10.46298/dmtcs.2978 https://dmtcs.episciences.org/2978 Richard, Adrien Adrien Richard A fixed point theorem for Boolean networks expressed in terms of forbidden subnetworks We 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. episciences.org feedback circuit Boolean network fixed point self-dual Boolean function discrete Jacobian matrix [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] 2023-03-22 2011-01-01 2011-01-01 en journal article https://hal.science/hal-01196145v1 1365-8050 https://dmtcs.episciences.org/2978/pdf VoR application/pdf Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AP, Automata 2011 - 17th International Workshop on Cellular Automata and Discrete Complex Systems Proceedings Researchers Students