Stochastic stabilization problem of complex networks without strong connectedness

This paper aims to study the stochastic stabilization problem of complex networks without strong connectedness (CNSC). To deal with large-scale complex networks which are not strongly connected, a hierarchical method and a hierarchical algorithm are given, respectively. Meanwhile, we construct a logarithmic Lyapunov function for CNSC. Next, based on the theory of asymptotically autonomous systems, the Lyapunov method and the graph theory, the whole complex network can be stabilized by stabilizing a part of nodes. Then, stability criteria of CNSC are given, whose conditions reflect the relationship between dynamic properties and topology structure clearly. Furthermore, the theoretical results are applied to coupled oscillators on CNSC to ensure the stability. Finally, a numerical example is provided to illustrate the effectiveness and practicability of the results.