Stochastic stabilization problem of complex networks without strong connectedness

Global Mittag-Leffler stability for fractional-order coupled systems on network without strong connectedness

“…On one hand, Theorem 1 ensures the existence and uniqueness of the solution of the considered coupled system without a time delay considering the contraction mapping principle. On the other hand, based on the hierarchical method [28], let (R ji , B ji ) denote the ith SCC of the jth layer of a network (G, A). Note that the SCC (R ji , B ji ) is considered in this study to be balanced.…”

“…Note that the methods and results reported in the above mentioned papers are all based on the same assumption that the digraph is strongly connected; however, coupled systems on networks without connectedness are more common in practice. Recently, Liu et al [28] studied the stabilization problem for integer-order coupled systems (IOCSs) on networks without strong connectedness using a hierarchical algorithm. According to this algorithm, a weighted graph can be divided into some strongly connected components (SCCs) as a single vertex to investigate the stability problem.…”

Global stability for a new predator–prey model with cross-dispersal among patches based on graph theory

“…In this section, some definitions and theorems are listed that will be used in the later sections (see [2,12,13]). We denote a weighted digraph as (G, A).…”

“…The cross-dispersal matrix is listed as follows: After a calculation, we obtain the positive equilibrium for system (12) as…”

“…However, to the best of the author's knowledge, graphs without strong connectedness are universal in reality. Due to dealing with large-scale complex networks without strong connectedness, a hierarchical method and a hierarchical algorithm were proposed in [12]. Based on the hierarchical algorithm and the theory of asymptotically autonomous systems, stability theorems for a new fractional-order coupled system on a network without strong connectedness were obtained in [13].…”