# A fixed point theorem for Boolean networks expressed in terms of forbidden subnetworks

1 Laboratoire d'Informatique, Signaux, et Systèmes de Sophia-Antipolis (I3S) / Equipe BIOINFO
Laboratoire I3S - MDSC - Modèles Discrets pour les Systèmes Complexes
Abstract : 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.
Keywords :
Document type :
Conference papers
Domain :

Cited literature [12 references]

https://hal.inria.fr/hal-01196145
Contributor : Coordination Episciences Iam <>
Submitted on : Wednesday, September 9, 2015 - 11:15:06 AM
Last modification on : Wednesday, October 14, 2020 - 4:24:35 AM
Long-term archiving on: : Monday, December 28, 2015 - 11:02:22 PM

### File

dmAP0101.pdf
Publisher files allowed on an open archive

### Identifiers

• HAL Id : hal-01196145, version 1

### Citation

Adrien Richard. A fixed point theorem for Boolean networks expressed in terms of forbidden subnetworks. 17th International Workshop on Celular Automata and Discrete Complex Systems, 2011, Santiago, Chile. pp.1-16. ⟨hal-01196145⟩

Record views