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Entropy 2015, 17(11), 7658-7679; doi:10.3390/e17117658

Neighborhood Approximations for Non-Linear Voter Models

1
Chair of Systems Design, ETH Zürich, Weinbergstrasse 58, 8092 Zürich, Switzerland
2
Department of Electrical Engineering, Indian Institute of Technology, 208016 Kanpur, India
*
Author to whom correspondence should be addressed.
Academic Editor: J.A. Tenreiro Machado
Received: 2 September 2015 / Revised: 27 October 2015 / Accepted: 4 November 2015 / Published: 10 November 2015
(This article belongs to the Section Complexity)
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Abstract

Non-linear voter models assume that the opinion of an agent depends on the opinions of its neighbors in a non-linear manner. This allows for voting rules different from majority voting. While the linear voter model is known to reach consensus, non-linear voter models can result in the coexistence of opposite opinions. Our aim is to derive approximations to correctly predict the time dependent dynamics, or at least the asymptotic outcome, of such local interactions. Emphasis is on a probabilistic approach to decompose the opinion distribution in a second-order neighborhood into lower-order probability distributions. This is compared with an analytic pair approximation for the expected value of the global fraction of opinions and a mean-field approximation. Our reference case is averaged stochastic simulations of a one-dimensional cellular automaton. We find that the probabilistic second-order approach captures the dynamics of the reference case very well for different non-linearities, i.e., for both majority and minority voting rules, which only partly holds for the first-order pair approximation and not at all for the mean-field approximation. We further discuss the interesting phenomenon of a correlated coexistence, characterized by the formation of large domains of opinions that dominate for some time, but slowly change. View Full-Text
Keywords: opinion dynamics; voter model; pair approximation; higher-order probability distribution; cellular automata opinion dynamics; voter model; pair approximation; higher-order probability distribution; cellular automata
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Schweitzer, F.; Behera, L. Neighborhood Approximations for Non-Linear Voter Models. Entropy 2015, 17, 7658-7679.

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