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Conference Papers Discrete Mathematics and Theoretical Computer Science Year : 2011

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

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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.
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Dates and versions

hal-01196145 , version 1 (09-09-2015)

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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, ⟨10.46298/dmtcs.2978⟩. ⟨hal-01196145⟩
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