Interaction graphs of isomorphic automata networks II: universal dynamics

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 $[n]$ that contains an arc from $j$ to $i$ if $f_i$ depends on input $j$. What can be said on the set $\mathbb{G}(f)$ 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 $K_n\in \mathbb{G}(f)$ whenever $f$ is not constant nor the identity (and $n\geq 5$). In this paper, taking the opposite direction, we prove that there exist universal automata networks $f$, in that $\mathbb{G}(f)$ contains all the digraphs on $[n]$, excepted the empty one. Actually, we prove that the presence of only three specific digraphs in $\mathbb{G}(f)$ 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 $q\geq 3$, there exists almost universal functions, that is, functions $f:Q^n\to Q^n$ such that the probability that a random digraph belongs to $\mathbb{G}(f)$ tends to $1$ as $n\to\infty$. We do not know if this holds in the binary case $q=2$, providing only partial results.