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 $\left[n\right]$ 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. Here, we report some basic facts. First, we prove that if $n\ge 5$ or $q\ge 3$ and f is neither the identity nor constant, then $G\left(f\right)$ always contains the complete digraph $K_n$, with $n^2$ arcs. Then, we prove that $G\left(f\right)$ 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 $G\left(f\right)$ cannot only contain $K_n$. However, we prove that $G\left(f\right)$ can contain only dense digraphs, with at least $\lfloor n^2/4\rfloor$ arcs.