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

TL;DR

This paper investigates the set of interaction graphs of automata networks isomorphic to a given network, revealing conditions under which the complete digraph appears and bounds on minimum in-degree.

Contribution

It establishes that for large enough networks or alphabet sizes, the interaction graph set always includes the complete digraph and bounds the minimum in-degree of graphs in the set.

Findings

The set of interaction graphs always contains the complete digraph under certain conditions.

Bounds on the minimum in-degree of graphs in the interaction graph set are established.

The interaction graph set can include only dense digraphs with at least half the maximum number of arcs.

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 $[n]$ 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.