Dynamical properties of disjunctive Boolean networks

TL;DR

This paper reviews and extends the understanding of the dynamics of disjunctive Boolean networks, focusing on their fixed, periodic, and image points, and discusses open problems and future research directions.

Contribution

It provides a comprehensive review, new fundamental results, and highlights open problems regarding the dynamical properties of disjunctive Boolean networks.

Findings

Characterization of fixed points, periodic points, and image points.

Methods to represent disjunctive networks using graphs, matrices, or relations.

Identification of open problems and future research avenues.

Abstract

A Boolean network is a mapping $f :\{0,1\}^n \to \{0,1\}^n$, which can be used to model networks of $n$ interacting entities, each having a local Boolean state that evolves over time according to a deterministic function of the current configuration of states. In this paper, we are interested in disjunctive networks, where each local function is simply the disjunction of a set of variables. As such, this network is somewhat homogeneous, though the number of variables may vary from entity to entity, thus yielding a generalised cellular automaton. The aim of this paper is to review some of the main results, derive some additional fundamental results, and highlight some open problems, on the dynamics of disjunctive networks. We first review the different defining characteristics of disjunctive networks and several ways of representing them using graphs, Boolean matrices, or binary relations. We then focus on three dynamical properties of disjunctive networks: their image points, their periodic points, and their fixed points. For each class of points, we review how they can be characterised and study how many they could be. The paper finishes with different avenues for future work on the dynamics of disjunctive networks and how to generalise them.