Weiguo Xia

Pinning synchronization of delayed dynamical networks via periodically intermittent control

This paper investigates the synchronization problem for a class of complex delayed dynamical networks by pinning periodically intermittent control. Based on a general model of complex delayed dynamical networks, using the Lyapunov stability theory and periodically intermittent control method, some simple criteria are derived for the synchronization of such dynamical networks. Furthermore, a Barabasi-Albert network consisting of coupled delayed Chua oscillators is finally given as an example to verify the effectiveness of the theoretical results.

Structural Balance and Opinion Separation in Trust–Mistrust Social Networks

Structural balance theory has been developed in sociology and psychology to explain how interacting agents, for example, countries, political parties, opinionated individuals, with mixed trust and mistrust relationships evolve into polarized camps. Recent results have shown that structural balance is necessary for polarization in networks with fixed, strongly connected neighbor relationships when the opinion dynamics are described by DeGroot-type averaging rules. We develop this line of research in this paper in two steps. First, we consider fixed, not necessarily strongly connected, neighbor relationships. It is shown that if the network includes a strongly connected subnetwork containing mistrust, which influences the rest of the network, then no opinion clustering is possible when that subnetwork is not structurally balanced; all of the opinions become neutralized in the end. In contrast, it is shown that when that subnetwork is indeed structurally balanced, the agents of the subnetwork evolve into two polarized camps and the opinions of all other agents in the network spread between these two polarized opinions. Second, we consider time-varying neighbor relationships. We show that the opinion separation criteria carry over if the conditions for fixed graphs are extended to joint graphs. The results are developed for both discrete-time and continuous-time models.

Adaptive synchronization of a switching system and its applications to secure communications

This paper studies the adaptive synchronization of a switching system with unknown parameters which switches between the Rössler system and a unified chaotic system. Using the Lyapunov stability theory and adaptive control method, the receiver system will achieve synchronization with the drive system and the unknown parameters would be estimated by the receiver. Then the proposed switching system is used for secure communications based on the communication schemes including chaotic masking, chaotic modulation, and chaotic shift key strategies. Since the system switches between two chaotic systems and the parameters are almost unknown, it is more difficult for the intruder to extract the useful message from the transmission channel. In addition, two new schemes in which the chaotic signal used to mask (or modulate) the transmitted signal switches between two components of a chaotic system are also presented. Finally, some simulation results are given to show the effectiveness of the proposed communication schemes.

Stability of Positive Switched Linear Systems: Weak Excitation and Robustness to Time-Varying Delay

This article investigates the stability of positive switched linear systems. We start from motivating examples and focus on the case when each switched subsystem is marginally stable (in the sense that all the eigenvalues of the subsystem matrix are in the closed left-half plane with those on the imaginary axis simple) instead of asymptotically stable. A weak excitation condition is first proposed such that the considered positive switched linear system is exponentially stable. An extension to the case without dwell time assumption is also presented. Then, we study the influence of time-varying delay on the stability of the considered positive switched linear system. We show that the proposed weak excitation condition for the delay-free case is also sufficient for the asymptotic stability of the positive switched linear system under unbounded time-varying delay. In addition, it is shown that the convergence rate is exponential if there exists an upper bound for the delay, irrespective of the magnitude of this bound. The motivating examples are revisited to illustrate the theoretical results.

Cluster synchronization algorithms

This paper presents two approaches to achieving cluster synchronization in dynamical multi-agent systems. In contrast to the widely studied synchronization behavior, where all the coupled agents converge to the same value asymptotically, in the cluster synchronization problem studied in this paper, we require that all the interconnected agents to evolve into several clusters and each agent only to synchronize within its cluster. The first approach is to add a constant forcing to the dynamics of each agent that are determined by positive diffusive couplings; and the other is to introduce both positive and negative couplings between the agents. Some sufficient and/or necessary conditions are constructed to guarantee n-cluster synchronization behavior. Simulation results are presented to illustrate the effectiveness of the theoretical analysis.

Convergence of max-min consensus algorithms

In this paper, we propose a distributed max-min consensus algorithm for a discrete-time n -node system. Each node iteratively updates its state to a weighted average of its own state together with the minimum and maximum states of its neighbors. In order for carrying out this update, each node needs to know the positive direction of the state axis, as some additional information besides the relative states from the neighbors. Various necessary and/or sufficient conditions are established for the proposed max-min consensus algorithm under time-varying interaction graphs. These convergence conditions do not rely on the assumption on the positive lower bound of the arc weights.

Analysis and applications of spectral properties of grounded Laplacian matrices for directed networks

In-depth understanding of the spectral properties of grounded Laplacian matrices is critical for the analysis of convergence speeds of dynamical processes over complex networks, such as opinion dynamics in social networks with stubborn agents. We focus on grounded Laplacian matrices for directed graphs and show that their eigenvalues with the smallest real part must be real. Power and upper bounds for such eigenvalues are provided utilizing tools from nonnegative matrix theory. For those eigenvectors corresponding to such eigenvalues, we discuss two cases when we can identify the vertex that corresponds to the smallest eigenvector component. We then discuss an application in leader-follower social networks where the grounded Laplacian matrices arise naturally. With the knowledge of the vertex corresponding to the smallest eigenvector component for the smallest eigenvalue, we prove that by removing or weakening specic directed couplings pointing to the vertex having the smallest eigenvector component, all the states of the other vertices converge faster to that of the leading vertex. This result is in sharp contrast to the well-known fact that when the vertices are connected together through undirected links, removing or weakening links does not accelerate and in general decelerates the converging process.

New stability results for switched discrete-time systems with application to consensus problems

This paper investigates the asymptotic stability of a class of switched linear time-invariant discrete-time systems. Without the restriction of switching signals, a novel result is first proposed to guarantee uniform global asymptotic stability. Then, it is applied to studying some consensus problems of linear multi-agent systems under switching network topology. The proposed result modifies some insufficient condition assumed in a related paper and establishes the exponentially fast convergence rate of the multi-agent system. An interesting example illustrates the effectiveness of the derived results.

Products of generalized stochastic Sarymsakov matrices

In the set of stochastic, indecomposable, aperiodic (SIA) matrices, the class of stochastic Sarymsakov matrices is the largest known subset (i) that is closed under matrix multiplication and (ii) the infinitely long left-product of the elements from a compact subset converges to a rank-one matrix. In this paper, we show that a larger subset with these two properties can be derived by generalizing the standard definition for Sarymsakov matrices. The generalization is achieved either by introducing an “SIA index”, whose value is one for Sarymsakov matrices, and then looking at those stochastic matrices with larger SIA indices, or by considering matrices that are not even SIA. Besides constructing a larger set, we give sufficient conditions for generalized Sarymsakov matrices so that their products converge to rank-one matrices. The new insight gained through studying generalized Sarymsakov matrices and their products has led to a new understanding of the existing results on consensus algorithms and will be helpful for the design of network coordination algorithms.

Synchronizing clocks in distributed networks

While various time synchronization protocols for clocks in wired and/or wireless networks are under development, recently it has been shown by Freris, Graham and Kumar that clocks in distributed networks cannot be synchronized precisely even in idealized situations. In this paper by determining the clock synchronization errors in the similar settings of the impossibility result just mentioned, we are able to show that the clocks can get synchronized within an acceptable level of accuracy. After studying the basic case of synchronizing two clocks with asymmetric time delays in the two-way message passing process, we first analyze the directed ring networks, in which neighboring clocks are likely to experience severe asymmetric time delays. We then discuss connected undirected networks with two-way message passing between each pair of adjacent nodes. In the end, we expand the discussions to networks with directed topologies that are strongly connected.