Fractal opinions among interacting agents

TL;DR

This paper models opinion dynamics among large groups of agents, deriving a mean-field equation, and finds that under certain conditions, the equilibrium opinions form fractal structures, illuminating opinion fragmentation.

Contribution

It introduces a new opinion model with a mean-field limit and reveals fractal structures in equilibrium distributions, advancing understanding of opinion fragmentation phenomena.

Findings

Derivation of a mean-field kinetic equation for the model

Proof of convergence to a unique equilibrium distribution

Identification of fractal structures in the equilibrium support

Abstract

We investigate an opinion model consisting of a large group of interacting agents, whose opinions are represented as numbers in $[-1,1]$. At each update time, two random agents are selected, and the opinion of the first agent is updated based on the opinion of the second (the ``persuader''). We derive the mean-field kinetic equation describing the large population limit of the model, and we provide several quantitative results establishing convergence to the unique equilibrium distribution. Surprisingly, in some range of the model parameters, the support of the equilibrium distribution exhibits a fractal structure. This provides a new mathematical description for the so-called opinion fragmentation phenomenon.