Stochastic Synchronization of Nonlinear Networks With Directed Graphs and Degenerate Noise

Noise is ubiquitous in network systems and has profound impacts on network dynamics. Despite substantial progress in stochastic synchronization of complex networks, most studies limit their focus to nondegenerate noise. However, the influence of degenerate noise on synchronization has not been studied properly. Here, we tackle this challenging problem by introducing a network of nonlinear units with a deterministic communication digraph and a noisy communication digraph. The communication links in the noisy communication digraph are affected by degenerate noise in a relative-dependent manner. Two types of deterministic communication digraphs are considered: 1) the case of strongly connected graphs and 2) the case of graphs containing a spanning tree. By using the algebraic graph theory and the stability theory of stochastic differential equations, algebraic sufficient conditions for both cases are derived, which depend on node dynamics, network structure, and coupling strengths. These criteria demonstrate that strong deterministic communication can compensate for the detrimental effect of degenerate noise, thus ensuring the stochastic synchronization in the almost sure sense. Numerical simulations support and illustrate the proposed theoretical results.