Consensus Convergence with Stochastic Effects

TL;DR

This paper analyzes a stochastic opinion dynamics model, using stability analysis of a Fokker-Planck equation to predict cluster formation, their properties, and long-term behavior, supported by numerical simulations.

Contribution

It introduces a novel analytical framework combining stability analysis and numerical simulations to study opinion clustering under stochastic influences.

Findings

Number of clusters estimated analytically

Critical randomness threshold for cluster formation identified

Numerical simulations confirm analytical predictions

Abstract

We consider a stochastic, continuous state and time opinion model where each agent's opinion locally interacts with other agents' opinions in the system, and there is also exogenous randomness. The interaction tends to create clusters of common opinion. By using linear stability analysis of the associated nonlinear Fokker-Planck equation that governs the empirical density of opinions in the limit of infinitely many agents, we can estimate the number of clusters, the time to cluster formation and the critical strength of randomness so as to have cluster formation. We also discuss the cluster dynamics after their formation, the width and the effective diffusivity of the clusters. Finally, the long term behavior of clusters is explored numerically. Extensive numerical simulations confirm our analytical findings.