A particle method for continuous Hegselmann-Krause opinion dynamics

TL;DR

This paper introduces a particle method for solving a differential-integral equation modeling opinion dynamics, demonstrating second-order convergence and a concentration inequality, thus advancing numerical analysis in social dynamics modeling.

Contribution

It derives a new differential-integral equation for opinion dynamics and proposes a particle method with proven convergence and a novel concentration inequality.

Findings

Second-order convergence of the particle method

Equivalence of the integral equation to a system of differential equations

Monotonic increase in a measure of concentration

Abstract

We derive a differential-integral equation akin to the Hegselmann-Krause model of opinion dynamics, and propose a particle method for solving the equation. Numerical experiments demonstrate second-order convergence of the method in a weak sense. We also show that our differential-integral equation can equivalently be stated as a system of differential equations. An integration-by-parts argument that would typically yield an energy dissipation inequality in physical problems then yields a concentration inequality, showing that a natural measure of concentration increases monotonically.