Equilibrium behavior in a nonequilibrium system: Ising-doped voter model on complete graph

TL;DR

This paper investigates a mixed opinion model combining Ising and voter agents on complete and random graphs, revealing that magnetization behavior mimics the Ising model despite nonequilibrium conditions, with susceptibility and ordering influenced by agent concentration.

Contribution

It demonstrates that a mixture of Ising and voter agents exhibits Ising-like magnetization behavior on complete graphs even when detailed balance is broken.

Findings

Magnetization in the mixed model follows the Ising equation for any p>0.

Susceptibility diverges at the critical temperature where magnetization vanishes.

A small concentration of Ising agents can induce ferromagnetic ordering on random graphs.

Abstract

While the Ising model belongs to the realm of equilibrium statistical mechanics, the voter model is an example of a nonequilibrium system. We examine an opinion formation model, which is a mixture of Ising and voter agents with concentrations $p$ and $1-p$, respectively. Although in our model for $p<1$ a detailed balance is violated, on a complete graph the average magnetization in the stationary state for any $p>0$ is shown to satisfy the same equation as for the pure Ising model ($p=1$). Numerical simulations confirm such a behavior, but the equivalence with the pure Ising model apparently holds only for magnetization. Susceptibility in our model diverges at the temperature at which magnetization vanishes, but its values depend on the concentration~$p$. Simulations on a random graph also show that a small concentration of Ising agents is sufficient to induce a ferromagnetic ordering.