A simple proof of asymptotic consensus in the Hegselmann-Krause and Cucker-Smale models with renormalization and delay

TL;DR

This paper introduces a straightforward proof technique for asymptotic consensus in discrete Hegselmann-Krause and Cucker-Smale models with renormalization and delay, avoiding restrictions on delay length or initial conditions.

Contribution

It provides a simple, optimal proof method that applies to models with arbitrary delays and extends to mean-field limits, improving upon previous approaches.

Findings

Proof does not restrict maximal delay or initial data.

Applicable to mean-field limits of the models.

Ensures asymptotic consensus and flocking under broad conditions.

Abstract

We present a simple proof of asymptotic consensus in the discrete Hegselmann-Krause model and flocking in the discrete Cucker-Smale model with renormalization and variable delay. It is based on convexity of the renormalized communication weights and a Gronwall-Halanay-type inequality. The main advantage of our method, compared to previous approaches to the delay Hegselmann-Krause model, is that it does not require any restriction on the maximal time delay, or the initial data, or decay rate of the influence function. From this point of view the result is optimal. For the Cucker-Smale model it provides an analogous result in the regime of unconditonal flocking with sufficiently slowly decaying communication rate, but still without any restriction on the length of the maximal time delay. Moreover, we demonstrate that the method can be easily extended to the mean-field limits of both the Hegselmann-Krause and Cucker-Smale systems, using appropriate stability results on the measure-valued solutions.