A Necessary and Sufficient Condition of an Interfering Reverse Edge for a Directed Acyclic Graph

Abstract

A directed acyclic graph (DAG) is a common topology in biological, engineering, and social networks. A network topology is critical in determining a collective behavior of a network dynamic system. For example, the convergence rate of a consensus behavior in a multiagent system relies on the eigenvalues of the Laplacian associated with the network topology. This article aims to analyze the influence of adding a reverse edge into a DAG on convergence rate. It reveals the existence of the so-called interfering reverse edges; adding one single edge in this category can reduce the so-called dominant convergence rate even for a large network. More specifically, a necessary and sufficient condition of an interfering reverse edge is explicitly constructed. According to the condition, a computationally efficient method is proposed to assess an interfering reverse edge.