Intrinsic properties of Boolean dynamics in complex networks.

We study intrinsic properties of attractor in Boolean dynamics of complex networks with scale-free topology, comparing with those of the so-called Kauffman's random Boolean networks. We numerically study both frozen and relevant nodes in each attractor in the dynamics of relatively small networks (20 View Article and Find Full Text PDF

Boolean dynamics of Kauffman models with a scale-free network.

We studied 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 approximately 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.

Rugged fitness landscapes of Kauffman models with a scale-free network.

We study the nature of the fitness landscapes of a "quenched" Kauffman's Boolean model with a scale-free network. We have numerically calculated the rugged fitness landscapes, the distributions, their tails, and the correlation between the fitness of local optima and their Hamming distance from the highest optimum found, respectively. We have found that (a) there is an interesting difference between random and scale-free networks such that the statistics of the rugged fitness landscapes is Gaussian for the random network while it is non-Gaussian with a tail for the scale-free network; (b) as the average degree [k] increases, there is a phase transition at the critical value of [k]=[k]c=2, below which there is a global order and above which the order goes away.

Vibrational modes and spectrum of oscillators on a scale-free network.

We study vibrational modes and spectrum of a model system of atoms and springs on a scale-free network where we assume that the atoms and springs are distributed as nodes and links of a scale-free network. To understand the nature of excitations with many degrees of freedom on the scale-free network, we adopt a particular model that we assign the mass M(i) and the specific oscillation frequency omega(i) of the ith atom and the spring constant K(ij) between the ith and jth atoms. We show that the density of states of the spectrum follows a scaling law P (omega(2)) proportional, variant (omega(2))(-gamma), where gamma = 3 and that as the number of nodes N is increasing, the maximum eigenvalue grows as fast as sqrt[N].

Exactly solvable scale-free network model.

We study a deterministic scale-free network recently proposed by Barabási, Ravasz, and Vicsek. We find that there are two types of nodes: the hub and rim nodes, which form a bipartite structure of the network. We first derive the exact numbers P (k) of nodes with degree k for the hub and rim nodes in each generation of the network, respectively.