On synchronization of random nonlinear complex networks

Traditionally, stochastic disturbances arising in complex networks are often assumed to be drawn from a Wiener process, potentially limiting their applicability in real engineering scenarios. To address this limitation, we incorporate randomness to quantify the stochastic disturbances within a group of participating individuals, thereby establishing random nonlinear complex networks in a directed interacting setting. Subsequently, we demonstrate that the maximal existence interval of the unique solution to the underlying systems is determined by the properties of the associated noise and the specified Lipschitz constant. Building on this, we further show that, by making use of supermartingale and Lyapunov-based techniques, the almost sure synchronization condition of the investigated random complex system is determined by the communication topology, weight gain, and the number of participating agents. Additionally, we discuss synchronization problems within strongly connected and undirected graphs. Finally, we validate the proposed method using Chen systems.