# On the Connectivity of Unions of Random Graphs

## TL;DR

This paper investigates how many Erdős-Rényi random graphs need to be unioned to ensure connectivity and algebraic connectivity thresholds, providing bounds on probability and expected connectivity for multi-agent communication networks.

## Contribution

It introduces bounds on the number of random graphs required for union connectivity and algebraic connectivity, filling a gap in understanding random graph unions in multi-agent systems.

## Key findings

- Bound on the number of graphs for expected algebraic connectivity to exceed threshold
- Probability bounds for union of random graphs being connected
- Bounds on expectation and variance of algebraic connectivity

## Abstract

Graph-theoretic tools and techniques have seen wide use in the multi-agent systems literature, and the unpredictable nature of some multi-agent communications has been successfully modeled using random communication graphs. Across both network control and network optimization, a common assumption is that the union of agents' communication graphs is connected across any finite interval of some prescribed length, and some convergence results explicitly depend upon this length. Despite the prevalence of this assumption and the prevalence of random graphs in studying multi-agent systems, to the best of our knowledge, there has not been a study dedicated to determining how many random graphs must be in a union before it is connected. To address this point, this paper solves two related problems. The first bounds the number of random graphs required in a union before its expected algebraic connectivity exceeds the minimum needed for connectedness. The second bounds the probability that a union of random graphs is connected. The random graph model used is the Erd\H{o}s-R\'enyi model, and, in solving these problems, we also bound the expectation and variance of the algebraic connectivity of unions of such graphs. Numerical results for several use cases are given to supplement the theoretical developments made.