Convergence Rate Estimates for Consensus over Random Graphs

TL;DR

This paper develops new convergence rate estimates for consensus algorithms operating over random graphs, applicable to both designed and external randomness scenarios, supported by explicit eigenvalue calculations and simulations.

Contribution

It introduces novel methods to estimate convergence rates for consensus over random graphs, including explicit eigenvalue-based bounds without restrictive assumptions.

Findings

Provides bounds on disagreement decay rate

Estimates probability of deviation from consensus

Supports results with simulation data

Abstract

Multi-agent coordination algorithms with randomized interactions have seen use in a variety of settings in the multi-agent systems literature. In some cases, these algorithms can be random by design, as in a gossip-like algorithm, and in other cases they are random due to external factors, as in the case of intermittent communications. Targeting both of these scenarios, we present novel convergence rate estimates for consensus problems solved over random graphs. Established results provide asymptotic convergence in this setting, and we provide estimates of the rate of convergence in two forms. First, we estimate decreases in a quadratic Lyapunov function over time to bound how quickly the agents' disagreement decays, and second we bound the probability of being at least a given distance from the point of agreement. Both estimates rely on (approximately) computing eigenvalues of the expected matrix exponential of a random graph's Laplacian, which we do explicitly in terms of the network's size and edge probability, without assuming that any relationship between them holds. Simulation results are provided to support the theoretical developments made.