Interaction graphs of isomorphic automata networks II: Universal dynamics

Abstract

An automata network with n components over a finite alphabet Q of size q is a discrete dynamical system described by the successive iterations of a function $f:Q^n\to Q^n$. In most applications, the main parameter is the interaction graph of f: the digraph with vertex set $\left[n\right]$ that contains an arc from j to i if $f_i$ depends on input j. What can be said on the set $G\left(f\right)$ of the interaction graphs of the automata networks isomorphic to f? It seems that this simple question has never been studied. In a previous paper, we prove that the complete digraph $K_n$, with $n^2$ arcs, is universal in that $K_n\in G\left(f\right)$ whenever f is not constant nor the identity (and $n\ge 5$). In this paper, taking the opposite direction, we prove that there exist universal automata networks f, in that $G\left(f\right)$ contains all the digraphs on $\left[n\right]$, excepted the empty one. Actually, we prove that the presence of only three specific digraphs in $G\left(f\right)$ implies the universality of f, and we prove that this forces the alphabet size q to have at least n prime factors (with multiplicity). However, we prove that for any fixed $q\ge 3$, there exists almost universal functions, that is, functions $f:Q^n\to Q^n$ such that the probability that a random digraph belongs to $G\left(f\right)$ tends to 1 as $n\to \infty$. We do not know if this holds in the binary case $q=2$, providing only partial results.