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.
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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⟩

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