Coalescing random walks and voting on connected graphs

TL;DR

This paper analyzes the expected coalescence time of particles performing random walks on connected graphs, providing bounds based on spectral properties and degree variability, with implications for distributed consensus algorithms.

Contribution

It derives a bound on coalescence time for general graphs, linking it to spectral gap and degree distribution, extending understanding of coalescing random walks.

Findings

Coalescence time bound depends on spectral gap and degree variability.

For regular graphs, the bound simplifies to a function of n and log n.

The results inform distributed consensus and self-stabilizing algorithms.

Abstract

In a coalescing random walk, a set of particles make independent random walks on a graph. Whenever one or more particles meet at a vertex, they unite to form a single particle, which then continues the random walk through the graph. Coalescing random walks can be used to achieve consensus in distributed networks, and is the basis of the self-stabilizing mutual exclusion algorithm of Israeli and Jalfon. Let G=(V,E), be an undirected, connected n vertex graph with m edges. Let C(n) be the expected time for all particles to coalesce, when initially one particle is located at each vertex of an n vertex graph. We study the problem of bounding the coalescence time C(n) for general classes of graphs. Our main result is that C(n)= O(1/(1-lambda_2))*((log n)^4 +n/A)), where lambda_2 is the absolute value of the second largest eigenvalue of the transition matrix of the random walk, A= (sum d^2(v))/(d^2 n), d(v) is the degree of vertex v, and d is the average node degree. The parameter A is an indicator of the variability of node degrees. Thus 1 <= A =O(n), with A=1 for regular graphs.