is stable over the course of several months as well as over all conventional EEG frequency bands. Moreover, it correlates with the severity of the illness characterized by positive and negative syndrome scales. We also reveal the EEG features specific to early Alzheimer's disease by applying multivariate phase synchronization method. Our analyses result in a specific map characterized by a decrease in the values of phase synchronization over the fronto-temporal and an increase over temporo-parieto-occipital region predominantly of the left hemisphere. These abnormalities in the synchronization maps correlate with the clinical scores associated to the patients and are able to discriminate patients from normal control subjects with high precision.

Intelligent systems cover a wide range of technologies related to hard sciences, such as modeling and control theory, and soft sciences, such as the artificial intelligence (AI). Intelligent systems, including neural networks (NNs), fuzzy logic (FL), and wavelet techniques, utilize the concepts of biological systems and human cognitive capabilities. These three systems have been recognized as a robust and attractive alternative to the some of the classical modeling and control methods. The application of classical NNs, FL, and wavelet technology to dynamic system modeling and control has been constrained by the non-dynamic nature of their popular architectures. The major drawbacks of these architectures are the curse of dimensionality, such as the requirement of too many parameters in NNs, the use of large rule bases in FL, the large number of wavelets, and the long training times, etc. These problems can be overcome with dynamic network structures, referred to as dynamic neural networks (DNNs), dynamic fuzzy networks (DFNs), and dynamic wavelet networks (DWNs), which have unconstrained connectivity and dynamic neural, fuzzy, and wavelet processing units, called “neurons”, “feurons”, and “wavelons”, respectively. The structure of dynamic networks are based on Hopfield networks. Here, we present a comparative study of DNNs, DFNs, and DWNs for non-linear dynamical system modeling. All three dynamic networks have a lag dynamic, an activation function, and interconnection weights. The network weights are adjusted using fast training (optimization) algorithms (quasi-Newton methods). Also, it has been shown that all dynamic networks can be effectively used in non-linear system modeling, and that DWNs result in the best capacity. But all networks have non-linearity properties in non-linear systems. In this study, all dynamic networks are considered as a non-linear optimization with dynamic equality constraints for non-linear system modeling. They encapsulate and generalize the target trajectories. The adjoint theory, whose computational complexity is significantly less than the direct method, has been used in the training of the networks. The updating of weights (identification of network parameters) is based on Broyden–Fletcher–Goldfarb–Shanno method. First, phase portrait examples are given. From this, it has been shown that they have oscillatory and chaotic properties. A dynamical system with discrete events is modeled using the above network structure. There is a localization property at discrete event instants for time and frequency in this example.

In this paper a method for enhancing the synchronizability of dynamical networks is discussed. The method considers both local structural properties, i.e. node degrees, and global properties, i.e. node and edge betweenness centrality measures. Its performance is compared with that of some other heuristic methods and the evidence for its superior behavior is discussed. As an index for the synchronizability of a dynamical network, the eigenratio of the Laplacian matrix of the connection graph is considered. As network structures, scale-free, Watts-Strogatz and random networks of different sizes and topological properties are used. Numerical calculations of the eigenratio in the networks weighted through different methods support the conclusion.

A variety of methods have been proposed for modeling and mining dynamic complex networks, in which the topological structure varies with time. As the most popular and successful network model, the stochastic block model (SBM) has been extended and applied to community detection, link prediction, anomaly detection, and evolution analysis of dynamic networks. However, all current models based on the SBM for modeling dynamic networks are designed at the community level, assuming that nodes in each community have the same dynamic behavior, which usually results in poor performance on temporal community detection and loses the modeling of node abnormal behavior. To solve the above-mentioned problem, this article proposes a hierarchical Bayesian dynamic SBM (HB-DSBM) for modeling the node-level and community-level dynamic behavior in a dynamic network synchronously. Based on the SBM, we introduce a hierarchical Dirichlet generative mechanism to associate the global community evolution with the microscopic transition behavior of nodes near-perfectly and generate the observed links across the dynamic networks. Meanwhile, an effective variational inference algorithm is developed and we can easy to infer the communities and dynamic behaviors of the nodes. Furthermore, with the two-level evolution behaviors, it can identify nodes or communities with abnormal behavior. Experiments on simulated and real-world networks demonstrate that HB-DSBM has achieved state-of-the-art performance on community detection and evolution. In addition, abnormal evolutionary behavior and events on dynamic networks can be effectively identified by our model.

Synchronizability of dynamical networks can be defined as the ease by which the network synchronizes its activity. We propose a method for enhancing the synchronizability of dynamical networks by efficient rewirings. The method is based on the eigenvectors corresponding to the second smallest and the largest eigenvalue of the Laplacian matrix and a modified version of the simulated annealing approach is used to perform the optimization task. Starting from a simple network, i.e. an undirected and unweighted network, and at each step, an edge is selected for disconnection and two non-adjacent nodes for creating at edge in between. The effectiveness of the algorithm is tested on artificially constructed networks such as random, Watts-Strogatz, and scale-free ones. We also investigate the coincidence of two measures of synchronizability, i.e. the eigen-ratio of the Laplacian matrix and the cost of synchronization, in the optimized networks.

Community structure is one of the most important properties in social networks, and community detection has received an enormous amount of attention in recent years. In dynamic networks, the communities may evolve over time so that pose more challenging tasks than in static ones. Community detection in dynamic networks is a problem which can naturally be formulated with two contradictory objectives and consequently be solved by multiobjective optimization algorithms. In this paper, a novel multiobjective immune algorithm is proposed to solve the community detection problem in dynamic networks. It employs the framework of nondominated neighbor immune algorithm to simultaneously optimize the modularity and normalized mutual information, which quantitatively measure the quality of the community partitions and temporal cost, respectively. The problem-specific knowledge is incorporated in genetic operators and local search to improve the effectiveness and efficiency of our method. Experimental studies based on four synthetic datasets and two real-world social networks demonstrate that our algorithm can not only find community structure and capture community evolution more accurately but also be more steadily than the state-of-the-art algorithms.

In online social media systems users are not only posting, consuming, and resharing content, but also creating new and destroying existing connections in the underlying social network. While each of these two types of dynamics has individually been studied in the past, much less is known about the connection between the two. How does user information posting and seeking behavior interact with the evolution of the underlying social network structure? Here, we study ways in which network structure reacts to users posting and sharing content. We examine the complete dynamics of the Twitter information network, where users post and reshare information while they also create and destroy connections. We find that the dynamics of network structure can be characterized by steady rates of change, interrupted by sudden bursts. Information diffusion in the form of cascades of post re-sharing often creates such sudden bursts of new connections, which significantly change users' local network structure. We also explore the effect of the information content on the dynamics of the network and find evidence that the appearance of new topics and real-world events can lead to significant changes in edge creations and deletions. Lastly, we develop a model that quantifies the dynamics of the network and the occurrence of these bursts as a function of the information spreading through the network. The model can successfully predict which information diffusion events will lead to bursts in network dynamics.

Synchronization in complex dynamical networks with nonsymmetric coupling

The master stability function is a powerful tool for determining synchrony in high-dimensional networks of coupled limit cycle oscillators. In part, this approach relies on the analysis of a low-dimensional variational equation around a periodic orbit. For smooth dynamical systems, this orbit is not generically available in closed form. However, many models in physics, engineering and biology admit to non-smooth piece-wise linear caricatures, for which it is possible to construct periodic orbits without recourse to numerical evolution of trajectories. A classic example is the McKean model of an excitable system that has been extensively studied in the mathematical neuroscience community. Understandably, the master stability function cannot be immediately applied to networks of such non-smooth elements. Here, we show how to extend the master stability function to non-smooth planar piece-wise linear systems, and in the process demonstrate that considerable insight into network dynamics can be obtained. In illustration, we highlight an inverse period-doubling route to synchrony, under variation in coupling strength, in globally linearly coupled networks for which the node dynamics is poised near a homoclinic bifurcation. Moreover, for a star graph, we establish a mechanism for achieving so-called remote synchronisation (where the hub oscillator does not synchronise with the rest of the network), even when all the oscillators are identical. We contrast this with node dynamics close to a non-smooth Andronov–Hopf bifurcation and also a saddle node bifurcation of limit cycles, for which no such bifurcation of synchrony occurs.

The emergence of synchronization in a network of coupled oscillators is a fascinating topic in various scientific disciplines. A widely adopted model of a coupled oscillator network is characterized by a population of heterogeneous phase oscillators, a graph describing the interaction among them, and diffusive and sinusoidal coupling. It is known that a strongly coupled and sufficiently homogeneous network synchronizes, but the exact threshold from incoherence to synchrony is unknown. Here, we present a unique, concise, and closed-form condition for synchronization of the fully nonlinear, nonequilibrium, and dynamic network. Our synchronization condition can be stated elegantly in terms of the network topology and parameters or equivalently in terms of an intuitive, linear, and static auxiliary system. Our results significantly improve upon the existing conditions advocated thus far, they are provably exact for various interesting network topologies and parameters; they are statistically correct for almost all networks; and they can be applied equally to synchronization phenomena arising in physics and biology as well as in engineered oscillator networks, such as electrical power networks. We illustrate the validity, the accuracy, and the practical applicability of our results in complex network scenarios and in smart grid applications.

Synchronization in complex delayed dynamical networks with impulsive effects

Connection graph stability method for synchronized coupled chaotic systems

Complex network analysis has been widely applied in various fields such as social system, information system, and biological system. As the most popular model for analyzing complex network, Stochastic Block Model can perform network reconstruction, community detection, link prediction, anomaly detection, and other tasks. However, for the dynamic complex networks which are always modeling as a series of snapshot networks, the existing works for dynamic networks analysis which are based on the stochastic block model always analyze the evolution of dynamic networks by introducing probability transition matrix, then, the scale-free characteristic (power law of the degree distribution) of the network, is ignoring. So in order to overcome this limitation, we propose a fully Bayesian generation model, which incorporates the heterogeneity of the degree of nodes to model dynamic complex networks. Then we present a new dynamic stochastic block model for community detection and evolution tracking under a unified framework. We also propose an effective variational inference algorithm to solve the proposed model. The model is tested on the simulated datasets and the real-world datasets, and the experimental results show that the performance of it is superior to the baselines of community detection in dynamic networks.

Superspreaders and superblockers based community evolution tracking in dynamic social networks

Complex systems provide an opportunity to analyze the essence of phenomena by studying their intricate connections. The networks formed by these connections, known as complex networks, embody the underlying principles governing the system’s behavior. While complex networks have been previously applied in the field of evolutionary computation, prior studies have been limited in their ability to reach conclusive conclusions. Based on our investigations, we are against the notion that there is a direct link between the complex network structure of an algorithm and its performance, and we demonstrate this experimentally. In this paper, we address these limitations by analyzing the dynamic complex network structures of five algorithms across three different problems. By incorporating mathematical distributions utilized in prior research, we not only generate novel insights but also refine and challenge previous conclusions. Specifically, we introduce the biased Poisson distribution to describe the algorithm’s exploration capability and the biased power-law distribution to represent its exploitation potential during the convergence process. Our aim is to redirect research on the interplay between complex networks and evolutionary computation towards dynamic network structures, elucidating the essence of exploitation and exploration in the black-box optimization process of evolutionary algorithms via dynamic complex networks.