Gadouleau, Maximilien and Richard, Adrien and Riis, Søren (2015) 'Fixed points of Boolean networks, guessing graphs, and coding theory.', SIAM journal on discrete mathematics., 29 (4). pp. 2312-2335.
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.
|Keywords:||Boolean networks, Fixed points, Signed digraphs, Error-correcting codes, Guessing number.|
|Full text:||(AM) Accepted Manuscript|
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|Publisher Web site:||http://dx.doi.org/10.1137/140988358|
|Publisher statement:||© 2015 Society for Industrial and Applied Mathematics|
|Date accepted:||17 September 2015|
|Date deposited:||15 December 2015|
|Date of first online publication:||24 November 2015|
|Date first made open access:||No date available|
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