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 ℤ 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 Γ, 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.