Asymptotic Behaviour Of The Noisy Voter Model Density Process

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 ∈ [0,1]. In this work we analyse the density process, defined as the stationary mass of vertices with opinion 1, that is, St = Σx∈V π(x)ξt(x), where π is the stationary distribution of P, and ξt(x) is the opinion of vertex x at time t. We investigate the asymptotic behaviour of St 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 St converges to a Gaussian random variable, while for large enough p, St 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.