Majority versus minority dynamics: Phase transition in an interacting two-state spin system

TL;DR

This paper introduces a simple opinion dynamics model with a phase transition at a critical probability, showing how local majority or minority influence affects global consensus or mixed states, with analytical and numerical results.

Contribution

It presents a new model of opinion dynamics exhibiting a phase transition at a critical influence probability p_c, connecting majority/minority influence with voter model behavior.

Findings

Phase transition at p_c=2/3 for group size G=3 in all dimensions.

For p>p_c, the system reaches a global majority consensus.

At p=p_c, the system exhibits voter model-like coarsening and algebraic decay of correlations.

Abstract

We introduce a simple model of opinion dynamics in which binary-state agents evolve due to the influence of agents in a local neighborhood. In a single update step, a fixed-size group is defined and all agents in the group adopt the state of the local majority with probability p or that of the local minority with probability 1-p. For group size G=3, there is a phase transition at p_c=2/3 in all spatial dimensions. For p>p_c, the global majority quickly predominates, while for p<p_c, the system is driven to a mixed state in which the densities of agents in each state are equal. For p=p_c, the average magnetization (the difference in the density of agents in the two states) is conserved and the system obeys classical voter model dynamics. In one dimension and within a Kirkwood decoupling scheme, the final magnetization in a finite-length system has a non-trivial dependence on the initial magnetization for all p.ne.p_c, in agreement with numerical results. At p_c, the exact 2-spin correlation functions decay algebraically toward the value 1 and the system coarsens as in the classical voter model.