Isomorphic Boolean networks and dense interaction graphs
| Autor: | Marchetto, Aymeric Picard, Richard, Adrien |
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| Rok vydání: | 2021 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 13 pages |
| Databáze: | arXiv |
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