Freezing and Slow Evolution in a Constrained Opinion Dynamics Model

TL;DR

This paper investigates a constrained opinion dynamics model where interactions lead to consensus or frozen states, revealing how initial conditions influence long-term opinion distributions and the emergence of frozen populations.

Contribution

It introduces a spin-1 Ising model mapping to analyze opinion formation with a focus on frozen states and domain growth dynamics.

Findings

Frozen states depend on initial centrist density rho_0

Domain size grows as L^{2*psi} with psi related to rho_0

Long-time approach follows a t^{-psi} tail

Abstract

We study opinion formation in a population that consists of leftists, centrists, and rightist. In an interaction between neighboring agents, a centrist and a leftist can become both centrists or leftists (and similarly for a centrist and a rightist). In contrast, leftists and rightists do not affect each other. The initial density of centrists rho_0 controls the evolution. With probability rho_0 the system reaches a centrist consensus, while with probability 1-rho_0 a frozen population of leftists and rightists results. In one dimension, we determine this frozen state and the opinion dynamics by mapping the system onto a spin-1 Ising model with zero-temperature Glauber kinetics. In the frozen state, the length distribution of single-opinion domains has an algebraic small-size tail x^{-2(1-psi)} and the average domain size grows as L^{2*psi}, where L is the system length. The approach to this frozen state is governed by a t^{-psi} long-time tail with psi-->2*rho_0/pi as rho_0-->0.