Anti-Diffusion in Continuous Opinion Dynamics

TL;DR

This paper develops a novel nonlinear PDE model for continuous opinion dynamics, revealing how diffusion and anti-diffusion interplay to generate opinion clusters and initial instabilities.

Contribution

It introduces a new nonlinear PDE framework for opinion evolution, enabling analytical approximations of opinion clustering and stability analysis.

Findings

Anti-diffusion contributes to opinion clustering.

Initial instabilities depend on population distribution regions.

The model captures steady state behaviors between consensus and fragmentation.

Abstract

Considerable effort using techniques developed in statistical physics has been aimed at numerical simulations of agent-based opinion models and analysis of their results. Such work has elucidated how various rules for interacting agents can give rise to steady state behaviors in the agent populations that vary between consensus and fragmentation. At the macroscopic population level, analysis has been limited due to the lack of an analytically tractable governing macro-equation for the continuous population state. We use the integro-differential equation that governs opinion dynamics for the continuous probability distribution function of agent opinions to develop a novel nonlinear partial differential equation for the evolution of opinion distributions. The highly nonlinear equation allows for the generation of a system of approximations. We consider three initial population distributions and determine their small-time behavior. Our analysis reveals how the generation of clusters results from the interplay of diffusion and anti-diffusion and how initial instabilities arise in different regions of the population distribution.