Further results on consensus formation in the Deffuant model

TL;DR

This paper extends the analysis of the Deffuant model for opinion dynamics, identifying phase transition thresholds for consensus formation across various initial opinion distributions and different lattice structures.

Contribution

It generalizes a key result on the critical threshold for consensus in the Deffuant model to broader initial opinion distributions and lattice types.

Findings

Determined the critical threshold for consensus on $ heta$ for various initial opinion distributions.

Extended phase transition analysis to higher-dimensional lattices and percolation clusters.

Provided theoretical insights into the long-term behavior of the Deffuant model.

Abstract

The so-called Deffuant model describes a pattern for social interaction, in which two neighboring individuals randomly meet and share their opinions on a certain topic, if their discrepancy is not beyond a given threshold $\theta$. The major focus of the analyses, both theoretical and based on simulations, lies on whether these single interactions lead to a global consensus in the long run or not. First, we generalize a result of Lanchier for the Deffuant model on $\mathbb{Z}$, determining the critical value for $\theta$ at which a phase transition of the long term behavior takes place, to other distributions of the initial opinions than i.i.d.\ uniform on $[0,1]$. Then we shed light on the situations where the underlying line graph $\mathbb{Z}$ is replaced by higher-dimensional lattices $\mathbb{Z}^d,\ d\geq2$, or the infinite cluster of supercritical i.i.d.\ bond percolation on these lattices.