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Register a new userThe q-voter model on the torus
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In the $q$-voter model, the voter at $x$ changes its opinion at rate $f_x^q$, where $f_x$ is the fraction of neighbors with the opposite opinion. Mean-field calculations suggest that there should be coexistence between opinions if $q<1$ and clustering if $q>1$. This model has been extensively studied by physicists, but we do not know of any rigorous results. In this paper, we use the machinery of voter model perturbations to show that the conjectured behavior holds for $q$ close to 1. More precisely, we show that if $q<1$, then for any $m<infty$ the process on the three-dimensional torus with $n$ points survives for time $n^m$, and after an initial transient phase has a density that it is always close to 1/2. If $q>1$, then the process rapidly reaches fixation on one opinion. It is interesting to note that in the second case the limiting ODE (on its sped up time scale) reaches 0 at time $log n$ but the stochastic process on the same time scale dies out at time $(1/3)log n$.
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Inspired by the spread of discontent as in the 2016 presidential election, we consider a voter model in which 0s are ordinary voters and 1s 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 $d(x)$ then at rate $d(x)p_k$ the individual at $x$ consults $kge 1$ 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 zealots survive and when they will survive locally.
We introduce a non-linear variant of the voter model, the q-voter model, in which q neighbors (with possible repetition) are consulted for a voter to change opinion. If the q neighbors agree, the voter takes their opinion; if they do not have an unanimous opinion, still a voter can flip its state with probability $epsilon$. We solve the model on a fully connected network (i.e. in mean-field) and compute the exit probability as well as the average time to reach consensus. We analyze the results in the perspective of a recently proposed Langevin equation aimed at describing generic phase transitions in systems with two ($Z_2$ symmetric) absorbing states. We find that in mean-field the q-voter model exhibits a disordered phase for high $epsilon$ and an ordered one for low $epsilon$ with three possible ways to go from one to the other: (i) a unique (generalized voter-like) transition, (ii) a series of two consecutive Ising-like and directed percolation transition, and (iii) a series of two transitions, including an intermediate regime in which the final state depends on initial conditions. This third (so far unexplored) scenario, in which a new type of ordering dynamics emerges, is rationalized and found to be specific of mean-field, i.e. fluctuations are explicitly shown to wash it out in spatially extended systems.
Given a continuous time Markov Chain ${q(x,y)}$ on a finite set $S$, the associated noisy voter model is the continuous time Markov chain on ${0,1}^S$, which evolves in the following way: (1) for each two sites $x$ and $y$ in $S$, the state at site $x$ changes to the value of the state at site $y$ at rate $q(x,y)$; (2) each site rerandomizes its state at rate 1. We show that if there is a uniform bound on the rates ${q(x,y)}$ and the corresponding stationary distributions are almost uniform, then the mixing time has a sharp cutoff at time $log|S|/2$ with a window of order 1. Lubetzky and Sly proved cutoff with a window of order 1 for the stochastic Ising model on toroids; we obtain the special case of their result for the cycle as a consequence of our result. Finally, we consider the model on a star and demonstrate the surprising phenomenon that the time it takes for the chain started at all ones to become close in total variation to the chain started at all zeros is of smaller order than the mixing time.
We present a detailed investigation of the behavior of the nonlinear q-voter model for opinion dynamics. At the mean-field level we derive analytically, for any value of the number q of agents involved in the elementary update, the phase diagram, the exit probability and the consensus time at the transition point. The mean-field formalism is extended to the case that the interaction pattern is given by generic heterogeneous networks. We finally discuss the case of random regular networks and compare analytical results with simulations.
We study the scaling limit of a large class of voter model perturbations in one dimension, including stochastic Potts models, to a universal limiting object, the continuum voter model perturbation. The perturbations can be described in terms of bulk and boundary nucleations of new colors (opinions). The discrete and continuum (space) models are obtained from their respective duals, the discrete net with killing and Brownian net with killing. These determine the color genealogy by means of reduced graphs. We focus our attention on models where the voter and boundary nucleation dynamics depend only on the colors of nearest neighbor sites, for which convergence of the discrete net with killing to its continuum analog was proved in an earlier paper by the authors. We use some detailed properties of the Brownian net with killing to prove voter model perturbations convergence to its continuum counterpart. A crucial property of reduced graphs is that even in the continuum, they are finite almost surely. An important issue is how vertices of the continuum reduced graphs are strongly approximated by their discrete analogues.
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