Synchronization of stochastic complex networks with time-delayed coupling

Noise and time delay are inevitable in real-world networks. In this article, the framework of master stability function is generalized to stochastic complex networks with time-delayed coupling. The focus is on the effects of noise, time delay, and their inner interactions on the network synchronization. It is found that when there exists time-delayed coupling in the network and noise diffuses through all state variables of nodes, appropriately increasing the noise intensity can effectively improve the network synchronizability; otherwise, noise can be either beneficial or harmful. For stochastic networks, large time delays will lead to desynchronization. These findings provide valuable references for designing optimal complex networks in practical applications.