Local negative circuits and fixed points in Boolean networks

Adrien Richard 1
1 Laboratoire d'Informatique, Signaux, et Systèmes de Sophia-Antipolis (I3S) / Equipe BIOINFO
Laboratoire I3S - MDSC - Modèles Discrets pour les Systèmes Complexes
Abstract : To each Boolean function $F$ from ${0,1}^n$ to itself and each point x in ${0,1}^n$, we associate the signed directed graph $G_F(x)$ of order $n$ that contains a positive (resp. negative) arc from $j$ to $i$ if the partial derivative of $f_i$ with respect of $x_j$ is positive (resp. negative) at point $x$. We then focus on the following open problem: Is the absence of a negative circuit in $G_F(x)$ for all $x$ in ${0,1}^n$ a sufficient condition for $F$ to have at least one fixed point? As main result, we settle this problem under the additional condition that, for all x in ${0,1}^n$, the out-degree of each vertex of $G_F(x)$ is at most one.
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Adrien Richard. Local negative circuits and fixed points in Boolean networks. Discrete Applied Mathematics, Elsevier, 2011, 159 (11), pp.1085-1093. ⟨hal-01298851⟩

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