Push-sum on random graphs

TL;DR

This paper extends push-sum algorithms to random directed graphs, proving almost sure convergence to the average consensus under certain conditions and establishing convergence rates for specific graph sequences.

Contribution

It introduces a new convergence analysis for push-sum algorithms on random graphs, including a rate of convergence for B-irreducible graph sequences.

Findings

Almost sure convergence to average consensus

Convergence rates for B-irreducible graph sequences

Conditions involving directed infinite flow property

Abstract

In this paper, we study the problem of achieving average consensus over a random time-varying sequence of directed graphs by extending the class of so-called push-sum algorithms to such random scenarios. Provided that an ergodicity notion, which we term the directed infinite flow property, holds and the auxiliary states of agents are uniformly bounded away from zero infinitely often, we prove the almost sure convergence of the evolutions of this class of algorithms to the average of initial states. Moreover, for a random sequence of graphs generated using a time-varying B-irreducible probability matrix, we establish convergence rates for the proposed push-sum algorithm.