Boolean Dynamics of Kauffman Models with a Scale-Free Network

Abstract

We study the Boolean dynamics of the &#34;quenched&#34; 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.