Partial Convergence of Heterogeneous Hegselmann-Krause Opinion Dynamics

TL;DR

This paper proves partial convergence in heterogeneous Hegselmann-Krause opinion dynamics, showing some agents stabilize while others evolve, with implications for specific cases like large confidence thresholds or small initial differences.

Contribution

It establishes the first partial convergence results for general heterogeneous HK dynamics, advancing understanding of opinion evolution.

Findings

Some agents reach static states in finite time.

Opinions evolve with a minimum distance if no consensus is reached.

Results imply convergence under certain confidence thresholds or initial opinion differences.

Abstract

In opinion dynamics, the convergence of the heterogeneous Hegselmann-Krause (HK) dynamics has always been an open problem for years which looks forward to any essential progress. In this short note, we prove a partial convergence conclusion of the general heterogeneous HK dynamis. That is, there must be some agents who will reach static states in finite time, while the other opinions have to evolve between them with a minimum distance if all the opinions does not reach consensus. And this result leads to the convergence of two special case of heterogeneous HK dynamics: the minimum confidence threshold is large enough, or the initial opinion difference is small enough.