Voter Models On Graphs

Abstract

The voter model which describes the flow of information through interactions between neighbors has been widely studied in the field of probability. In this paper we study two variations of the voter model, one is called the Latent Voter Model and the other is called the Zealot Voter Model. Both models are implemented in a space that is a random graph.In the latent voter model, which models the spread of a technology through a social network, individuals who have just changedtheir choice have a latent period, which is exponential with rate $\lambda$,during which they will not buy a new device. We study site and edge versions of this model on random graphs generated by a configuration model in which the degrees $d(x)$ have $3 \le d(x) \le M$. We show that if the number ofvertices $n \to\infty$ and $\log n \ll \lambda_n \ll n$ then the fraction of 1's at time $\lambda_n t$ converges tothe solution of $du/dt = c_pu(1-u)(1-2u)$. Using this we showthe latent voter model has a quasi-stationary state in which each opinion has probability $\approx 1/2$ and persists in this state for a time that is $\ge n^m$ for any $m<\infty$. Thus, even a very small latent period drastically changes the behavior of the voter model, which has a one parameter family of stationary distributions and reaches fixation in time $O(n)$.Inspired by the spread of discontent as in the 2016 presidential election, we consider a voter modelin which 0's are ordinary voters and 1's are zealots. Thinking of a social network, but desiring the simplicity of an infinite object that can have a nontrivial stationary distribution, space is represented by a tree. The dynamics are a variant of the biased voter: if $x$ has degree $d(x)$ thenat rate $d(x)p_k$ the individual at $x$ consults $k\ge 1$ neighbors.If at least one neighbor is 1, they adopt state 1, otherwise they become 0. In addition at rate $p_0$individuals with opinion 1 change to 0. As in the contact process on trees, we are interested in determining when the zealotssurvive and when they will survive locally, i.e., the root of the tree is in state 1 infinitely often.