Local negative circuits and fixed points in Boolean networks

2009/10/05 by Adrien Richard, Richard, Adrien
Computer Science · #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #cs.DM

paper · pdf · doi:10.48550/arxiv.0910.0750

19 pages

arxiv created 2009/10/05 · arxiv updated 2009/12/01

Abstract

To each Boolean function F from 0,1ⁿ to itself and each point x in 0,1ⁿ, 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 0,1ⁿ 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 0,1ⁿ, the out-degree of each vertex of G_F(x) is at most one.

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