A simple mean field model for social interactions: dynamics, fluctuations, criticality

TL;DR

This paper introduces a mean field spin-flip model for social interactions, analyzing its dynamics, phase transitions, and fluctuations, highlighting how inhomogeneities influence critical behavior and fluctuation timescales.

Contribution

It provides a novel analysis of a non-reversible, inhomogeneous mean field model capturing social dynamics, including phase transitions and fluctuation behavior.

Findings

Phase transition occurs in the model.

Inhomogeneities accelerate critical fluctuations.

Long-time fluctuations differ at critical parameter values.

Abstract

We study the dynamics of a spin-flip model with a mean field interaction. The system is non reversible, spacially inhomogeneous, and it is designed to model social interactions. We obtain the limiting behavior of the empirical averages in the limit of infinitely many interacting individuals, and show that phase transition occurs. Then, after having obtained the dynamics of normal fluctuations around this limit, we analize long time fluctuations for critical values of the parameters. We show that random inhomogeneities produce critical fluctuations at a shorter time scale compared to the homogeneous system.