Hegselmann--Krause model with environmental noise

TL;DR

This paper analyzes a stochastic version of the Hegselmann-Krause opinion model, incorporating environmental noise, and derives a limiting McKean-Vlasov equation with well-posedness results.

Contribution

It introduces a continuous-time stochastic opinion dynamics model with environmental noise and establishes the propagation of chaos and well-posedness of the limiting equations.

Findings

Derivation of a McKean-Vlasov SDE as the number of agents tends to infinity.

Proof of existence and uniqueness of solutions for the McKean-Vlasov SDE.

Establishment of well-posedness for the associated stochastic Fokker-Planck equation.

Abstract

We study a continuous-time version of the Hegselmann-Krause model describing the opinion dynamics of interacting agents subject to random perturbations. Mathematically speaking, the opinion of agents is modelled by an interacting particle system with a non-Lipschitz continuous interaction force, perturbed by idiosyncratic and environmental noises. Sending the number of agents to infinity, we derive a McKean-Vlasov stochastic differential equation as the limiting dynamic, by establishing propagation of chaos for regularized versions of the noisy opinion dynamics. To that end, we prove the existence of a unique strong solution to the McKean-Vlasov stochastic differential equation as well as well-posedness of the associated non-local, non-linear stochastic Fokker-Planck equation.