Circuits, Attractors and Reachability in Mixed-K Kauffman Networks

We study the intrinsic properties of attractors in the Boolean dynamics in complex network with scale-free topology, comparing with those of the so-called random Kauffman networks. We have numerically investigated the frozen and relevant nodes for each attractor, and the robustness of the attractors to the perturbation that flips the state of a single node of attractors in the relatively small network ($N=30 \sim 200$). It is shown that the rate of frozen nodes in the complex...

The critical Kauffman model with connectivity one is the simplest class of critical Boolean networks. Nevertheless, it exhibits intricate behavior at the boundary of order and chaos. We introduce a formalism for expressing the dynamics of multiple loops as a product of the dynamics of individual loops. Using it, we prove that the number of attractors scales as $2^m$, where $m$ is the number of nodes in loops - as fast as possible, and much faster than previously believed.

We show that the mean number of attractors in a critical Boolean network under asynchronous stochastic update grows like a power law and that the mean size of the attractors increases as a stretched exponential with the system size. This is in strong contrast to the synchronous case, where the number of attractors grows faster than any power law.

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...

The Kauffman model of genetic computation highlights the importance of criticality at the border of order and chaos. But our understanding of its behavior is incomplete, and much of what we do know relies on heuristic arguments. To better understand the model and obtain more rigorous insights, we show that there are fundamental links between the critical Kauffman model and aspects of number theory. Using these connections, we prove that the number of attractors and the mean a...

We investigate the dynamics of network minority games on Kauffman's NK networks (Kauffman nets), growing directed networks (GDNets), as well as growing directed networks with a small fraction of link reversals (GDRNets). We show that the dynamics and the associated phase structure of the game depend crucially on the structure of the underlying network. The dynamics on GDNets is very stable for all values of the connection number $K$, in contrast to the dynamics on Kauffman's ...

Boolean variables are such that they take only values on $ \mathbb{Z}_2 \cong \left\{0, 1 \right\} $. \textit{NK}-Kauffman networks are dynamical deterministic systems of $ N $ Boolean functions that depend only on $ K \leq N $ Boolean variables. They were proposed by Kauffman as a first step to understand cellular behaviour [Kauffman, S.A.; {\rm The Large Scale Structure and Dynamics of Gene Control Circuits: An Ensemble Approach}. {\it J. Theoret. Biol.} {\bf 44} (1974) 167...

We derive analytically the scaling behavior in the thermodynamic limit of the number of nonfrozen and relevant nodes in the most general class of critical Kauffman networks for any number of inputs per node, and for any choice of the probability distribution for the Boolean functions. By defining and analyzing a stochastic process that determines the frozen core we can prove that the mean number of nonfrozen nodes in any critical network with more than one input per node scal...

We derive mostly analytically the scaling behavior of the number of nonfrozen and relevant nodes in critical Kauffman networks (with two inputs per node) in the thermodynamic limit. By defining and analyzing a stochastic process that determines the frozen core we can prove that the mean number of nonfrozen nodes scales with the network size N as N^{2/3}, with only N^{1/3} nonfrozen nodes having two nonfrozen inputs. We also show the probability distributions for the numbers o...

We apply in this article (non rigorous) statistical mechanics methods to the problem of counting long circuits in graphs. The outcomes of this approach have two complementary flavours. On the algorithmic side, we propose an approximate counting procedure, valid in principle for a large class of graphs. On a more theoretical side, we study the typical number of long circuits in random graph ensembles, reproducing rigorously known results and stating new conjectures.