Behavior of social dynamical models ii: Clustering for some multitype particle systems with confidence threshold

Abstract

We generalize the clustering theorem by Lanchier (2012) on the infinite one-dimensional integer lattice & for the constrained voter model and the two-feature two-trait Axelrod model to multitype biased models with confidence threshold. Types are represented by a connected graph ;&, and dynamics is described as follows. At independent exponential times for each site of type i, one of the neighboring sites is chosen randomly, and its type j is adopted if i, j are adjacent on γ. Starting from a product measure with positive type densities, the clustering theorem dictates that fluctuation and clustering occurs, i.e., each site changes type at arbitrary large times and looking at a finite interval consensus is reached asymptotically with probability 1, if there is one or two vertices of γ adjacent to all other vertices but each other. Additionally, we propose a simple definition of clustering on a finite set, in which case one can apply the clustering theorem that justifies known previous claims. © 2012 Springer-Verlag Berlin Heidelberg.