Convergence Properties of the Heterogeneous Deffuant-Weisbuch Model

TL;DR

The paper proves almost-sure convergence for the heterogeneous Deffuant-Weisbuch model with weighting parameter $\mu\in[\tfrac{1}{2},1)$, showing that each agent's opinion converges to a limit $x^*$ and that, in the limit, any pair of opinions are either equal or separated by more than the corresponding confidence bounds. The authors introduce a DW-control system and maximal-confidence clusters to structure the proof, and they design finite-time control sequences that force the system toward a topology amenable to contraction analysis, yielding an exponential rate in mean-square convergence. They also derive sufficient (and in some cases necessary) conditions for almost-sure consensus, notably showing that $r_1\ge1$ guarantees consensus for random initial states with positive density, and they provide a probabilistic consensus analysis under bounded density assumptions. The results advance the mathematical understanding of heterogenous BC models and offer a framework that could inform both theory and simulations in opinion dynamics with state-dependent interaction structures.

Abstract

The Deffuant-Weisbuch (DW) model is a bounded-confidence opinion dynamics model that has attracted much recent interest. Despite its simplicity and appeal, the DW model has proved technically hard to analyze and its most basic convergence properties, easy to observe numerically, are only conjectures. This paper solves the convergence problem for the heterogeneous DW model with the weighting factor not less than $1/2$. We establish that, for any positive confidence bounds and initial values, the opinion of each agent will converge to a limit value almost surely, and the convergence rate is exponential in mean square. Moreover, we show that the limiting opinions of any two agents either are the same or have a distance larger than the confidence bounds of the two agents. Finally, we provide some sufficient conditions for the heterogeneous DW model to reach consensus.

Figures (4)

• Figure 1: A convergent simulation of the heterogeneous DW model
• Figure 2: The estimated consensus probability with respect to the maximal confidence bound $r_{\max}$, where the error bars denote the standard deviations of the estimated probability of reaching consensus at the points of $r_{\max}$.
• Figure 3: Two MC clusters $C_1(x)$ and $C_2(x)$. The distance between two adjacent nodes in $C_1(x)$ (or $C_2(x)$) is not bigger than the $r_1$ (or $r_2$), but the distance between the agents $3$ and $2$ is bigger than $r_1$.
• Figure 4: An example for the relation of $C_i(x(0))$, $\underline{A}(0)$, and agents $j$, $k$, $l$, and $i_0'$.