Abstract
Emergence of hierarchies is found in wide range of the societies or
animal clusters. In these phenomena, a little difference of each
individual is enhanced by some causes, and the classes are organized
spontaneously.
We analyze these phenomena by use of a simple agent based model originally proposed by Bonabeau
et al.Physica A 217, 373 (1995). In their model each individual is assumed to have
power and diffuses on the square lattice. If two individuals meet, they fight and then the winner increases its power, while the power of loser decreases. In competition with this battle effect, the powers of
all individuals relax toward zero gradually. This model found to exhibit
a transition from the homogeneous equal society to a heterogeneous
hierarchical society.
We generalize the mean-field analysis of Bonabeau et al. to societies obeying
complex diffusion rules where each individual selects a moving
direction following their power rankings , and we apply this
analysis to the pacifist society model recently
investigated by use of Monte Carlo simulation Physica A 367,
435 (2006). In this society, all individuals hope not to fight
as much as possible. Therefore it always moves to a vacant site if it
exists around them. If all of the nearest neighbor sites are occupied,
it moves to a site occupied by an individual whose power is the smallest
among the neighbors.
We show analytically that the self-organization of hierarchies occurs in two steps as the individual density is increased.
There are three phase, one egalitarian and two hierarchical
states. A difference
between the first and the second hierarchical state is whether winners
exist or not; all individuals belong to either middle class or losers in
the first hierarchical state. We also highlight that the transition from the egalitarian
phase to the first hierarchical is a continuous change in the order
parameter and the second transition causes a discontinuous jump in the
order parameter.
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