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

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 via simulations over 20 years, its convergence proof is open. Our previous paper (Chen et al., 2020) solves the problem for the case of uniform weighting factors greater than or equal to 1/2, but the general case remains unresolved. This article 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 article 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.