# Fixed Points of Boolean Networks, Guessing Graphs, and Coding Theory

2 Laboratoire d'Informatique, Signaux, et Systèmes de Sophia-Antipolis (I3S) / Equipe BIOINFO
Laboratoire I3S - MDSC - Modèles Discrets pour les Systèmes Complexes
Abstract : In this paper, we are interested in the number of fixed points of functions $f:A^n\to A^n$ over a finite alphabet $A$ defined on a given signed digraph $D$. We first use techniques from network coding to derive some lower bounds on the number of fixed points that only depends on $D$. We then discover relationships between the number of fixed points of $f$ and problems in coding theory, especially the design of codes for the asymmetric channel. Using these relationships, we derive upper and lower bounds on the number of fixed points, which significantly improve those given in the literature. We also unveil some interesting behaviour of the number of fixed points of functions with a given signed digraph when the alphabet varies. We finally prove that signed digraphs with more (disjoint) positive cycles actually do not necessarily have functions with more fixed points.
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Journal articles
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Cited literature [35 references]

https://hal.archives-ouvertes.fr/hal-01298045
Contributor : Adrien Richard <>
Submitted on : Tuesday, April 5, 2016 - 1:33:10 PM
Last modification on : Friday, April 19, 2019 - 11:12:03 AM
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Maximilien Gadouleau, Adrien Richard, Søren Riis. Fixed Points of Boolean Networks, Guessing Graphs, and Coding Theory. Siam Journal on Discrete Mathematics, Society for Industrial and Applied Mathematics, 2015, ⟨10.1137/140988358⟩. ⟨hal-01298045⟩

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