Dynamics of non-conservative voters

We study a family of opinion formation models in one dimension where the propensity for a voter to align with its local environment depends non-linearly on the fraction of disagreeing neighbors. Depending on this non-linearity in the voting rule, the population may exhibit a bias toward zero magnetization or toward consensus, and the average magnetization is generally not conserved. We use a decoupling approximation to truncate the equation hierarchy for multi-point spin correlations and thereby derive the probability to reach a final state of ↑ consensus as a function of the initial magnetization. The case when voters are influenced by more distant voters is also considered by investigating the Sznajd model. Copyright c © EPLA, 2008 It often happens that individuals change their attitudes, behaviors and/or morals, to conform to those of their acquaintances. Perhaps the simplest description of this conformity is the voter model [1], where each node of a graph (i.e., the social network) is occupied by a voter that has one of two opinions, ↑ or ↓. In the voter model, the population evolves by: i) picking a random voter; ii) the selected voter adopts the state of a randomly chosen neighbor; iii) repeating these steps ad infinitum or until a finite system necessarily reaches consensus. Figuratively, voters have no self-confidence and merely follow one of their neighbors. With this dynamics, a voter changes opinion with a probability pf that equals the fraction f of disagreeing neighbors. This proportionality rule leads to the conservation of the average opinion in the system, a feature that renders the voter model soluble in all dimensions [1,2]. However, the specific rule pf = f is one among many possible and socially plausible relations between pf and f . In this work, we generalize the voter model so that pf depends non-linearly on f . Non-linear voter models have been discussed previously, primarily by numerical simulations in two dimensions [3]. Here we focus on one dimension, where the range of possibilities for the nonlinearity is limited. In one dimension, a voter may be confronted by 0, 1, or 2 disagreeing neighbors. It is (a)E-mail: [email protected] (b)E-mail: [email protected] Fig. 1: Update illustration. A random voter changes state with probability p1 if it has 1 disagreeing neighbor (left), and with probability p2 if it has 2 disagreeing neighbors (right). natural to impose p0 = 0, so that no evolution occurs when there is local consensus. Then the most general description of the system requires two parameters, p1 and p2 (fig. 1). One parameter, which we choose to be p1, determines the overall time scale and is thus immaterial. The only relevant parameter then is γ = p2/p1. When γ = 2, one recovers the classical voter model. When γ > 2, the combined effect of two neighbors is more than twice that of one neighbor. Equivalently, voters can be viewed as having a conviction for their opinion and that strong peer pressure is needed to change opinion. As γ→∞, voters only change opinion when are confronted by a unanimity of opposite-opinion voters [4]. In contrast, when γ < 2, one disagreeing neighbor is more effective in triggering an opinion change than in the classical voter model. When γ = 1, one recovers the vacillating voter model [5,6]