https://hal.inria.fr/hal-01196145Richard, AdrienAdrienRichardLaboratoire d'Informatique, Signaux, et Systèmes de Sophia-Antipolis (I3S) / Equipe BIOINFO - Laboratoire I3S - MDSC - Modèles Discrets pour les Systèmes Complexes - I3S - Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis - UNS - Université Nice Sophia Antipolis (1965 - 2019) - COMUE UCA - COMUE Université Côte d'Azur (2015-2019) - CNRS - Centre National de la Recherche Scientifique - UCA - Université Côte d'AzurA fixed point theorem for Boolean networks expressed in terms of forbidden subnetworksHAL CCSD2011Boolean networkfixed pointself-dual Boolean functiondiscrete Jacobian matrixfeedback circuit[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM][MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS][NLIN.NLIN-CG] Nonlinear Sciences [physics]/Cellular Automata and Lattice Gases [nlin.CG][MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]Episciences Iam, CoordinationFatès, Nazim and Goles, Eric and Maass, Alejandro and Rapaport, Iván2015-09-09 11:15:062023-03-24 14:53:012015-09-09 11:16:45enConference papershttps://hal.inria.fr/hal-01196145/document10.46298/dmtcs.2978application/pdf1We are interested in fixed points in Boolean networks, $\textit{i.e.}$ functions $f$ from $\{0,1\}^n$ to itself. We define the subnetworks of $f$ as the restrictions of $f$ to the hypercubes contained in $\{0,1\}^n$, and we exhibit a class $\mathcal{F}$ of Boolean networks, called even or odd self-dual networks, satisfying the following property: if a network $f$ has no subnetwork in $\mathcal{F}$, then it has a unique fixed point. We then discuss this "forbidden subnetworks theorem''. We show that it generalizes the following fixed point theorem of Shih and Dong: if, for every $x$ in $\{0,1\}^n$, there is no directed cycle in the directed graph whose the adjacency matrix is the discrete Jacobian matrix of $f$ evaluated at point $x$, then $f$ has a unique fixed point. We also show that $\mathcal{F}$ contains the class $\mathcal{F'}$ of networks whose the interaction graph is a directed cycle, but that the absence of subnetwork in $\mathcal{F'}$ does not imply the existence and the uniqueness of a fixed point.