Scaling laws in critical random Boolean networks with general in- and out-degree distributions

TL;DR

This paper analytically and numerically investigates the size of the frozen core and scaling laws in critical Boolean networks with power-law in- and out-degree distributions, revealing diverse scaling behaviors based on degree distribution parameters.

Contribution

It generalizes an existing method to include power-law out-degree distributions and derives new scaling laws for critical Boolean networks with various degree exponents and cutoffs.

Findings

Scaling laws depend on degree distribution exponents and cutoffs.

Divergence of the second moment influences the nonfrozen node scaling.

Numerical results confirm analytical predictions.

Abstract

We evaluate analytically and numerically the size of the frozen core and various scaling laws for critical Boolean networks that have a power-law in- and/or out-degree distribution. To this purpose, we generalize an efficient method that has previously been used for conventional random Boolean networks and for networks with power-law in-degree distributions. With this generalization, we can also deal with power-law out-degree distributions. When the power-law exponent is between 2 and 3, the second moment of the distribution diverges with network size, and the scaling exponent of the nonfrozen nodes depends on the degree distribution exponent. Furthermore, the exponent depends also on the dependence of the cutoff of the degree distribution on the system size. Altogether, we obtain an impressive number of different scaling laws depending on the type of cutoff as well as on the exponents of the in- and out-degree distributions. We confirm our scaling arguments and analytical considerations by numerical investigations.