# Exploring the Dynamic Organization of Random and Evolved Boolean Networks
## 1 Introduction  
Many natural, social or artificial systems can be described as networks of interacting elements, where the dynamical variables are associated to the nodes of the network, and directed links represent the influence of a variable upon another one. While networks can represent different kinds of systems and relationships, let us consider here for the sake of definiteness a dynamical system. A variable \(x_{i}(t)\) (\(i=1\ldots\)\(N\)), which can take either continuous or discrete values, is associated to every node \(i\) of the network at time \(t\), and a deterministic law (e.g., a differential or finite difference equation) determines the time behavior of \(x_{i}\). If this equation contains at least one term which depends upon \(x_{j}\) then there is a link from node \(j\) to node \(i\). The study of the properties of networks has become one of the major threads in complexity science. The discovery of the widespread presence of some types of topological relationships, associated to important generic properties (like e.g., the presence of hubs), prompted several studies on topology [1; 2; 3]. It soon became clear that the properties of these complex systems are neither determined by the topology of the interaction network alone, nor by the dynamical laws in isolation, but they are the outcome of the interplay between topology and dynamics [4; 5; 6; 7; 8; 9]. This implies that it is difficult to tell in advance what the properties of a complex system are, unless of course it belongs to a class of already well-studied cases. We will loosely use the term \"dynamical organization\" to refer to the emergent properties of the interactions between topology and dynamics. It goes without saying that there may be different organizational levels, and that in general a simple hierarchical scheme may not be adequate. Different methods to explore the dynamical organization of complex systems can be proposed. In this paper, we make use of one such method, which was introduced by us some years ago [10; 11] and which has later been improved [12] and applied to different classes of systems [13; 14; 15; 16]. The method identifies integrated groups of variables, by means of measures based upon information theory (the Relevance Index methodology--RI in the following), which is briefly recalled in Section 2.1 (referring the interested reader to the literature [12] for further details). In this paper, we concentrate on an important class of models, i.e., the so-called random Boolean networks (RBNs) which were initially conceived by [PERSON] as a simplified model of genetic regulatory networks, and which turned out to be one of the most general and most widely used models in the field of Complex Systems Science. The definition and properties of RBNs are briefly reviewed in Section 2.2. Due to their randomness, the study of single RBNs makes little sense, and it is necessary to study the behavior of families of networks, built at random while keeping some parameters fixed--in a way which closely parallels the ensemble approach of statistical mechanics [17; 18; 19]. A very important feature of families of RBNs is that their \"typical\" behaviors can be ordered or disordered, depending upon the value of some parameter. This makes them well-suited to study the transitions between order and chaos and, for this reason, they have been analyzed in-depth by several groups [20]. It can indeed be shown that a single parameter, the so-called Derrida parameter \(\lambda\) (which is sometimes called the dynamical sensitivity) distinguishes between ordered (\(\lambda<1\)) and disordered (\(\lambda>1\)) systems. In ordered systems, a slight perturbation of the initial conditions tends to die out, while in chaotic networks it initially increases. In so-called critical systems, i.e., those which are neither ordered nor chaotic (\(\lambda\equiv 1\)), a small perturbation tends on average to keep the same size unaltered.