The Number of Attractors in Kauffman Networks

Random Boolean networks, originally invented as models of genetic regulatory networks, are simple models for a broad class of complex systems that show rich dynamical structures. From a biological perspective, the most interesting networks lie at or near a critical point in parameter space that divides ``ordered'' from ``chaotic'' attractor dynamics. In the ordered regime, we show rigorously that the average number of relevant nodes (the ones that determine the attractor dyna...

Fixed points are fundamental states in any dynamical system. In the case of gene regulatory networks (GRNs) they correspond to stable genes profiles associated to the various cell types. We use Kauffman's approach to model GRNs with random Boolean networks (RBNs). We start this paper by proving that, if we fix the values of the source nodes (nodes with in-degree 0), the expected number of fixed points of any RBN is one (independently of the topology we choose). For finding su...

Boolean Networks have been used to study numerous phenomena, including gene regulation, neural networks, social interactions, and biological evolution. Here, we propose a general method for determining the critical behavior of Boolean systems built from arbitrary ensembles of Boolean functions. In particular, we solve the critical condition for systems of units operating according to canalizing functions and present strong numerical evidence that our approach correctly predic...

A model of cellular metabolism due to S. Kauffman is analyzed. It consists of a network of Boolean gates randomly assembled according to a probability distribution. It is shown that the behavior of the network depends very critically on certain simple algebraic parameters of the distribution. In some cases, the analytic results support conclusions based on simulations of random Boolean networks, but in other cases, they do not.

We describe systems using Kauffman and similar networks. They are directed funct ioning networks consisting of finite number of nodes with finite number of discr ete states evaluated in synchronous mode of discrete time. In this paper we introduce the notion and phenomenon of `structural tendencies'. Along the way we expand Kauffman networks, which were a synonym of Boolean netw orks, to more than two signal variants and we find a phenomenon during network g rowth which we ...

For years, we have been building models of gene regulatory networks, where recent advances in molecular biology shed some light on new structural and dynamical properties of such highly complex systems. In this work, we propose a novel timing of updates in Random and Scale-Free Boolean Networks, inspired by recent findings in molecular biology. This update sequence is neither fully synchronous nor asynchronous, but rather takes into account the sequence in which genes affect ...

The deterministic dynamics of randomly connected neural networks are studied, where a state of binary neurons evolves according to a discreet-time synchronous update rule. We give a theoretical support that the overlap of systems' states between the current and a previous time develops in time according to a Markovian stochastic process in large networks. This Markovian process predicts how often a network revisits one of previously visited states, depending on the system siz...

We investigate analytically and numerically the critical line in undirected random Boolean networks with arbitrary degree distributions, including scale-free topology of connections $P(k)\sim k^{-\gamma}$. We show that in infinite scale-free networks the transition between frozen and chaotic phase occurs for $3<\gamma < 3.5$. The observation is interesting for two reasons. First, since most of critical phenomena in scale-free networks reveal their non-trivial character for $\...

Boolean networks at the critical point have been a matter of debate for many years as, e.g., scaling of number of attractor with system size. Recently it was found that this number scales superpolynomially with system size, contrary to a common earlier expectation of sublinear scaling. We here point to the fact that these results are obtained using deterministic parallel update, where a large fraction of attractors in fact are an artifact of the updating scheme. This limits t...

We show that to correctly describe the position of the critical line in the Kauffman random boolean networks one must take into account percolation phenomena underlying the process of damage spreading. For this reason, since the issue of percolation transition is much simpler in random undirected networks, than in the directed ones, we study the Kauffman model in undirected networks. We derive the mean field formula for the critical line in the giant component of these networ...