Number of Fixed Points and Disjoint Cycles in Monotone Boolean Networks

Abstract : Given a digraph G, a lot of attention has been deserven on the maximum number φ(G) of fixed points in a Boolean network f : {0, 1} n → {0, 1} n with G as interaction graph. In particular, a central problem in network coding consists in studying the optimality of the feedback bound φ(G) ≤ 2 τ , where τ is the minimum size of a feedback vertex set of G. In this paper, we study the maximum number φ m (G) of fixed points in a monotone Boolean network with interaction graph G. We establish new upper and lower bounds on φ m (G) that depends on the cycle structure of G. In addition to τ , the involved parameters are the maximum number ν of vertex-disjoint cycles, and the maximum number ν * of vertex-disjoint cycles verifying some additional technical conditions. We improve the feedback bound 2 τ by proving that φ m (G) is at most the largest sub-lattice of {0, 1} τ without chain of size ν + 2, and without another forbidden pattern described by two disjoint antichains of size ν * + 1. Then, we prove two optimal lower bounds: φ m (G) ≥ ν + 1 and φ m (G) ≥ 2 ν *. As a consequence, we get the following characterization: φ m (G) = 2 τ if and only if ν * = τ. As another consequence, we get that if c is the maximum length of a chordless cycle of G then 2 ν/3 c ≤ φ m (G) ≤ 2 cν. Finally, with the techniques introduced, we establish an upper bound on the number of fixed points of any Boolean network according to its signed interaction graph.
Type de document :
Article dans une revue
Siam Journal on Discrete Mathematics, Society for Industrial and Applied Mathematics, 2017, 31 (3), pp.1702 - 1725. 〈10.1137/16M1060868〉
Liste complète des métadonnées

Littérature citée [31 références]  Voir  Masquer  Télécharger

https://hal.archives-ouvertes.fr/hal-01630477
Contributeur : Adrien Richard <>
Soumis le : mardi 7 novembre 2017 - 16:23:13
Dernière modification le : vendredi 10 novembre 2017 - 01:17:50

Fichier

2017_04_12_maxFP_Monotonous_R1...
Fichiers produits par l'(les) auteur(s)

Identifiants

Collections

Citation

Julio Aracena, Adrien Richard, Lilian Salinas. Number of Fixed Points and Disjoint Cycles in Monotone Boolean Networks. Siam Journal on Discrete Mathematics, Society for Industrial and Applied Mathematics, 2017, 31 (3), pp.1702 - 1725. 〈10.1137/16M1060868〉. 〈hal-01630477〉

Partager

Métriques

Consultations de la notice

13

Téléchargements de fichiers

2