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

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.

Figure 1

Graphical representation of the nine cases listed in Table : (a) exponent $x$; (b) exponent $\frac{y}{x}$.

Figure 2

Typical example for the probability distribution of nonfrozen nodes. This sample is for critical networks with a scale free in- and out-degree distribution with $\gamma =2.3$ and a cutoff scaling as $k^{\max }\propto N^{\frac{1}{\gamma }}$. The number of realizations was $2\times {10}^5$, and number of nodes in the network was $N=2^{10}$ for the solid curve and $N=2^{11}$ for the dotted curve. The curves were collapsed using the exponent $0.635$, which is the best value our algorithm found for these two curves.

Figure 3

The quality of the data collapse (color coded) of the distributions $P\left(N_{\mathrm{n}f}\right)$ obtained for two $N$ values that differ by a factor of 2, as a function of $N$ ($x$ axis: log scale from ${10}^3$ to ${10}^8$) and the exponent used for the collapse ($y$ axis: linear scale from 0 to 1). The darker the color, the better the collapse. The different plots are for different combinations of the in- and out-degree distributions. When the in-degree distribution was identical to the out-degree distribution, we generated the out-degree distribution by using exactly the same sample as the one obtained for the in-degree distribution. The lines give the value 2/3 (white dotted line), the theoretically predicted exponent (white solid line), and the best exponent for the collapse of the pairs of curves (black solid line).

Figure 4

Example for the color coding in Fig. 3. Each subplot is similar to Fig. 2. The only difference is the number of nodes, which was in this case $2^{17}$ and $2^{18}$. The numbers above the graphs give the exponents used for scaling the two curves, and the color indicates the quality of the collapse. The theoretical exponent is used in (i).