  We propose a stochastic process of interacting many agents, which is inspired
by rank-based supplanting dynamics commonly observed in a group of Japanese
macaques. In order to characterize the breaking of permutation symmetry with
respect to agents' rank in the stochastic process, we introduce a
rank-dependent quantity, overlap centrality, which quantifies how often a given
agent overlaps with the other agents. We give a sufficient condition in a wide
class of the models such that overlap centrality shows perfect correlation in
terms of the agents' rank in zero-supplanting limit. We also discuss a
singularity of the correlation in the case of interaction induced by a Potts
energy.
