Discordant edges for the voter model on regular random graphs

TL;DR

This paper studies the evolution of discordant edges in the voter model on regular random graphs, revealing how they decrease and stabilize over different time scales, with implications for understanding consensus formation.

Contribution

It introduces a novel approach to analyze discordant edges' dynamics using coalescing random walks and the First Visit Time Lemma, providing new insights into the voter model's behavior.

Findings

Fraction of discordant edges stabilizes depending on initial opinion distribution and degree d.

On time scale n, discordant edges decay exponentially to zero.

New concentration technique for weak-dependent coalescing random walks.

Abstract

We consider the two-opinion voter model on a regular random graph with n vertices and degree $d \geq 3$. It is known that consensus is reached on time scale n and that on this time scale the volume of the set of vertices with one opinion evolves as a Fisher-Wright diffusion. We are interested in the evolution of the number of discordant edges (i.e., edges linking vertices with different opinions), which can be thought as the perimeter of the set of vertices with one opinion, and is the key observable capturing how consensus is reached. We show that if initially the two opinions are drawn independently from a Bernoulli distribution with parameter $u \in (0, 1)$, then on time scale 1 the fraction of discordant edges decreases and stabilises to a value that depends on d and u, and is related to the meeting time of two random walks on an infinite tree of degree d starting from two neighbouring vertices. Moreover, we show that on time scale n the fraction of discordant edges moves away from the constant plateau and converges to zero in an exponential fashion. Our proofs exploit the classical dual system of coalescing random walks and use ideas from Cooper et al. (2010) built on the so-called First Visit Time Lemma. We further introduce a novel technique to derive concentration properties from weak-dependence of coalescing random walks on moderate time scales.