Fixed points of Boolean networks, guessing graphs, and coding theory

Gadouleau, Maximilien; Richard, Adrien; Riis, Søren

Abstract

n 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 behavior 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.

Citation

Gadouleau, M., Richard, A., & Riis, S. (2015). Fixed points of Boolean networks, guessing graphs, and coding theory. SIAM Journal on Discrete Mathematics, 29(4), 2312-2335. https://doi.org/10.1137/140988358

• Journal Article Type: Article
• Acceptance Date: Sep 17, 2015
• Online Publication Date: Nov 24, 2015
• Publication Date: Nov 24, 2015
• Deposit Date: Oct 14, 2015
• Publicly Available Date: Dec 15, 2015
• Journal: SIAM Journal on Discrete Mathematics
• Print ISSN: 0895-4801
• Electronic ISSN: 1095-7146
• Publisher: Society for Industrial and Applied Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 29
• Issue: 4
• Pages: 2312-2335
• DOI: https://doi.org/10.1137/140988358
• Keywords: Boolean networks, Fixed points, Signed digraphs, Error-correcting codes, Guessing number.
• Public URL: https://durham-repository.worktribe.com/output/1400701