Behavior of social dynamical models i: Fixation in the symmetric cyclic system (with paradoxical effect in the six-color automaton)

Abstract

We modify the transition rule of the N-color cyclic particle system, such that the arising system is applied for social dynamics. An asynchronous update is examined, where each two neighboring sites of a social network interact at exponential rate, and one of the sites adopts the color (opinion) of a randomly chosen neighbor, provided that their colors are adjacent on the cycle C N that represents the colors. We show that starting from independent and uniformly distributed colors on the integer lattice, each site fixates to a final color with probability 1 if N∈≥∈5, which is sharp due to Lanchier (2012). Moreover, we conduct simulations with appropriate cellular automata and find that the frequency of the long observed Condorcet's paradox of voting increases in the fixation region of the six-color automaton. © 2012 Springer-Verlag Berlin Heidelberg.