Tight Upper Bound for the Scrambling Constant of Uniformly Jointly Connected Directed Graphs With Application to Consensus of Multiagent Systems

Abstract

In this article, we give upper bounds for the scrambling constant (SC) of the composition of general uniformly jointly connected (UJC) directed graphs. Some bounds are tight as they can be achieved by the UJC complete $N$ -layer graphs. Moreover, this article establishes new and explicit connections between the upper bounds and the number of consecutive graphs needed to composite a neighbor-shared graph. Then, the upper bounds are exploited to find new lower bounds for the convergence rate of consensus protocols over UJC communication graphs. The simulation shows that consensus over UJC complete $N$ -layer graphs may have the slowest convergence rate over all unweighted switching graphs.