Critical Boolean networks with scale-free in-degree distribution

TL;DR

This paper analyzes the dynamical behavior of critical Boolean networks with scale-free in-degree distributions, revealing how the number of non-frozen nodes scales with system size depending on the distribution's exponent.

Contribution

It provides an analytical and numerical study of critical Boolean networks with power-law in-degree distributions, extending understanding beyond fixed in-degree models.

Findings

For exponents > 3, non-frozen nodes scale as N^{2/3}.

For exponents between 2 and 3, non-frozen nodes scale as N^x, with x between 0 and 2/3.

Results explain previous simulation findings.

Abstract

We investigate analytically and numerically the dynamical properties of critical Boolean networks with power-law in-degree distributions. When the exponent of the in-degree distribution is larger than 3, we obtain results equivalent to those obtained for networks with fixed in-degree, e.g., the number of the non-frozen nodes scales as $N^{2/3}$ with the system size $N$. When the exponent of the distribution is between 2 and 3, the number of the non-frozen nodes increases as $N^x$, with $x$ being between 0 and 2/3 and depending on the exponent and on the cutoff of the in-degree distribution. These and ensuing results explain various findings obtained earlier by computer simulations.