Scaling in critical random Boolean networks

Abstract

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 of these nodes. Using a different stochastic process, we determine the scaling behavior of the number of relevant nodes. Their mean number increases for large N as N^{1/3}, and only a finite number of relevant nodes have two relevant inputs. 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.

9 pages Revtex, 5 eps figures

References in corpus (6)

• Random Boolean Network Models and the Yeast Transcriptional Network
• Number and length of attractors in a critical Kauffman model with connectivity one
• Scaling in ordered and critical random Boolean networks
• Relevant elments, Magnetization and Dynamical Properties in Kauffman Networks: a Numerical Study
• The Modular Structure of Kauffman Networks
• On the number of attractors in random Boolean networks

Cited by in corpus (21)

• Boolean Delay Equations: A simple way of looking at complex systems
• Damage Spreading and Criticality in Finite Random Dynamical Networks
• Canalization and Symmetry in Boolean Models for Genetic Regulatory Networks
• Scaling in a general class of critical random Boolean networks
• Exhaustive percolation on random networks
• The properties of attractors of canalyzing random Boolean networks
• The phase diagram of random threshold networks
• Perturbation propagation in random and evolved Boolean networks
• Entropy of complex relevant components of Boolean networks
• Critical Kauffman networks under deterministic asynchronous update
• Kauffman networks with threshold functions
• Relevant components in critical random Boolean networks
• 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
• Optimal Stabilization of Boolean Networks through Collective Influence
• Response of Boolean networks to perturbations
• 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