Pattern Selection and Super-patterns in the Bounded Confidence Model

TL;DR

This paper analyzes pattern formation in the bounded confidence model of opinion dynamics, revealing how clusters form and stabilize with quasi-periodic spacing and modulations related to continued fractions.

Contribution

It provides an analytical description of cluster spacing and modulations in the bounded confidence model, linking pattern properties to continued fraction representations.

Findings

Clusters form with nearly constant separation L

Traveling wave propagates from boundary into uniform state

Modulations in cluster sizes follow a continued fraction pattern

Abstract

We study pattern formation in the bounded confidence model of opinion dynamics. In this random process, opinion is quantified by a single variable. Two agents may interact and reach a fair compromise, but only if their difference of opinion falls below a fixed threshold. Starting from a uniform distribution of opinions with compact support, a traveling wave forms and it propagates from the domain boundary into the unstable uniform state. Consequently, the system reaches a steady state with isolated clusters that are separated by distance larger than the interaction range. These clusters form a quasi-periodic pattern where the sizes of the clusters and the separations between them are nearly constant. We obtain analytically the average separation between clusters L. Interestingly, there are also very small quasi-periodic modulations in the size of the clusters. The spatial periods of these modulations are a series of integers that follow from the continued fraction representation of the irrational average separation L.