Effects of Adding Edges on the Consensus Convergence Rate of Weighted Directed Chain Networks

For a multi-agent system with a directed graph as its interaction topology, the consensus convergence rate is determined by the algebraic connectivity (the smallest real part of nonzero Laplacian eigenvalues) of its underlying network. In this paper, the effects of adding weighted edges to a weighted directed path on the algebraic connectivity are investigated. Firstly, it is proved that the Laplacian eigenvalues are only affected by local subgraphs containing the additional edges if some weighted edges are added. Secondly, considering the case of adding one weighted edge, it is shown that the algebraic connectivity is determined by the range and the weight of the added edge, as well as the distribution of weights along the path. Interestingly, if equal-weight edges are added to a directed path with each arc having equal weight, then the algebraic connectivity can be calculated by a formula of the weight and the maximum range of the edges, which means that the algebraic connectivity of the graph obtained from the path by adding some edges with the same weight is independent of the order of the directed path and the location of the edges added. Finally, numerical experiments are given to verify the theoretical results.