Isomorphic Boolean networks and dense interaction graphs (2105.01914v1)

Abstract: A Boolean network (BN) with $n$ components is a discrete dynamical system described by the successive iterations of a function $f:{0,1}^n\to{0,1}^n$. In most applications, the main parameter is the interaction graph of $f$: the digraph with vertex set ${1,\dots,n}$ that contains an arc from $j$ to $i$ if $f_i$ depends on input $j$. What can be said on the set $\mathcal{G}(f)$ of the interaction graphs of the BNs $h$ isomorphic to $f$, that is, such that $h\circ \pi=\pi\circ f$ for some permutation $\pi$ of ${0,1}^n$? It seems that this simple question has never been studied. Here, we report some basic facts. First, if $n\geq 5$ and $f$ is neither the identity or constant, then $\mathcal{G}(f)$ is of size at least two and contains the complete digraph on $n$ vertices, with $n^2$ arcs. Second, for any $n\geq 1$, there are $n$-component BNs $f$ such that every digraph in $\mathcal{G}(f)$ has at least $n^2/9$ arcs.