Convergence of the Heterogeneous Deffuant-Weisbuch Model: A Complete Proof and Some Extensions

TL;DR

This paper proves that in the heterogeneous Deffuant-Weisbuch opinion model, all agents' opinions converge to fixed points with probability one, resolving a long-standing open problem and analyzing factors affecting convergence speed.

Contribution

It provides a complete proof of convergence for the general heterogeneous DW model, extending previous results limited to specific cases.

Findings

Opinions of agents converge to fixed points with probability one.

Convergence speed can be arbitrarily slow under certain conditions.

Resolved the open conjecture for the general heterogeneous DW model.

Abstract

The Deffuant-Weisbuch (DW) model is a well-known bounded-confidence opinion dynamics that has attracted wide interest. Although the heterogeneous DW model has been studied by simulations over $20$ years, its convergence proof is open. Our previous paper \cite{GC-WS-WM-FB:20} solves the problem for the case of uniform weighting factors greater than or equal to $1/2$, but the general case remains unresolved. This paper considers the DW model with heterogeneous confidence bounds and heterogeneous (unconstrained) weighting factors and shows that, with probability one, the opinion of each agent converges to a fixed vector. In other words, this paper resolves the convergence conjecture for the heterogeneous DW model. Our analysis also clarifies how the convergence speed may be arbitrarily slow under certain parameter conditions.