This paper proposes and analyzes a new strategy to accelerate the process of reaching consensus in leader-follower networks. By removing or weakening specific directed couplings pointing to the first followers from the other followers, we prove that all the followers' states converge faster to that of the leader. This result is in sharp contrast to the well known fact that when the followers are coupled together through undirected links, removing or weakening links always decelerate the converging process. Simulation results are provided to illustrate this subtle, yet somewhat surprising, provably correct result.

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.

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 thesystem 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.

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 Barabási–Albert network consisting of coupled delayed Chua oscillators is finally given as an example to verify the effectiveness of the theoretical results.

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.

This paper discloses the similarities between the condition for realizing cluster synchronization and that for uncontrollability in diffusively coupled multi-agent networks, both of which are built upon the characteristics of the networks' topologies. We first generalize the notions of equitable partitions and almost equitable partitions to make them applicable to directed, weighted graphs. Consequently, we are enabled to characterize the controllable subspace of a given diffusively coupled multi-agent system using graph theoretic ideas. After comparing the condition to realize cluster synchronization and the condition for the network to be controllable, we conclude that those diffusively coupled multi-agent networks that are not controllable usually realize cluster synchronization asymptotically. Simulation results are provided to illustrate the theoretical results.

This paper shows how different mechanisms may lead to clustering behavior in connected networks consisting of diffusively coupled agents. In contrast to the widely studied synchronization processes, in which the states of all the coupled agents converge to the same value asymptotically, in the cluster synchronization problem studied in this paper, we require all the interconnected agents to evolve into several clusters and each agent only to synchronize within its cluster. The first mechanism is that agents have different self-dynamics, and those agents having the same self-dynamics may evolve into the same cluster. When the agents’ self-dynamics are identical, we present two other mechanisms under which cluster synchronization might be achieved. One is the presence of delays and the other is the existence of both positive and negative couplings between the agents. Some sufficient and/or necessary conditions are constructed to guarantee n-cluster synchronization. Simulation results are presented to illustrate the effectiveness of the theoretical analysis.

We provide new insight into the somewhat obscure definition of the Sarymsakov class of stochastic matrices and use it to construct a new necessary and sufficient condition for the convergence of products of stochastic matrices. Such convergence result is critical in establishing the effectiveness of distributed coordination algorithms for multi-agent systems and enables us to investigate a specific coordination task with asynchronous update events. The set of scrambling stochastic matrices, a subclass of the Sarymsakov class, is utilized to establish the convergence of the agents' states even when there is no common clock for the agents to synchronize their update actions.

The convergence of products of stochastic matrices has proven to be critical in establishing the effectiveness of distributed coordination algorithms for multi-agent systems. After reviewing some classic and recent results on infinite backward products of stochastic matrices, we provide a new necessary and sufficient condition for the convergence in terms of matrices from the Sarymsakov class of stochastic matrices, which complements the known other necessary and sufficient conditions. To gain insight into the somewhat obscure definition of the Sarymsakov class, we generalize some conditions in the definition and prove that the resulted set of matrices is exactly the set of indecomposable, aperiodic, stochastic matrices that has been extensively studied in the past. To apply the gained knowledge about the Sarymsakov class to the coordination of multi-agent systems, we investigate a specific coordination task with asynchronous update events. Then the set of scrambling stochastic matrices, a subclass of the Sarymsakov class, is utilized to establish the convergence of the system's state even when there is no common clock for the agents to synchronize their update actions.

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.

Structural balance theory has been developed in sociology and psychology to explain how interacting agents, e.g., 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 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.

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.

In this paper we deal with robust synchronization problems for directed Lur'e networks subject to incrementally passive nonlinearities and incrementally sector bounded non-linearities, respectively. By making use of general algebraic connectivities of strongly connected graphs and subgraphs, sufficient synchronization conditions are obtained for diffusively interconnected identical Lur'e systems on both the strongly connected interconnection topology and the topology containing a directed spanning tree. The static feedback gain matrices are determined by the matrices defining the individual agent dynamics and the general algebraic connectivities. The synchronization criteria obtained in the present paper extend those for undirected Lur'e networks in our previous work.