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. We have found that for the relatively large N approximately 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 the existence of the transition at k(c)=2 derived in the annealed 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.

Localization property in the disordered few-chain DNA systems with a long-range correlation is numerically investigated. We apply the chain system with the correlated disorder in the interchain and/or intrachain hoppings to the simple model of a double strand of DNA. Numerical results for the density of states and the Lyapunov exponent of the wave function in the two-or three-chain models are given. It is found that the correlation effect enhances the localization length (the inverse least nonnegative Lyapunov exponent) around the band center.

The effect of dynamical perturbation on the quantum localization phenomenon in a one-dimensional disordered quantum system (1DDS) is investigated systematically by a numerical method. The dynamical perturbation is modeled by an oscillatory driving force containing M independent (mutually incommensurate) frequency components. For M>or=2 a diffusive behavior emerges and in the presence of the finite localization length of the asymptotic wave packet can no longer be detected numerically. The diffusive motion obeys a subdiffusion law characterized by the exponent alpha as xi(t)(2) proportional t(alpha), where xi(t)(2) is the mean square displacement of the wave packet at time t. With an increase in M and/or the perturbation strength, the exponent alpha rapidly approaches 1, which corresponds to normal diffusion. Moreover, the space-time (x-t) dependence of the distribution function P(x,t) is reduced to a scaled form decided by alpha and another exponent beta such that P(x,t) approximately exp(-constx(x/t(alpha/2))(beta)), which contains the two extreme limits, i.e., the localization limit (alpha=0, beta=1) and the normal-diffusion limit (alpha=1, beta=2) in a unified manner. Some 1DDSs driven by the oscillatory perturbation in different ways are examined and compared.

We investigate the localization property of an electron in the disordered two-and three-chain systems (ladder model) with long-range correlation as a simple model for electronic property in a double strand of DNA. The chains are constructed by repetition of the sugar-phosphate sites, and the inter-chain hopping at the sugar sites come from nucleotide pairs, i.e., A − T or G − C pairs. It has been found that some DNA sequences have long-range correlation. In this paper we investigate the localization properties of the electronic states in some actual DNA sequences such as bacteriophages of escherichia coli, human chromosome 22 and histone protein. We will present some numerical results for the Lyapunov exponent (inverse localization length) of the wave function in the cases in comparison to the results for artificial sequence generated by an asymmetric modified Bernoulli map. It is shown that the correlation and asymmetry of the sequence affect on the localization in both the artificial and the real DNA sequences.