The Role of Reverse Edges on Consensus Performance of Chain Networks

The minimal real part of all the nonzero eigenvalues of Laplacian matrix, also known as the dominant convergence rate, characterizes the consensus performance of multiagent systems on a directed graph. The effect of adding the second reverse edge to a directed chain graph on the dominant convergence rate is investigated. According to the relative positions of the two reverse edges, three cases are discussed, respectively. It is revealed that the dominant convergence rate of the whole network is determined only by the ranges and the relative positions of the reverse edges. Moreover, the consensus performance will not get better with the new reverse edge being added, and it decreases as the reverse range increases. Finally, the theoretical results are illustrated by some numerical examples.