Local negative circuits and fixed points in Boolean networks

TL;DR

This paper investigates the relationship between local negative circuits in Boolean networks and the existence of fixed points, providing a solution under specific out-degree constraints.

Contribution

It resolves an open problem by establishing that absence of negative circuits in certain graphs guarantees fixed points when out-degree is limited to one.

Findings

Absence of negative circuits implies fixed points under out-degree constraints

Introduces a new condition linking local graph structure to fixed points

Provides a partial answer to a longstanding open problem in Boolean network theory

Abstract

To each Boolean function F from {0,1}^n to itself and each point x in {0,1}^n, we associate the signed directed graph G_F(x) of order n that contains a positive (resp. negative) arc from j to i if the partial derivative of f_i with respect of x_j is positive (resp. negative) at point x. We then focus on the following open problem: Is the absence of a negative circuit in G_F(x) for all x in {0,1}^n a sufficient condition for F to have at least one fixed point? As main result, we settle this problem under the additional condition that, for all x in {0,1}^n, the out-degree of each vertex of G_F(x) is at most one.