754 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 3, MARCH 2012 Synchronization in Complex Networks With Stochastically Switching Coupling Structures Bo Liu, Wenlian Lu, and Tianping Chen, Senior Member, IEEE Abstract—Synchronization in complex networks with time dependent coupling and stochastically switching coupling structure is discussed. A novel approach investigating synchronization based on the scramblingness property of the coupling matrix is proposed. Some sufficient condition for a network with general time-varying coupling structure to reach complete synchronization is provided. Based on the general theorem, networks with stochastically switching coupling structures is investigated. In particular, two kinds of stochastic switching coupling networks are addressed: (a) in- dependent and identically distributed switching processes and (b) Markov jump processes. In both cases, some sufficient condition for almost sure synchronization of the networks is given. Also, numerical simulations are provided to illustrate the theoretical results. Index Terms—Coupled system, stochastic systems, switched systems, synchronization, time-varying. I. INTRODUCTION Today, the study of synchronization in complex dynamical systems has become a subject of great interest due to its applications and po- tential applications in a variety of fields, such as communication [1], seismology [2], and neural networks [3]. Till now, many works are available engaging in the study of synchro- nization of complex networks. For example, in the pioneering work [4], Master Stability Function (MSF) method to study the local syn- chronization of coupled chaotic systems was proposed. In [5]–[7], the distance to the synchronization manifold was defined, and some suf- ficient conditions for an array of linearly coupled systems to synchro- nize were proposed. Most of the works on synchronization are focused on static networks, i.e., the coupling structure and coupling strength is constant in time. In such case, the criteria for synchronization have been well-established by analyzing the eigen-structure of the coupling matrix. Besides, there are also some papers concerning synchroniza- tion in dynamic networks, i.e., the network coupling structure and cou- pling strength is dynamically changing along with time. Here, we list some (not all) relating papers [9]–[13], [15], [20], etc. In [9], the authors studied global synchronization in a blinking network model. They used the connection graph stability method developed in [8]. In [20], the au- thors investigated synchronization in networks of coupled Kuramoto oscillators with switching topologies and time delays. This model also can be viewed as nonlinear consensus (see [25]). In [12], sufficient conditions for fast switching synchronization in networks with time- varying topologies were given. In [15], the authors studied synchro- nization in networks with random switching topologies and gave suf- ficient conditions for almost sure local synchronization. Both [12] and [15] indicate that under the assumption of fast switching, the synchro- nization of the time-varying network can be deduced from their time- average system. Manuscript received June 09, 2010; revised January 24, 2011; accepted July 16, 2011. Date of publication September 01, 2011; date of current version Feb- ruary 29, 2012. This work was supported by the National Science Founda- tion of China under Grant 60974015, the Graduate Innovation Foundation of Fudan University under Grant EYH1411040. Recommended by Associate Ed- itor M. Egerstedt. The authors are with the School of Mathematical Sciences, Fudan University, Shanghai 200433, China (e-mail: 071018024@fuan.edu.cn; wenlian@fuan.edu.cn; tchen@fuan.edu.cn). Digital Object Identifier 10.1109/TAC.2011.2166665 Another topic closely relating to synchronization problem is con- sensus problem in networks of multiagents. [16]–[19], [21]–[30] are a few among them. Though consensus problem is a special case of synchronization problem, some efficient approach used in the consensus problem can still be applied to investigation of synchronization problem. It is also known that if the networks are time-varying or with switching topologies, the synchronization or consensus becomes complicated [10], [14], [28]. Because it is difficult to construct a Lyapunov function when the coupling matrices are time varying, except the switching occurs among several strongly connected and balanced graphs. In such case, it is proved that for arbitrary switching, average consensus can be reached exponentially (see [16]). Instead, if the graph is not node balanced, it is difficult to find a common Lyapunov function so that this approach can not apply. Another efficient approach comes from the theory of nonhomoge- nous Markov chains by reducing the convergence of consensus algo- rithm to the ergodicity of infinite products of stochastic matrices. This method is widely used in [21]–[23], [27] and others. It was based on the work of Hajnal back to the 1950’s [31]. Hajnal investigated the weak er- godicity of non-homogenous Markov chains and proposed scrambling matrix, which plays an important role in the convergence of products of stochastic matrices. Similar method has also been used to study con- sensus problem in continuous time networks in [17], [18], [28], etc. This method was also used to discuss synchronization in [13]. It is natural to ask if this method can be further extended to more general cases in synchronization analysis. This is the aim of this tech- nical note. In the following, we first address global synchronization in net- works with a general time-varying topology and sufficient condition for global synchronization is given. To this purpose, we extend the concept of Hajnal’s scrambling property from stochastic matrices to matrices with nonnegative off-diagonal entries. Then we will turn to stochastic switching networks. Particularly, we will study syn- chronization for networks with two kinds of stochastically switching topologies. That is: (a) the switching sequence are independent and identically distributed; (b) the switching sequence forms a Markov chain. In both cases, we give sufficient conditions for the network to synchronize almost surely. In previous works, the authors considered either special node dy- namics such as Kuramoto model in [20], or linear dynamics such as in [13], or local synchronization as in [15]. Instead, in this note, we con- sider global synchronization for continuous-time networks with gen- eral nonlinear node dynamics and general time varying topologies. In case the stochastic switching topologies, we don’t require the network to switch fast enough, while this requirement is assumed in [9], [12]. We also point out that if there is a nonzero probability of a scrambling coupling matrix, then the network will synchronize almost surely if the coupling strength is strong enough. The rest of the technical note is organized as follows. In Section II, we study networks with general time-varying topologies and provide a sufficient condition for such networks to achieve synchronization. In Section III, we study stochastic network and give sufficient conditions for almost sure synchronization. An example with numerical simula- tions are provided in Section IV, and the technical note is concluded in Section V. II. GENERAL THEORY In this section, we discuss synchronization in networks with general time dependent coupling. 0018-9286/$26.00 © 2011 IEEE