Non-Asymptotic Connectivity of Random Graphs and Their Unions

TL;DR

This paper provides probabilistic bounds on the connectivity of individual and union of random graphs, generalizing Erdos-Renyi models to better understand finite-size network connectivity in multi-agent systems.

Contribution

It introduces bounds on the probability of connectivity for individual and union of random graphs, extending classic models to include edges that may never appear.

Findings

Bounds on the probability of individual graph connectivity.

Bounds on the probability of union graph connectivity.

Numerical results validating the analytical bounds.

Abstract

Graph-theoretic methods have seen wide use throughout the literature on multi-agent control and optimization. When communications are intermittent and unpredictable, such networks 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 results that determine the number of finite-size random graphs needed to attain a connected union. Therefore, this paper 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 Erdos-Renyi model which allows some edges never to appear. Numerical results are presented to illustrate the analytical developments made.