Time-delayed opinion dynamics with leader-follower interactions: consensus, stability, and mean-field limits

TL;DR

This paper analyzes a time-delayed opinion formation model with leaders and followers, proving consensus convergence and exploring mean-field limits for large populations.

Contribution

It introduces a novel time-delayed opinion dynamics model with leader-follower interactions and provides rigorous convergence and mean-field analysis.

Findings

Exponential convergence to consensus without small delay assumptions.

Existence and uniqueness of mean-field models in two regimes.

Explicit decay estimates for the macroscopic models.

Abstract

We study a time-delayed variant of the Hegselmann-Krause opinion formation model featuring a small group of leaders and a large group of non-leaders. In this model, leaders influence all agents but only interact among themselves. At the same time, non-leaders update their opinions via interactions with their peers and the leaders, with time delays accounting for communication and decision-making lags. We prove the exponential convergence to consensus of the particle system, without imposing smallness assumptions on the delay parameters. Furthermore, we analyze the mean-field limit in two regimes: (i) with a fixed number of leaders and an infinite number of non-leaders, and (ii) with both populations tending to infinity, obtaining existence, uniqueness, and exponential decay estimates for the corresponding macroscopic models.