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

TL;DR

This paper investigates the fixed points of Boolean network functions on signed digraphs, establishing bounds and relationships with coding theory, revealing complex behaviors and challenging previous assumptions about positive cycles.

Contribution

It introduces new bounds on fixed points using network coding techniques and uncovers novel links between fixed points and coding theory, improving upon existing results.

Findings

Derived lower bounds on fixed points based on digraph structure

Established relationships between fixed points and coding theory problems

Showed that more positive cycles do not guarantee more fixed points

Abstract

In 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 behaviour 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.