# Exploring the Dynamic Organization of Random and Evolved Boolean Networks
## 2 Materials and Methods
### Random Boolean Networks  
Random Boolean Networks are a well-known model of gene regulatory dynamics [17; 18; 19; 20; 24; 30; 39; 40; 41] able to perform computation [42]. In this section, we present a synthetic description of the RBN model, referring the reader to [17; 18; 20; 23; 24] for more details. Several variants of the model have been presented and discussed [43; 44; 45; 46], but we will restrict our attention here to the \"classical\" model. A RBN, as introduced by [PERSON] [23; 24; 39], is a dynamic system whose components (nodes in an oriented graph, each one representing a gene of a genetic regulatory network) can assume only two levels (0 and 1). If the state of a single node can be represented by \(x_{i}(t)\), the state of the whole network can be represented by the vector \(X(t)=[x_{1}(t),\,x_{2}(t),\,\dots\,x_{N}(t)]\). In \"classic\" RBNs, each node has the same incoming connectivity, while multiple connections and auto-connections are prohibited. Each node is associated with a Boolean function, which represents the response to the signals (proteins--not represented in the model) coming from the upstream nodes. In this paper, in case of random generation of the Boolean functions, for each entry, we extract a 1 with probability \(p\) (a quantity called \"bias\")--and consequently the frequency of the values equal to 0 is \((p-1)\). The topology and Boolean function associated to each node do not change in time, and the network dynamics is discrete and synchronous, so fixed points and cycles are the only possible asymptotic states in finite networks. The RBN's average degree of connectivity and the type of Boolean functions influence the behavior of the system, which can show ordered or disordered dynamic regimes. In ordered regimes, there are few attractors, whose period scales as a power law with the number of nodes N of the system, while in disordered regimes, the period of the very numerous attractors scales as a more than polynomial function [23; 24]. The disordered regime also shows a strong sensitivity to initial conditions, which is not present for the ordered regime. As mentioned in Section 1, these different behaviors can be related to the value of a single index, often called the Derrida parameter \(\lambda\): on average, a small perturbation tends to vanish in \"ordered\" networks (\(\lambda<1\)), while in chaotic networks (\(\lambda>1\)) it increases in time, and in critical networks (\(\lambda=1\)) it tends to maintain its size. The Derrida parameter can be determined by studying the average behavior in time of the distance between two close initial states of a network and taking its limiting value when the initial distance approaches its minimum value (dynamical sensitivity) [20; 47; 48]. However, direct measurement of the dynamical Derrida parameter in physical biological systems may be prohibitive, due to their transient nature, but it is sometimes possible, under plausible assumptions, to circumvent this problem and to infer the value of \(\lambda\) from other data. This is the case of the effects of gene perturbations, induced by permanently inhibiting the expression of a single gene (knock-out) and by observing the changes of the expression levels of the other genes. In synthetic networks (by means of simulations), it is possible to evaluate these effects by comparing the behavior of a non-perturbed system (\"wild type\", briefly WT) with the behavior of a perturbed system, which differs from the previous one in that one of the genes, which are active in the WT system, is silenced (its Boolean function is fixed at False--\"knock-out\" event). It is possible to observe the two systems over time, and to mark nodes having different values in the different systems as \"perturbed\" nodes: the number of such nodes (different from the silenced node) provides the size of the \"avalanche of changes\" (or more briefly \"avalanche\") to which these nodes belong.