Stochastic Minority on Graphs

TL;DR

This paper explores how asynchronous updates and different graph structures influence the behavior and convergence time of the classical Minority rule in cellular automata, revealing complex dynamics beyond regular grids.

Contribution

It provides a comparative analysis of asynchronous Minority dynamics on various graph topologies, highlighting the impact of topology on convergence and stability.

Findings

Convergence time varies significantly with graph topology.

Trees exhibit non-trivial behavior under asynchronous Minority.

Asynchronism can induce complex dynamics beyond regular grid cases.

Abstract

Cellular automata have been mainly studied on very regular graphs carrying the vertices (like lines or grids) and under synchronous dynamics (all vertices update simultaneously). In this paper, we study how the asynchronism and the graph act upon the dynamics of the classical Minority rule. Minority has been well-studied for synchronous updates and is thus a reasonable choice to begin with. Yet, beyond its apparent simplicity, this rule yields complex behaviors when asynchronism is introduced. We investigate the transitory part as well as the asymptotic behavior of the dynamics under full asynchronism (also called sequential: only one random vertex updates at each time step) for several types of graphs. Such a comparative study is a first step in understanding how the asynchronous dynamics is linked to the topology (the graph). Previous analyses on the grid [1,2] have observed that Minority seems to induce fast stabilization. We investigate here this property on arbitrary graphs using tools such as energy, particles and random walks. We show that the worst case convergence time is, in fact, strongly dependent on the topology. In particular, we observe that the case of trees is non trivial.