Outer-totalistic cellular automata on graphs

TL;DR

This paper introduces a formalism for implementing cellular automata on arbitrary graph topologies, identifies symmetries, and analyzes how network structure influences the complexity of automata patterns.

Contribution

It provides a novel formalism for cellular automata on arbitrary graphs and explores the impact of topology on automata complexity and symmetry.

Findings

Symmetry operation identified in elementary cellular automata

Number of complex patterns decreases with larger neighborhoods in regular graphs

Scale-free and metabolic networks can generate complex dynamics

Abstract

We present an intuitive formalism for implementing cellular automata on arbitrary topologies. By that means, we identify a symmetry operation in the class of elementary cellular automata. Moreover, we determine the subset of topologically sensitive elementary cellular automata and find that the overall number of complex patterns decreases under increasing neighborhood size in regular graphs. As exemplary applications, we apply the formalism to complex networks and compare the potential of scale-free graphs and metabolic networks to generate complex dynamics.