Pattern Selection and Super-patterns in the Bounded Confidence Model
E. Ben-Naim and A.Scheel
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 {\it integers} that follow from
the continued fraction representation of the {\it irrational}
average separation $L$.
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