(Biased) Majority Rule Cellular Automata

TL;DR

This paper studies biased and unbiased majority rule cellular automata on a 2D torus, revealing threshold behaviors with two phase transitions that determine whether the system becomes all red, all blue, or coexists.

Contribution

It proves that both models exhibit two phase transitions on a 2D torus, characterizing the thresholds for color dominance and coexistence.

Findings

Threshold behavior with two phase transitions identified

System reaches monochromatic or coexistence states in O(n^2) steps

Different initial probabilities lead to distinct long-term configurations

Abstract

Consider a graph $G=(V,E)$ and a random initial vertex-coloring, where each vertex is blue independently with probability $p_{b}$, and red with probability $p_r=1-p_b$. In each step, all vertices change their current color synchronously to the most frequent color in their neighborhood and in case of a tie, a vertex conserves its current color; this model is called majority model. If in case of a tie a vertex always chooses blue color, it is called biased majority model. We are interested in the behavior of these deterministic processes, especially in a two-dimensional torus (i.e., cellular automaton with (biased) majority rule). In the present paper, as a main result we prove both majority and biased majority cellular automata exhibit a threshold behavior with two phase transitions. More precisely, it is shown that for a two-dimensional torus $T_{n,n}$, there are two thresholds $0\leq p_1, p_2\leq 1$ such that $p_b \ll p_1$, $p_1 \ll p_b \ll p_2$, and $p_2 \ll p_b$ result in monochromatic configuration by red, stable coexistence of both colors, and monochromatic configuration by blue, respectively in $\mathcal{O}(n^2)$ number of steps