Effect of initial concentration and spatial heterogeneity of active agent distribution on opinion dynamics

We analyze the effect of spatial heterogeneity in the initial spin distribution on spin dynamics in a three-state square lattice divided into spatial cells (districts). In the spirit of the statistical mechanics of social impact, we introduce a dominant influence rule (DIR), according to which, in a single update step, a chosen node adopts the state determined by the influence of its discussion group formed by the node itself and its neighbors within one or more coordination spheres. In contrast to models based on some form of majority rule (MR), a system governed by the DIR is easily trapped in a stable non-consensus state, if all nodes of the discussion group have the same weight of influence. To ensure that a consensus in the DIR system is necessarily reached, we need to put a stochastic process in the update rule. Further, the stochastic DIR model is used as a starting point for understanding the effect of spatial heterogeneity of active agent (non-zero spin) distribution on the exit probabilities. Initially, the positive and negative spins (active agents) are assigned to some nodes with non-uniform spatial distributions; while the rest of the nodes remain in the state with spin zero (uncommitted voters). By varying the relative means and skewness of the initial spin distributions, we observe critical behaviors of exit probabilities in finite size systems. The critical exponents are obtained by Monte Carlo simulations. The results of numerical simulations are discussed in the context of social dynamics.