  We study the Boolean dynamics of the "quenched" Kauffman models with a
directed scale-free network, comparing with that of the original directed
random Kauffman networks and that of the directed exponential-fluctuation
networks. We have numerically investigated the distributions of the state cycle
lengths and its changes as the network size $N$ and the average degree $<k>$ of
nodes increase. In the relatively small network ($N \sim 150$), the median, the
mean value and the standard deviation grow exponentially with $N$ in the
directed scale-free and the directed exponential-fluctuation networks with $<k
> =2 $, where the function forms of the distributions are given as an almost
exponential. We have found that for the relatively large $N \sim 10^3$ the
growth of the median of the distribution over the attractor lengths
asymptotically changes from algebraic type to exponential one as the average
degree $<k>$ goes to $<k > =2$. The result supports an existence of the
transition at $<k >_c =2$ derived in the annealed model.
