Critical Boolean networks with scale-free in-degree distribution.

We investigate analytically and numerically the dynamical properties of critical Boolean networks with power-law in-degree distributions and for two choices of update functions. 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 nonfrozen 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 nonfrozen 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.