Optimal condition for asymptotic consensus in the Hegselmann-Krause model with finite speed of information propagation

TL;DR

This paper establishes the optimal conditions under which the Hegselmann-Krause model with finite information speed guarantees global consensus, highlighting the necessity of positive influence functions and agent speeds below the information speed.

Contribution

It proves that global consensus is achieved under minimal assumptions on the influence function and agent speeds, considering finite information propagation speed.

Findings

Global consensus always reached under specified conditions.

Necessary influence function positivity for consensus.

Agent speeds below information propagation speed are essential.

Abstract

We prove that asymptotic global consensus is always reached in the Hegselmann-Krause model with finite speed of information propagation $\mathfrak{c}>0$ under minimal (i.e., necessary) assumptions on the influence function. In particular, we assume that the influence function is globally positive, which is necessary for reaching global consensus, and such that the agents move with speeds strictly less than $\mathfrak{c}$, which is necessary for well-posedness of solutions. From this point of view, our result is optimal. The proof is based on the fact that the state-dependent delay, induced by the finite speed of information propagation, is uniformly bounded.