Kazumoto Iguchi | Search PubMed

PubMed2005

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.

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PubMed2005

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. Using this, we obtain the exact exponents of the distribution function P (k) of nodes with k degree in the asymptotic limit of k-->infinity . We show that the degree distribution for the hub nodes exhibits the scale-free nature, P (k) proportional to k(-gamma) with gamma=ln 3/ln 2=1.584 962 , while the degree distribution for the rim nodes is given by P(k) proportional to e(-gamma'k) with gamma' =ln (3/2) =0.405 465 . Second, we analytically calculate the second-order average degree of nodes, d(-) . Third, we numerically as well as analytically calculate the spectra of the adjacency matrix A for representing topology of the network. We also analytically obtain the exact number of degeneracies at each eigenvalue in the network. The density of states (i.e., the distribution function of eigenvalues) exhibits the fractal nature with respect to the degeneracy. Fourth, we study the mathematical structure of the determinant of the eigenequation for the adjacency matrix. Fifth, we study hidden symmetry, zero modes, and its index theorem in the deterministic scale-free network. Finally, we study the nature of the maximum eigenvalue in the spectrum of the deterministic scale-free network. We will prove several theorems for it, using some mathematical theorems. Thus, we show that most of all important quantities in the network theory can be analytically obtained in the deterministic scale-free network model of Barabási, Ravasz, and Vicsek. Therefore, we may call this network model the exactly solvable scale-free network.

Journal Article↗

PubMed2005

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].

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