Asymptotic behaviour of the noisy voter model density process

Abstract

Given a transition matrix P indexed by a finite set V of vertices, the voter model is a discrete-time Markov chain in ${\{0,1\}}^{\mathit{V}}$ where at each time-step a randomly chosen vertex x imitates the opinion of vertex y with probability $\mathit{P}(\mathit{x},\mathit{y})$. 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 $\mathit{p}\in [0,1]$.

In this work we analyse the density process, defined as the stationary mass of vertices with opinion 1, that is, ${\mathit{S}}_{\mathit{t}}={\sum }_{\mathit{x}\in \mathit{V}}\mathit{\pi }(\mathit{x}){\mathit{\xi }}_{\mathit{t}}(\mathit{x})$, where π is the stationary distribution of P, and ${\mathit{\xi }}_{\mathit{t}}(\mathit{x})$ is the opinion of vertex x at time t. We investigate the asymptotic behaviour of ${\mathit{S}}_{\mathit{t}}$ 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 $|\mathit{V}|$, we show that, under appropriate conditions and small enough p a normalised version of ${\mathit{S}}_{\mathit{t}}$ converges to a Gaussian random variable, while for large enough p, ${\mathit{S}}_{\mathit{t}}$ 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 $|\mathit{V}|$) that separates these two asymptotic behaviours.