Variance Analysis of Randomized Consensus in Switching Directed Networks

TL;DR

This paper derives explicit formulas for the mean and variance of the consensus value in distributed algorithms over switching directed Erdos-Renyi networks, enhancing understanding of their asymptotic behavior.

Contribution

It provides the first analytical expressions for the distribution of the asymptotic consensus value in such networks, specifically the mean and variance.

Findings

Closed-form expressions for mean and variance of consensus value

Validation through numerical simulations

Insights into the impact of network size and communication probability

Abstract

In this paper, we study the asymptotic properties of distributed consensus algorithms over switching directed random networks. More specifically, we focus on consensus algorithms over independent and identically distributed, directed Erdos-Renyi random graphs, where each agent can communicate with any other agent with some exogenously specified probability $p$. While it is well-known that consensus algorithms over Erdos-Renyi random networks result in an asymptotic agreement over the network, an analytical characterization of the distribution of the asymptotic consensus value is still an open question. In this paper, we provide closed-form expressions for the mean and variance of the asymptotic random consensus value, in terms of the size of the network and the probability of communication $p$. We also provide numerical simulations that illustrate our results.