The voter model on random regular graphs with random rewiring

TL;DR

This work analyzes the binary voter model on random $d$-regular graphs that evolve by random edge rewiring at rate $\nu$, proving that the fraction of opinion-1 converges, on time scale $n$, to a Fisher–Wright diffusion with an explicit diffusion constant $\vartheta_{d,\nu}$ given by a convergent continued-fraction expansion. The diffusion constant encodes the interplay between network degree and rewiring speed and recovers the static and mean-field limits as $\nu\to0$ or $\nu\to\infty$, or $d\to\infty$. A central component is the homogenisation of discordances, which links the opinion-density dynamics to the meeting times of two independent random walks on the evolving graph; the latter are shown to converge to an exponential law with rate $2\vartheta_{d,\nu}$. The paper also provides a detailed spectral-type analysis of $\vartheta_{d,\nu}$, including its asymptotics and monotonicity, establishing how rewiring speeds up consensus. Overall, the work delivers the first explicit diffusion constant for a voter model in a dynamic random environment and offers a robust framework for comparing dynamic-network effects across graph parameters.

Abstract

We consider the voter model with binary opinions on a random regular graph with $n$ vertices of degree $d \geq 3$, subject to a rewiring dynamics in which pairs of edges are rewired, i.e., broken into four half-edges and subsequently reconnected at random. A parameter $ν\in (0,\infty)$ regulates the frequency at which the rewirings take place, in such a way that any given edge is rewired exponentially at a rate $ν$ in the limit as $n\to\infty$. We show that, under the joint law of the random rewiring dynamics and the random opinion dynamics, the fraction of vertices with either one of the two opinions converges on time scale $n$ to the Fisher-Wright diffusion with an explicit diffusion constant $\vartheta_{d,ν}$ in the limit as $n\to\infty$. In particular, we identify $\vartheta_{d,ν}$ in terms of a continued-fraction expansion and analyse its dependence on $d$ and $ν$. A key role in our analysis is played by the set of discordant edges, which constitutes the boundary between the sets of vertices carrying the two opinions.

Figures (8)

• Figure 2.1: Numerical computation of $\nu\mapsto\vartheta_{d,\nu}$ for $d=3,4,5$ (from left to right). The blue line has height $1$, the orange line has height $\vartheta_{d,0}=(d-2)/(d-1)$. The green line is the numerical approximation of $\nu\mapsto \vartheta_{d,\nu}$ obtained after retaining 10 terms in the continued fraction.
• Figure 2.2: Simulation of the meeting time of two independent random walks in the case $d=3$ and with rewiring rate $\nu = 0.3$. The size of the graph is $n = 1500$. The empirical histogram in blue is obtained by running $1000$ independent simulations. The orange curve is the probability density function of ${\rm Exp}(2\vartheta_{3,0.3}/n)$.
• Figure 5.1: Visual representation of the transition in step (6-a), where an arrival of $\mathfrak{T}^{\rm rw}_{X_s,3}$ is portrayed, and $\hslash = 3$. Edges represented in gray in the second plot are those that are not part of $\mathcal{B}_{\hslash,t}(X_t)\cup\mathcal{B}_{\hslash,t}(Y_t)$.
• Figure 5.2: Visual representation of the transition in step (6-b), where an arrival of both $\hat{\mathfrak{T}}^{\rm dyn}_{z,2}$ and $\hat{\mathfrak{T}}^{\rm dyn}_{v,1}$ is portrayed, with $v=Y_s$. Edges represented in gray in the second plot are those that are not part of $\mathcal{B}_{\hslash,t}(X_t)\cup\mathcal{B}_{\hslash,t}(Y_t)$, and $\hslash = 3$. Edges represented in brown are those in $V^-(\sigma_{z,2}, \sigma_{v,1}, E_s)$.
• Figure 5.3: Visual representation of the transition in step (6-b), where an arrival of both $\hat{\mathfrak{T}}^{\rm dyn}_{z,3}$ and $\hat{\mathfrak{T}}^{\rm dyn}_{v,2}$is portrayed, where $z=X_s$. In this case, the set $V^{-}$ is empty. Edges represented in gray in the second plot are those that are not part of $\mathcal{B}_{\hslash,t}(X_t)\cup\mathcal{B}_{\hslash,t}(Y_t)$, where $\hslash = 3$.

Theorems & Definitions (49)

• Definition 2.1
• Remark 2.2
• Remark 2.3
• Theorem 2.4
• Theorem 2.5
• Remark 2.6
• Proposition 2.7
• Proposition 2.8
• Lemma 2.9
• proof