The Deffuant model on $\mathbb{Z}$ with higher-dimensional opinion spaces

TL;DR

This paper extends the Deffuant bounded confidence model to higher-dimensional opinion spaces on the integer lattice, analyzing phase transitions based on a critical confidence threshold.

Contribution

It introduces a vector-valued opinion model with general metrics on $ ext{Z}$, generalizing prior real-valued models and identifying phase transition phenomena.

Findings

Existence of a critical confidence threshold for phase transition

Extension of the model to vector-valued opinions and general metrics

Similar phase transition behavior as in the univariate case

Abstract

When it comes to the mathematical modelling of social interaction patterns, a number of different models have emerged and been studied over the last decade, in which individuals randomly interact on the basis of an underlying graph structure and share their opinions. A prominent example of the so-called bounded confidence models is the one introduced by Deffuant et al.: Two neighboring individuals will only interact if their opinions do not differ by more than a given threshold $\theta$. We consider this model on the line graph $\mathbb{Z}$ and extend the results that have been achieved for the model with real-valued opinions by considering vector-valued opinions and general metrics measuring the distance between two opinion values. Just as in the univariate case, there exists a critical value for $\theta$ at which a phase transition in the long-term behavior takes place.