Adaptive dynamical networks via neighborhood information: synchronization and pinning control

In this paper, we introduce a model of an adaptive dynamical network by integrating the complex network model and adaptive technique. In this model, the adaptive updating laws for each vertex in the network depend only on the state information of its neighborhood, besides itself and external controllers. This suggests that an adaptive technique be added to a complex network without breaking its intrinsic existing network topology. The core of adaptive dynamical networks is to design suitable adaptive updating laws to attain certain aims. Here, we propose two series of adaptive laws to synchronize and pin a complex network, respectively. Based on the Lyapunov function method, we can prove that under several mild conditions, with the adaptive technique, a connected network topology is sufficient to synchronize or stabilize any chaotic dynamics of the uncoupled system. This implies that these adaptive updating laws actually enhance synchronizability and stabilizability, respectively. We find out that even though these adaptive methods can succeed for all networks with connectivity, the underlying network topology can affect the convergent rate and the terminal average coupling and pinning strength. In addition, this influence can be measured by the smallest nonzero eigenvalue of the corresponding Laplacian. Moreover, we provide a detailed study of the influence of the prior parameters in this adaptive laws and present several numerical examples to verify our theoretical results and further discussion.     

一种基于元胞自动机的自适应网络病毒传播模型

自适应网络是节点动力学和网络动力学相互作用和反馈的演化网络. 基于元胞自动机建立自适应网络中易感-感染-易感(susceptible-infected-susceptible)的病毒传播模型,研究节点为了规避病毒传播所采取的多种网络重连规则对病毒传播及网络统计特征的影响. 结果表明:自适应网络中的重连规则可以有效减缓病毒传播速度,降低病毒传播规模;随机重连规则使得网络统计特征趋于随机网络;基于元胞自动机建立的传播模型清晰地表达了病毒在传播过程中的双稳态现象. 关键词： 自适应网络 传播动力学 网络动力学 元胞自动机     

Fault diagnosis of time-delay complex dynamical networks using output signals

This paper proposes a novel approach for fault diagnosis of a time-delay complex dynamical network.Unlike the other methods,assuming that the dynamics of the network can be described by a linear stochastic model,or using the state variables of nodes in the network to design an adaptive observer,it only uses the output variable of the nodes to design an observer and an adaptive law of topology matrix in the observer of a complex network,leading to simple design of the observer and easy realisation of topology monitoring for the complex networks in real engineering.The proposed scheme can monitor any changes of the topology structure of a time-delay complex network.The effectiveness of this method is successfully demonstrated by virtue of a complex networks with Lorenz model.     

Information diffusion on adaptive network

Based on the adaptive network, the feedback mechanism and interplay between the network topology and the diffusive process of information are studied. The results reveal that the adaptation of network topology can drive systems into the scale-free one with the assortative or disassortative degree correlations, and the hierarchical clustering. Meanwhile, the processes of the information diffusion are extremely speeded up by the adaptive changes of network topology.     

Network evolution driven by dynamics applied to graph coloring

An evolutionary network driven by dynamics is studied and applied to the graph coloring problem. From an initial structure, both the topology and the coupling weights evolve according to the dynamics. On the other hand, the dynamics of the network are determined by the topology and the coupling weights, so an interesting structure-dynamics co-evolutionary scheme appears. By providing two evolutionary strategies, a network described by the complement of a graph will evolve into several clusters of nodes according to their dynamics. The nodes in each cluster can be assigned the same color and nodes in different clusters assigned different colors. In this way, a co-evolution phenomenon is applied to the graph coloring problem. The proposed scheme is tested on several benchmark graphs for graph coloring.     

Voter model on adaptive networks

Voter model is an important basic model in statistical physics. In recent years, it has been more and more used to describe the process of opinion formation in sociophysics. In real complex systems, the interactive network of individuals is dynamically adjusted, and the evolving network topology and individual behaviors affect each other. Therefore, we propose a linking dynamics to describe the coevolution of network topology and individual behaviors in this paper, and study the voter model on the adaptive network. We theoretically analyze the properties of the voter model, including consensus probability and time. The evolution of opinions on dynamic networks is further analyzed from the perspective of evolutionary game. Finally, a case study of real data is shown to verify the effectiveness of the theory.     

Topology identification for a class of complex dynamical networks using output variables

A problem of topology identification for complex dynamical networks is investigated in this paper. An adaptive observer is proposed to identify the topology of a complex dynamical networks based on the Lyapunov stability theory. Here the output of the network and the states of the observer are used to construct the updating law of the topology such that the communication resources from the network to its observer are saved. Some convergent criteria of the adaptive observer are derived in the form of linear inequality matrices. Several numerical examples are shown to demonstrate the effectiveness of the proposed observer.     

Cyclic dominance in adaptive networks

The Rock-Paper-Scissors (RPS) game is a paradigmatic model for cyclic dominance in biological systems. Here we consider this game in the social context of competition between opinions in a networked society. In our model, every agent has an opinion which is drawn from the three choices: rock, paper or scissors. In every timestep a link is selected randomly and the game is played between the nodes connected by the link. The loser either adopts the opinion of the winner or rewires the link. These rules define an adaptive network on which the agents’ opinions coevolve with the network topology of social contacts. We show analytically and numerically that nonequilibrium phase transitions occur as a function of the rewiring strength. The transitions separate four distinct phases which differ in the observed dynamics of opinions and topology. In particular, there is one phase where the population settles to an arbitrary consensus opinion. We present a detailed analysis of the corresponding transitions revealing an apparently paradoxical behavior. The system approaches consensus states where they are unstable, whereas other dynamics prevail when the consensus states are stable.     

Studies on controllability of directed networks with extremal optimization

Almost all natural, social and man-made-engineered systems can be represented by a complex network to describe their dynamic behaviors. To make a real-world complex network controllable with its desired topology, the study on network controllability has been one of the most critical and attractive subjects for both network and control communities. In this paper, based on a given directed–weighted network with both state and control nodes, a novel optimization tool with extremal dynamics to generate an optimal network topology with minimum control nodes and complete controllability under Kalman’s rank condition has been developed. The experimental results on a number of popular benchmark networks show the proposed tool is effective to identify the minimum control nodes which are sufficient to guide the whole network’s dynamics and provide the evolution of network topology during the optimization process. We also find the conclusion: “the sparse networks need more control nodes than the dense, and the homogeneous networks need fewer control nodes compared to the heterogeneous” (Liu et al., 2011  [18]), is also applicable to network complete controllability. These findings help us to understand the network dynamics and make a real-world network under the desired control. Moreover, compared with the relevant research results on structural controllability with minimum driver nodes, the proposed solution methodology may also be applied to other constrained network optimization problems beyond complete controllability with minimum control nodes.     

Topology identification of weighted complex dynamical networks

Recently, various papers investigated the geometry features, synchronization and control of complex network provided with certain topology. While, the exact topology of a network is sometimes unknown or uncertain. Using Lyapunov theory, we propose an adaptive feedback controlling method to identify the exact topology of a rather general weighted complex dynamical network model. By receiving the network nodes evolution, the topology of such kind of network with identical or different nodes, or even with switching topology can be monitored. Experiments show that the methods presented in this paper are of high accuracy with good performance.     

Interplay between topology and dynamics in the World Trade Web

We present an empirical analysis of the network formed by the trade relationships between all world countries, or World Trade Web (WTW). Each (directed) link is weighted by the amount of wealth flowing between two countries, and each country is characterized by the value of its Gross Domestic Product (GDP). By analysing a set of year-by-year data covering the time interval 1950–2000, we show that the dynamics of all GDP values and the evolution of the WTW (trade flow and topology) are tightly coupled. The probability that two countries are connected depends on their GDP values, supporting recent theoretical models relating network topology to the presence of a `hidden' variable (or fitness). On the other hand, the topology is shown to determine the GDP values due to the exchange between countries. This leads us to a new framework where the fitness value is a dynamical variable determining, and at the same time depending on, network topology in a continuous feedback.     

Understanding the enhanced synchronization of delay-coupled networks with fluctuating topology

We study the dynamics of networks with coupling delay, from which the connectivity changes over time. The synchronization properties are shown to depend on the interplay of three time scales: the internal time scale of the dynamics, the coupling delay along the network links and time scale at which the topology changes. Concentrating on a linearized model, we develop an analytical theory for the stability of a synchronized solution. In two limit cases, the system can be reduced to an “effective” topology: in the fast switching approximation, when the network fluctuations are much faster than the internal time scale and the coupling delay, the effective network topology is the arithmetic mean over the different topologies. In the slow network limit, when the network fluctuation time scale is equal to the coupling delay, the effective adjacency matrix is the geometric mean over the adjacency matrices of the different topologies. In the intermediate regime, the system shows a sensitive dependence on the ratio of time scales, and on the specific topologies, reproduced as well by numerical simulations. Our results are shown to describe the synchronization properties of fluctuating networks of delay-coupled chaotic maps.     

Synchronization-based topology identification of weighted general complex dynamical networks with time-varying coupling delay

Many existing papers investigated the geometric features, control and synchronization of complex dynamical networks provided with certain topology. However, the exact topology of a network is sometimes unknown or uncertain. Based on LaSalle’s invariance principle, we propose an adaptive feedback technique to identify the exact topology of a weighted general complex dynamical network model with time-varying coupling delay. By receiving the network nodes evolution, the topology of such a kind of network with identical or different nodes, or even with varying topology can be monitored. In comparison with previous methods, time delay is taken into account in this simple, analytical and systematic synchronization-based technique. Particularly, the weight configuration matrix is not necessarily symmetric or irreducible, and the inner-coupling matrix need not be symmetric. Illustrative simulations are provided to verify the correctness and effectiveness of the proposed scheme.     

Speed of adaptation in structured populations

Adaptation of populations takes place with the occurrence and subsequent fixation of mutations that confer some selective advantage to the individuals which acquire it. For this reason, the study of the process of fixation of advantageous mutations has a long history in the population genetics literature. Particularly, the previous investigations aimed to find out the main evolutionary forces affecting the strength of natural selection in the populations. In the current work, we investigate the dynamics of fixation of beneficial mutations in a subdivided population. The subpopulations (demes) can exchange migrants among their neighbors, in a migration network which is assumed to have either a random graph or a scale-free topology. We have observed that the migration rate drastically affects the dynamics of mutation fixation, despite of the fact that the probability of fixation is invariant on the migration rate, accordingly to Maruyama's conjecture. In addition, we have noticed a topological dependence of the adaptive evolution of the population when clonal interference becomes effective.     

自适应网络中病毒传播的稳定性和分岔行为研究

自适应复杂网络是以节点状态与拓扑结构之间存在反馈回路为特征的网络. 针对自适应网络病毒传播模型, 利用非线性微分动力学系统研究病毒传播行为; 通过分析非线性系统对应雅可比矩阵的特征方程, 研究其平衡点的局部稳定性和分岔行为, 并推导出各种分岔点的计算公式. 研究表明, 当病毒传播阈值小于病毒存在阈值, 即R₀0^c时, 网络中病毒逐渐消除, 系统的无病毒平衡点是局部渐近稳定的; R₀^c0<1时, 网络出现滞后分岔, 产生双稳态现象, 系统存在稳定的无病毒平衡点、较大稳定的地方病平衡点和较小不稳定的地方病平衡点; R₀>1时, 网络中病毒持续存在, 系统唯一的地方病平衡点是局部渐近稳定的. 研究发现, 系统先后出现了鞍结分岔、跨临界分岔、霍普夫分岔等分岔行为. 最后通过数值仿真验证所得结论的正确性. 关键词： 自适应网络 稳定性 分岔 基本再生数     

Time correlations and persistence probability of a Brownian particle in a shear flow

This paper presents an adaptive lag synchronization based method for simultaneous identification of topology and parameters of uncertain general complex dynamical networks with and without time delays. Based on Lyapunov stability theorem and LaSalle??s invariance principle, an adaptive controller is designed to realize lag synchronization between drive and response systems, meanwhile, identification criteria of network topology and system parameters are obtained. Numerical simulations illustrate the effectiveness of the proposed method.     

Topology Identification of General Dynamical Network with Distributed Time Delays

Genera/dynamical networks with distributed time delays are studied. The topology of the networks are viewed as unknown parameters, which need to be identified. Some auxiliary systems （also called the network estimators） are designed to achieve this goal. Both linear feedback control and adaptive strategy are applied in designing these network estimators. Based on linear matrix inequalities and the Lyapunov function method, the sufficient condition for the achievement of topology identification is obtained. This method can also better monitor the switching topology of dynamical networks. Illustrative examples are provided to show the effectiveness of this method.     

A self-organized model for network evolution

Here we provide a detailed analysis, along with some extensions and additonal investigations, of a recently proposed [1] self-organized model for the evolution of complex networks. Vertices of the network are characterized by a fitness variable evolving through an extremal dynamics process, as in the Bak-Sneppen [2] model representing a prototype of Self-Organized Criticality. The network topology is in turn shaped by the fitness variable itself, as in the fitness network model [3]. The system self-organizes to a nontrivial state, characterized by a power-law decay of dynamical and topological quantities above a critical threshold. The interplay between topology and dynamics in the system is the key ingredient leading to an unexpected behaviour of these quantities.     

Emergent hierarchical structures in multiadaptive games

We investigate a game-theoretic model of a social system where both the rules of the game and the interaction structure are shaped by the behavior of the agents. We call this type of model, with several types of feedback couplings from the behavior of the agents to their environment, a multiadaptive game. Our model has a complex behavior with several regimes of different dynamic behavior accompanied by different network topological properties. Some of these regimes are characterized by heterogeneous, hierarchical interaction networks, where cooperation and network topology coemerge from the dynamics.