October 17, 2005
We study the Boolean dynamics of the "quenched" Kauffman models with a directed scale-free network, comparing with that of the original directed random Kauffman networks and that of the directed exponential-fluctuation networks. We have numerically investigated the distributions of the state cycle lengths and its changes as the network size $N$ and the average degree $<k>$ of nodes increase. In the relatively small network ($N \sim 150$), the median, the mean value and the standard deviation grow exponentially with $N$ in the directed scale-free and the directed exponential-fluctuation networks with $<k > =2 $, where the function forms of the distributions are given as an almost exponential. We have found that for the relatively large $N \sim 10^3$ the growth of the median of the distribution over the attractor lengths asymptotically changes from algebraic type to exponential one as the average degree $<k>$ goes to $<k > =2$. The result supports an existence of the transition at $<k >_c =2$ derived in the annealed model.
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August 20, 2007
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...
October 22, 2004
The Kauffman model describes a system of randomly connected nodes with dynamics based on Boolean update functions. Though it is a simple model, it exhibits very complex behavior for "critical" parameter values at the boundary between a frozen and a disordered phase, and is therefore used for studies of real network problems. We prove here that the mean number and mean length of attractors in critical random Boolean networks with connectivity one both increase faster than any ...
October 27, 1995
Kauffman net is a dynamical system of logical variables receiving two random inputs and each randomly assigned a boolean function. We show that the attractor and transient lengths exhibit scaleless behavior with power-law distributions over up to ten orders of magnitude. Our results provide evidence for the existence of the "edge of chaos" as a distinct phase between the ordered and chaotic regimes analogous to a critical point in statistical mechanics. The power-law distribu...
November 1, 2002
The Kauffman model describes a particularly simple class of random Boolean networks. Despite the simplicity of the model, it exhibits complex behavior and has been suggested as a model for real world network problems. We introduce a novel approach to analyzing attractors in random Boolean networks, and applying it to Kauffman networks we prove that the average number of attractors grows faster than any power law with system size.
July 13, 2007
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 $\...
July 3, 2005
We study the nature of fitness landscapes of 'quenched' Kauffman's Boolean model with a scale-free network. We have numerically calculated the rugged fitness landscapes, the distributions, its tails, and the correlation between the fitness of local optima and their Hamming distance from the highest optimum found, respectively. We have found that (a) there is an interesting difference between the random and the scale-free networks such that the statistics of the rugged fitness...
September 25, 2002
The dynamics of Boolean networks (the N-K model) with scale-free topology are studied here. The existence of a phase transition governed by the value of the scale-free exponent of the network is shown analytically by analyzing the overlap between two distinct trajectories. The phase diagram shows that the phase transition occurs for values of the scale-free exponent in the open interval (2,2.5). Since the Boolean networks under study are directed graphs, the scale-free topolo...
April 26, 2002
This paper reviews a class of generic dissipative dynamical systems called N-K models. In these models, the dynamics of N elements, defined as Boolean variables, develop step by step, clocked by a discrete time variable. Each of the N Boolean elements at a given time is given a value which depends upon K elements in the previous time step. We review the work of many authors on the behavior of the models, looking particularly at the structure and lengths of their cycles, the...
June 30, 2005
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...
June 23, 2006
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...