Interaction graphs of isomorphic automata networks I: complete digraph and minimum in-degree

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. Here, we report some basic facts. First, we prove that if $n\geq 5$ or $q\geq 3$ and $f$ is neither the identity nor constant, then $\mathbb{G}(f)$ always contains the complete digraph $K_n$, with $n^2$ arcs. Then, we prove that $\mathbb{G}(f)$ always contains a digraph whose minimum in-degree is bounded as a function of $q$. Hence, if $n$ is large with respect to $q$, then $\mathbb{G}(f)$ cannot only contain $K_n$. However, we prove that $\mathbb{G}(f)$ can contain only dense digraphs, with at least $\lfloor n^2/4 \rfloor$ arcs.