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

Gadouleau, Maximilien; Richard, Adrien; Riis, Søren

doi:10.1137/140988358

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

Gadouleau, Maximilien; Richard, Adrien; Riis, Søren

Authors

Dr Maximilien Gadouleau m.r.gadouleau@durham.ac.uk
Associate Professor

Adrien Richard

Søren Riis

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.