Dynamical properties of disjunctive Boolean networks

Abstract

A Boolean network is a mapping f : {0, 1} n → {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.The set of variables N (i) does depend on the entity and its size may vary.Since disjunctions are such special Boolean functions, disjunctive networks can be characterised in different ways. We first review these characterisations in Section 2.2. Also, disjunctive networks can be represented using graphs, Boolean matrices, or binary relations; again, we review these representations in Section 2.3.The dynamical properties of Boolean networks have been thoroughly studied, see [3,6,7,17,20,22,31] for example. Due to their different representations, disjunctive networks have attracted interest inside and outside of the Boolean network community. For instance, some early work on binary relations and Boolean matrices classifies convergent and idempotent disjunctive networks [10,30,35]. The main interests in the dynamical properties of disjunctive networks include the transient length [21], the characterisation of their cycle structure [21,28], and in particular determining when they present no oscillations [1,28]. It is worth noting that some results on disjunctive networks are included in works that consider related or more general classes of Boolean networks, e.g. [2,3,5,7]. Even though in this paper we focus on the synchronous dynamics (i.e. parallel updates), the reader interested in the asynchronous dynamics of disjunctive networks is directed to [18].In this paper, we are interested in the following three dynamical features of a disjunctive network. An image point of f is a reachable state; a periodic point is a recurring state ; and a fixed point is a stationary state. We first give characterisations of their sets of image points, periodic points, and fixed points, respectively in Section 3.1. We then consider the number of image, periodic, and fixed points of disjunctive networks...