Number of fixed points and disjoint cycles in monotone Boolean networks

TL;DR

This paper investigates the maximum number of fixed points in monotone Boolean networks based on the cycle structure of their interaction graphs, providing new bounds and characterizations related to feedback vertex sets and disjoint cycles.

Contribution

It introduces improved upper bounds and optimal lower bounds for fixed points in monotone Boolean networks, linking these to cycle parameters of the interaction graph.

Findings

Upper bounds depend on feedback vertex set and cycle disjointness

Lower bounds are established as functions of disjoint cycles

Characterization of when maximum fixed points equals classical bounds

Abstract

Given a digraph $G$, a lot of attention has been deserved on the maximum number $\phi(G)$ of fixed points in a Boolean network $f:\{0,1\}^n\to\{0,1\}^n$ with $G$ as interaction graph. In particular, a central problem in network coding consists in studying the optimality of the classical upper bound $\phi(G)\leq 2^{\tau}$, where $\tau$ is the minimum size of a feedback vertex set of $G$. In this paper, we study the maximum number $\phi_m(G)$ of fixed points in a {\em monotone} Boolean network with interaction graph $G$. We establish new upper and lower bounds on $\phi_m(G)$ that depends on the cycle structure of $G$. In addition to $\tau$, the involved parameters are the maximum number $\nu$ of vertex-disjoint cycles, and the maximum number $\nu^{*}$ of vertex-disjoint cycles verifying some additional technical conditions. We improve the classical upper bound $2^\tau$ by proving that $\phi_m(G)$ is at most the largest sub-lattice of $\{0,1\}^\tau$ without chain of size $\nu+1$, and without another forbidden-pattern of size $2\nu^{*}$. Then, we prove two optimal lower bounds: $\phi_m(G)\geq \nu+1$ and $\phi_m(G)\geq 2^{\nu^{*}}$. As a consequence, we get the following characterization: $\phi_m(G)=2^\tau$ if and only if $\nu^{*}=\tau$. As another consequence, we get that if $c$ is the maximum length of a chordless cycle of $G$ then $2^{\nu/3^c}\leq\phi_m(G)\leq 2^{c\nu}$. Finally, with the technics introduced, we establish an upper bound on the number of fixed points of any Boolean network according to its signed interaction graph.