Boolean Dynamics of Kauffman Models with a Scale-Free Network

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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 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 10³ 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.