Abstract
In the q-voter model, the voter at x changes its opinion at rate fxq, where fx 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<∞ the process on the three-dimensional torus with n points survives for time nm, and after an initial transient phase has a density that it is always close to 1/2. Readers familiar with long time survival results for the contact process and other praticle systems might expect the conjecture to say survival occurs for time exp(γn) with γ>0, however we show persistence does not hold for exp(nβ) with β>1∕3. 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 logn but the stochastic process on the same time scale dies out at time (1∕3)logn.
Highlights
In the linear voter model, the state at time t is ξt : Zd → {0, 1}, where 0 and 1 are two opinions
Vasconclos, Levin, and Pinheiro [29] have considered a version of the q-voter in which the powers q1 and q0 for flipping to 1 and 0 can be different. They did this to study complex contagions which have been used to model the spread of idioms and hashtags on Twitter [26] and in many other situations, see the book by Centola [7]
The rate at which a site x flips to 0 in the q-voter model is fxq, where fx is the fraction of neighbors with the opposite opinion
Summary
In the linear voter model, the state at time t is ξt : Zd → {0, 1}, where 0 and 1 are two opinions. We will instead consider the system on the complete graph in which each site interacts with all the others In this case, the frequency of 1’s, u, satisfies du/dt = −u(1 − u)q + (1 − u)uq = u(1 − u)g(u) where g(u) = uq−1 − (1 − u)q−1. Vasconclos, Levin, and Pinheiro [29] have considered a version of the q-voter in which the powers q1 and q0 for flipping to 1 and 0 can be different They did this to study complex contagions which have been used to model the spread of idioms and hashtags on Twitter [26] and in many other situations, see the book by Centola [7].
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