Consensus over a Random Network Generated by i.i.d. Stochastic Matrices

TL;DR

This paper establishes a unified necessary and sufficient condition for consensus in random networks generated by i.i.d. stochastic matrices across multiple convergence modes, using stability analysis in a projected subspace.

Contribution

It provides a comprehensive condition for consensus in such networks, showing equivalence across different convergence modes, which was previously not established.

Findings

Consensus conditions are equivalent across convergence modes

Necessary and sufficient condition derived via stability analysis

Condition applies to networks generated by i.i.d. stochastic matrices

Abstract

Our goal is to find a necessary and sufficient condition on the consensus over a random network, generated by i.i.d. stochastic matrices. We show that the consensus problem in three different convergence modes (almost surely, in probability, and in L1) are equivalent, thus have the same necessary and sufficient condition. We obtain the necessary and sufficient condition through the stability in a projected subspace.