On Asymptotic Consensus Value in Directed Random Networks

TL;DR

This paper analyzes the long-term behavior of distributed consensus algorithms over switching directed random networks, deriving formulas for the mean and variance of the consensus value.

Contribution

It provides the first closed-form expressions for the mean and variance bounds of the asymptotic consensus value in directed random networks.

Findings

Derived closed-form expressions for the mean of the consensus value.

Established an upper bound for the variance of the consensus value.

Validated results through numerical simulations.

Abstract

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 random graphs, where each agent can communicate with any other agent with some exogenously specified probability. While different aspects of consensus algorithms over random switching networks have been widely studied, a complete characterization of the distribution of the asymptotic value for general \textit{asymmetric} random consensus algorithms remains an open problem. In this paper, we derive closed-form expressions for the mean and an upper bound for the variance of the asymptotic consensus value, when the underlying network evolves according to an i.i.d. \textit{directed} random graph process. We also provide numerical simulations that illustrate our results.