Scaling in a general class of critical random Boolean networks

Abstract

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 scales with the network size $N$ as $N^{2/3}$, with only $N^{1/3}$ nonfrozen nodes having two nonfrozen inputs and the number of nonfrozen nodes with more than two inputs being finite in the thermodynamic limit. Using these results we can conclude that the mean number of relevant nodes increases for large $N$ as $N^{1/3}$, with only a finite number of relevant nodes having two relevant inputs, and a vanishing fraction of nodes having more than three of them. It follows that all relevant components apart from a finite number are simple loops, and that the mean number and length of attractors increases faster than any power law with network size.

11 pages

References in corpus (11)

• Random Boolean Network Models and the Yeast Transcriptional Network
• Number and length of attractors in a critical Kauffman model with connectivity one
• Percolation in Directed Scale-Free Networks
• Scaling in ordered and critical random Boolean networks
• Stable and unstable attractors in Boolean networks
• Relevant elments, Magnetization and Dynamical Properties in Kauffman Networks: a Numerical Study
• The dynamics of critical Kauffman networks under asynchronous stochastic update
• The Modular Structure of Kauffman Networks
• On the number of attractors in random Boolean networks
• Scaling in critical random Boolean networks
• The properties of attractors of canalyzing random Boolean networks

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• Dynamical Instability in Boolean Networks as a Percolation Problem
• Stability of Boolean networks: The joint effects of topology and update rules
• Broad edge of chaos in strongly heterogeneous Boolean networks
• Critical Boolean networks with scale-free in-degree distribution
• Critical Line in Random Threshold Networks with Inhomogeneous Thresholds
• Symmetry in Critical Random Boolean Network Dynamics
• Formation of the frozen core in critical Boolean Networks
• Feedback topology and XOR-dynamics in Boolean networks with varying input structure
• Scaling laws in critical random Boolean networks with general in- and out-degree distributions
• Predicting attractors from spectral properties of stylized gene regulatory networks