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This paper investigates the coupled systems of stochastic differential equations with variable delays (CSDDEs) on networks. We analyze the existence and uniqueness of solution by combining the method of graph theory with the Lyapunov function analysis. Furthermore, we utilize the graph theory technique and the nonnegative semimartingale convergence theorem to obtain the almost sure stability of sample solutions and the sufficient principles to locate their limit sets, which correlate closely with the topology property of CSDDEs. Finally we illustrate our main results by examples from population dynamics and vibration systems.

The almost sure stability of coupled system of stochastic delay differential equations on networks