2013/02/26 by Adrien Richard, Richard, Adrien
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · Neuroscience · #Discrete Mathematics (cs.DM) #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #Gene Regulatory Network Analysis #Graph theory and applications #Neural dynamics and brain function #Protein Structure and Dynamics #cs.DM #math.DS
paper · pdf · doi:10.48550/arxiv.1302.6346
40 pages
openalex publication_date 2013/02/26 · arxiv created 2014/12/04 · arxiv updated 2014/12/05 · openalex created_date 2022/12/19 · openalex updated_date 2026/07/28
We are interested in fixed points in Boolean networks, \em i.e. functions f from \0,1\n to itself. We define the subnetworks of f as the restrictions of f to the subcubes of \0,1\n, and we characterizes a class F of Boolean networks satisfying the following property: Every subnetwork of f has a unique fixed point if and only if f has no subnetwork in F. This characterization generalizes the fixed point theorem of Shih and Dong, which asserts that if for every x in \0,1\n there is no directed cycle in the directed graph whose the adjacency matrix is the discrete Jacobian matrix of f evaluated at point x, then f has a unique fixed point. Then, denoting by C+ (resp. C-) the networks whose the interaction graph is a positive (resp. negative) cycle, we show that the non-expansive networks of F are exactly the networks of C+∪ C-; and for the class of non-expansive networks we get a "dichotomization" of the previous forbidden subnetwork theorem: Every subnetwork of f has at most (resp. at least) one fixed point if and only if f has no subnetworks in C+ (resp. C-) subnetwork. Finally, we prove that if f is a conjunctive network then every subnetwork of f has at most one fixed point if and only if f has no subnetwork in C+.