Topology regulates pattern formation capacity of binary cellular automata on graphs

TL;DR

This paper investigates how different graph topologies influence the pattern formation and dynamic behavior of binary cellular automata, revealing topology-induced transitions between Wolfram classes and quantifying pattern complexity.

Contribution

It introduces degree-dependent rules on various graph topologies and demonstrates topology-driven transitions in cellular automata dynamics without changing update rules.

Findings

Topology variations induce transitions between Wolfram classes.

Pattern formation capacity varies with graph type and entropy measures.

A mean-field model explains dynamics on random graphs.

Abstract

We study the effect of topology variation on the dynamic behavior of a system with local update rules. We implement one-dimensional binary cellular automata on graphs with various topologies by formulating two sets of degree-dependent rules, each containing a single parameter. We observe that changes in graph topology induce transitions between different dynamic domains (Wolfram classes) without a formal change in the update rule. Along with topological variations, we study the pattern formation capacities of regular, random, small-world and scale-free graphs. Pattern formation capacity is quantified in terms of two entropy measures, which for standard cellular automata allow a qualitative distinction between the four Wolfram classes. A mean-field model explains the dynamic behavior of random graphs. Implications for our understanding of information transport through complex, network-based systems are discussed.