Isomorphic Boolean networks and dense interaction graphs

TL;DR

This paper investigates the set of interaction graphs of Boolean networks that are isomorphic to a given network, revealing that for most networks, this set is large and contains dense graphs, with some having at least a quarter of possible arcs.

Contribution

It introduces the study of the set of interaction graphs of isomorphic Boolean networks, providing foundational results on their size and density.

Findings

For networks with n≥5, the set of interaction graphs includes the complete digraph.

Existence of Boolean networks where all interaction graphs are densely connected.

The size of the set of interaction graphs can be at least two for certain networks.

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.