Direct proof of unconditional asymptotic consensus in the Hegselmann-Krause model with transmission-type delay

TL;DR

This paper provides a direct proof of asymptotic consensus in the Hegselmann-Krause model with transmission delay, using explicit diameter shrinkage estimates and minimal assumptions, applicable to various weight normalization schemes.

Contribution

It introduces a novel direct proof technique for consensus in delayed Hegselmann-Krause models without Lyapunov functionals, applicable to different weight normalization methods.

Findings

Proves asymptotic consensus under minimal assumptions.

Applicable to models with arbitrary delay lengths.

Works for both scaled and normalized communication weights.

Abstract

We present a direct proof of asymptotic consensus in the nonlinear Hegselmann-Krause model with transmission-type delay, where the communication weights depend on the particle distance in phase space. Our approach is based on an explicit estimate of the shrinkage of the group diameter on finite time intervals and avoids the usage of Lyapunov-type functionals or results from nonnegative matrix theory. It works for both the original formulation of the model with communication weights scaled by the number of agents, and the modification with weights normalized a'la Motsch-Tadmor. We pose only minimal assumptions on the model parameters. In particular, we only assume global positivity of the influence function, without imposing any conditions on its decay rate or monotonicity. Moreover, our result holds for any length of the delay.