Asymptotic behaviour of the noisy voter model density process

TL;DR

This paper studies the long-term behaviour of the density of opinions in the noisy voter model, showing how it converges to different distributions depending on the noise level and graph structure.

Contribution

It characterizes the asymptotic distribution of the opinion density in the noisy voter model, revealing phase transitions based on noise strength and graph topology.

Findings

For small noise, the density converges to a Gaussian distribution.

For large noise, the density converges to a Bernoulli distribution.

Identifies critical noise thresholds on various graphs.

Abstract

Given a transition matrix $P$ indexed by a finite set $V$ of vertices, the voter model is a discrete-time Markov chain in $\{0,1\}^V$ where at each time-step a randomly chosen vertex $x$ imitates the opinion of vertex $y$ with probability $P(x,y)$. The noisy voter model is a variation of the voter model in which vertices may change their opinions by the action of an external noise. The strength of this noise is measured by an extra parameter $p \in [0,1]$. In this work we analyse the density process, defined as the stationary mass of vertices with opinion 1, i.e. $S_t = \sum_{x\in V} \pi(x)\xi_t(x)$, where $\pi$ is the stationary distribution of $P$, and $\xi_t(x)$ is the opinion of vertex $x$ at time $t$. We investigate the asymptotic behaviour of $S_t$ when $t$ tends to infinity for different values of the noise parameter $p$. In particular, by allowing $P$ and $p$ to be functions of the size $|V|$, we show that, under appropriate conditions and small enough $p$ a normalised version of $S_t$ converges to a Gaussian random variable, while for large enough $p$, $S_t$ converges to a Bernoulli random variable. We provide further analysis of the noisy voter model on a variety of specific graphs including the complete graph, cycle, torus and hypercube, where we identify the critical rate $p$ (depending on the size $|V|$) that separates these two asymptotic behaviours.