Network Analysis of the State Space of Discrete Dynamical Systems

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.