Nonasymptotic Connectivity of Random Graphs and Their Unions

Abstract

Graph-theoretic methods have seen wide use throughout the literature on multiagent control and optimization. When communication networks are intermittent and unpredictable, they have been modeled using random communication graphs. When graphs are time varying, it is common to assume that their unions are connected over time, yet, to the best of our knowledge, there are not any results that determine the number of finite-size random graphs needed to attain a connected union. Therefore, this article bounds the probability that individual random graphs are connected and bounds the same probability for connectedness of unions of random graphs. The random graph model used is a generalization of the classic Erdös-Rényi model, which allows some edges to never appear. Numerical results are presented to illustrate the analytical developments made.