Convergence Time Analysis of Quantized Gossip Consensus on Digraphs

TL;DR

This paper analyzes the convergence time of quantized gossip algorithms for consensus on directed graphs, providing polynomial bounds and linking convergence to Markov chain hitting times.

Contribution

It introduces a detailed convergence time analysis for quantized gossip algorithms, connecting it to Markov chain hitting times and simplifying the analysis for complete graphs.

Findings

Polynomial upper bounds on convergence time for complete graphs.

Convergence time characterized by hitting times in Markov chains.

Analysis applicable to directed graphs with minimal connectivity requirements.

Abstract

We have recently proposed quantized gossip algorithms which solve the consensus and averaging problems on directed graphs with the least restrictive connectivity requirements. In this paper we study the convergence time of these algorithms. To this end, we investigate the shrinking time of the smallest interval that contains all states for the consensus algorithm, and the decay time of a suitable Lyapunov function for the averaging algorithm. The investigation leads us to characterizing the convergence time by the hitting time in certain special Markov chains. We simplify the structures of state transition by considering the special case of complete graphs, where every edge can be activated with an equal probability, and derive polynomial upper bounds on convergence time.