Isomorphic Boolean networks and dense interaction graphs

Aymeric Picard (1) , Adrien Richard (2, 1)

1 I3S - Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis
2 CNRS - Centre National de la Recherche Scientifique

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

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]

Aymeric Picard, Adrien Richard. Isomorphic Boolean networks and dense interaction graphs. Automata 2021, Jul 2021, Marseille, France. ⟨hal-03235769⟩