# Fixed point theorems for Boolean networks expressed in terms of forbidden subnetworks

Abstract : We are interested in fixed points in Boolean networks, {\em i.e.} functions $f$ from $\{0,1\}^n$ to itself. We define the subnetworks of $f$ as the restrictions of $f$ to the subcubes of $\{0,1\}^n$, and we characterizes a class $\mathcal{F}$ of Boolean networks satisfying the following property: Every subnetwork of $f$ has a unique fixed point if and only if $f$ has no subnetwork in $\mathcal{F}$. This characterization generalizes the fixed point theorem of Shih and Dong, which asserts that 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. Then, denoting by $\mathcal{C}^+$ (resp. $\mathcal{C}^-$) the networks whose the interaction graph is a positive (resp. negative) cycle, we show that the non-expansive networks of $\mathcal{F}$ are exactly the networks of $\mathcal{C}^+\mathcal{C}up \mathcal{C}^-$; and for the class of non-expansive networks we get a dichotomization'' of the previous forbidden subnetwork theorem: Every subnetwork of $f$ has at most (resp. at least) one fixed point if and only if $f$ has no subnetworks in $\mathcal{C}^+$ (resp. $\mathcal{C}^-$) subnetwork. Finally, we prove that if $f$ is a conjunctive network then every subnetwork of $f$ has at most one fixed point if and only if $f$ has no subnetworks in $\mathcal{C}^+$.
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Type de document :
Article dans une revue
Theoretical Computer Science, Elsevier, 2015, 583, pp.1-26. 〈http://www.sciencedirect.com/science/article/pii/S0304397515002650〉
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https://hal.archives-ouvertes.fr/hal-01298012
Contributeur : Adrien Richard <>
Soumis le : mardi 5 avril 2016 - 17:59:03
Dernière modification le : jeudi 17 mai 2018 - 12:52:03
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Adrien Richard. Fixed point theorems for Boolean networks expressed in terms of forbidden subnetworks. Theoretical Computer Science, Elsevier, 2015, 583, pp.1-26. 〈http://www.sciencedirect.com/science/article/pii/S0304397515002650〉. 〈hal-01298012〉

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