Uncover the Relation Between Consensus and Topology of Directed Network: The Minimum Node In-Degree of Directed Cycles

Directed networks widely exist in industrial, social, and natural networks, the consensus of which is characterized by the convergence rate. In this paper, a quantitative analysis for the inhibitory effect of directed cycles on the consensus of directed network is provided. Specifically, directed cycle necessarily inhibits network consensus when there is at least one node with in-degree of 1 on the cycle, and the more nodes with in-degree of 1, the greater the inhibition. The inhibition induced by multiple disjoint directed cycles is only determined by the directed cycle with the minimum node in-degree, independent of other directed cycles. Generally, the more intersected directed cycles, the smaller the convergence rate. Accordingly, to improve the consensus or synchronizability of a directed network, it is necessary to avoid the generation of multiple intersected directed cycles, especially those with a large number of in-degree 1.