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.Comment: 6 pages, 6 figure
