Network Analysis of the State Space of Discrete Dynamical Systems

TL;DR

This paper investigates the network structures derived from elementary cellular automata dynamics, analyzing how local and global properties scale with system size to distinguish between simple and complex behaviors.

Contribution

It introduces analytical scaling laws for node in-degree and defines path diversity as a new global network measure to differentiate CA complexity classes.

Findings

Scaling of largest node in-degree varies with CA rule

Path diversity correlates with CA complexity class

Complex CA exhibit non-trivial scaling in network measures

Abstract

We study networks representing the dynamics of elementary 1-d cellular automata (CA) on finite lattices. We analyze scaling behaviors of both local and global network properties as a function of system size. The scaling of the largest node in-degree is obtained analytically for a variety of CA including rules 22, 54 and 110. We further define the \emph{path diversity} as a global network measure. The co-appearance of non-trivial scaling in both hub size and path diversity separates simple dynamics from the more complex behaviors typically found in Wolfram's Class IV and some Class III CA.