Cluster synchronization of stochastic two-layer networks with infinite distributed delays via delayed pinning impulsive control

The cluster synchronization problem of stochastic two-layer networks with infinite distributed delays is concerned. Firstly, we study the cluster synchronization of the first layer (leader-layer) network with the average state of each cluster of sets as synchronization target. Secondly, we design a delayed pinning impulsive controller to synchronize the second layer (follower-layer) network to the first layer network in a mean square cluster sense. Based on stochastic impulsive analysis and Lyapunov stability theory, some sufficient conditions for cluster synchronization are obtained. Finally, the effectiveness of the theoretical results is verified through numerical simulations.