Consensus of the Hegselmann-Krause opinion formation model with time delay

TL;DR

This paper analyzes the Hegselmann-Krause opinion formation model with time delays, establishing conditions for exponential consensus, studying the mean-field limit, and validating results with numerical tests.

Contribution

It introduces a comprehensive analysis of consensus behavior in delayed Hegselmann-Krause models, including mean-field limits and numerical validation.

Findings

Exponential consensus achieved under small time delay.

Global existence and uniqueness of solutions for the continuum model.

Numerical tests support theoretical results.

Abstract

In this paper, we study Hegselmann-Krause models with a time-variable time delay. Under appropriate assumptions, we show the exponential asymptotic consensus when the time delay satisfies a suitable smallness assumption. Our main strategies for this are based on Lyapunov functional approach and careful estimates on the trajectories. We then study the mean-field limit from the many-individual Hegselmann-Krause equation to the continuity-type partial differential equation as the number N of individuals goes to infinity. For the limiting equation, we prove global-in-time existence and uniqueness of measure-valued solutions. We also use the fact that constants appearing in the consensus estimates for the particle system are independent of N to extend the exponential consensus result to the continuum model. Finally, some numerical tests are illustrated.