  We derive analytically the scaling behavior in the thermodynamic limit of the
number of nonfrozen and relevant nodes in the most general class of critical
Kauffman networks for any number of inputs per node, and for any choice of the
probability distribution for the Boolean functions. By defining and analyzing a
stochastic process that determines the frozen core we can prove that the mean
number of nonfrozen nodes in any critical network with more than one input per
node scales with the network size $N$ as $N^{2/3}$, with only $N^{1/3}$
nonfrozen nodes having two nonfrozen inputs and the number of nonfrozen nodes
with more than two inputs being finite in the thermodynamic limit. Using these
results we can conclude that the mean number of relevant nodes increases for
large $N$ as $N^{1/3}$, with only a finite number of relevant nodes having two
relevant inputs, and a vanishing fraction of nodes having more than three of
them. 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.
