Insights from number theory into the critical Kauffman model with connectivity one

The evaluation of the number of attractors in Kauffman networks by Samuelsson and Troein is generalized to critical networks with one input per node and to networks with two inputs per node and different probability distributions for update functions. A connection is made between the terms occurring in the calculation and between the more graphic concepts of frozen, nonfrozen and relevant nodes, and relevant components. Based on this understanding, a phenomenological argument...

This is the first of two papers about the structure of Kauffman networks. In this paper we define the relevant elements of random networks of automata, following previous work by Flyvbjerg and Flyvbjerg and Kjaer, and we study numerically their probability distribution in the chaotic phase and on the critical line of the model. A simple approximate argument predicts that their number scales as sqrt(N) on the critical line, while it is linear with N in the chaotic phase and in...

In this work we present a general mechanism by which simple dynamics running on networks become self-organized critical for scale free topologies. We illustrate this mechanism with a simple arithmetic model of division between integers, the division model. This is the simplest self-organized critical model advanced so far, and in this sense it may help to elucidate the mechanism of self-organization to criticality. Its simplicity allows analytical tractability, characterizing...

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.

The growth in number and nature of dynamical attractors in Kauffman NK network models are still not well understood properties of these important random boolean networks. Structural circuits in the underpinning graph give insights into the number and length distribution of attractors in the NK model. We use a fast direct circuit enumeration algorithm to study the NK model and determine the growth behaviour of structural circuits. This leads to an explanation and lower bound o...

This is the second paper of a series of two about the structural properties that influence the asymptotic dynamics of Random Boolean Networks. Here we study the functionally independent clusters in which the relevant elements, introduced and studied in our first paper, are subdivided. We show that the phase transition in Random Boolean Networks can also be described as a percolation transition. The statistical properties of the clusters of relevant elements (that we call modu...

Complexity theory as practiced by physicists and computational complexity theory as practiced by computer scientists both characterize how difficult it is to solve complex problems. Here it is shown that the parameters of a specific model can be adjusted so that the problem of finding its global energy minimum is extremely sensitive to small changes in the problem statement. This result has implications not only for studies of the physics of random systems but may also lead t...

Critical, or scale independent, systems are so ubiquitous, that gaining theoretical insights on their nature and properties has many direct repercussions in social and natural sciences. In this report, we start from the simplest possible growth model for critical systems and deduce constraints in their growth : the well-known preferential attachment principle, and, mainly, a new law of temporal scaling. We then support our scaling law with a number of calculations and simulat...

We determine the average number $ \vartheta (N, K) $, of \textit{NK}-Kauffman networks that give rise to the same binary function. We show that, for $ N \gg 1 $, there exists a connectivity critical value $ K_c $ such that $ \vartheta(N,K) \approx e^{\phi N} $ ($ \phi > 0 $) for $ K < K_c $ and $\vartheta(N,K) \approx 1 $ for $ K > K_c $. We find that $ K_c $ is not a constant, but scales very slowly with $ N $, as $ K_c \approx \log_2 \log_2 (2N / \ln 2) $. The problem of ge...