Power-law Distributions in the Kauffman Net

We study two types of simple Boolean networks, namely two loops with a cross-link and one loop with an additional internal link. Such networks occur as relevant components of critical K=2 Kauffman networks. We determine mostly analytically the numbers and lengths of cycles of these networks and find many of the features that have been observed in Kauffman networks. In particular, the mean number and length of cycles can diverge faster than any power law.

Stability is an important characteristic of network models that has implications for other desirable aspects such as controllability. The stability of a Boolean network depends on various factors, such as the topology of its wiring diagram and the type of the functions describing its dynamics. In this paper, we study the stability of linear Boolean networks by computing Derrida curves and quantifying the number of attractors and cycle lengths imposed by their network topologi...

Random Boolean networks, the Kauffman model, are revisited by means of a novel decimation algorithm, which reduces the networks to their dynamical cores. The average size of the removed part, the stable core, grows approximately linearly with N, the number of nodes in the original networks. We show that this can be understood as the percolation of the stability signal in the network. The stability of the dynamical core is investigated and it is shown that this core lacks the ...

It is an increasingly important problem to study conditions on the structure of a network that guarantee a given behavior for its underlying dynamical system. In this paper we report that a Boolean network may fall within the chaotic regime, even under the simultaneous assumption of several conditions which in randomized studies have been separately shown to correlate with ordered behavior. These properties include using at most two inputs for every variable, using biased and...

We present new applications of parity inversion and time-reversal to the emergence of complex behavior from simple dynamical rules in stochastic discrete models. Our parity-based encoding of causal relationships and time-reversal construction efficiently reveal discrete analogs of stable and unstable manifolds. We demonstrate their predictive power by studying decision-making in systems biology and statistical physics models. These applications underpin a novel attractor iden...

We present a renormalization-grouplike method performed in the state space for detecting the dynamical behaviors of large scale-free Boolean networks, especially for the chaotic regime as well as the edge of chaos. Numerical simulations with different coarse-graining level show that the state space networks of scale-free Boolean networks follow universal power-law distributions of in and out strength, in and out degree, as well as weight. These interesting results indicate sc...

Boolean networks have been the object of much attention, especially since S. Kauffman proposed them in the 1960's as models for gene regulatory networks. These systems are characterized by being defined on a Boolean state space and by simultaneous updating at discrete time steps. Of particular importance for biological applications are networks in which the indegree for each variable is bounded by a fixed constant, as was stressed by Kauffman in his original papers. An impo...

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...

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 ...

In this work we analyze scale-free networks with different power law spectra $N(k) \sim k^{-\gamma}$ under a boolean dynamic, where the boolean rule that each node obeys is a function of its connectivity $k$. This is done by using only two logical functions (AND and XOR) which are controlled by a parameter $q$. Using damage spreading technique we show that the Hamming distance and the number of 1's exhibit power law behavior as a function of $q$. The exponents appearing in th...