Quasi-Synchronization of Delayed Stochastic Multiplex Networks via Impulsive Pinning Control

The quasi-synchronization problems of the delayed multiplex networks with stochastic perturbations are investigated in this article. Different from previous studies on complete synchronization for the delayed multiplex networks, quasi-synchronization, which is more practical and more challenging than complete synchronization, is discussed. To adopt a better cost-saving control, an impulsive pinning scheme is designed. That is, the key nodes in the multiplex networks are pinned and controlled at impulsive instants. At the same time, a virtual leader is introduced. Based on the theory of stochastic impulsive differential equations and the Lyapunov–Krasovskii functional method, the sufficient conditions are derived for the error systems to converge into a bounded region in the sense of the mean square. Therefore, stochastic quasi-synchronization is guaranteed. Finally, the two-layer Newman–Watts (NW) small-world networks and two-layer Barabási–Albert (BA) scale-free networks are provided to illustrate the effectiveness of the theoretical results.