Diffusion in scale-free networks with annealed disorder

The scale-free (SF) networks that have been studied so far contained quenched disorder generated by random dilution which does not vary with the time. In practice, if a SF network is to represent, for example, the worldwide web, then the links between its various nodes may temporarily be lost, and re-established again later on. This gives rise to SF networks with annealed disorder. Even if the disorder is quenched, it may be more realistic to generate it by a dynamical process that is happening in the network. In this paper, we study diffusion in SF networks with annealed disorder generated by various scenarios, as well as in SF networks with quenched disorder which, however, is generated by the diffusion process itself. Several quantities of the diffusion process are computed, including the mean number of distinct sites visited, the mean number of returns to the origin, and the mean number of connected nodes that are accessible to the random walkers at any given time. The results including, (1) greatly reduced growth with the time of the mean number of distinct sites visited; (2) blocking of the random walkers; (3) the existence of a phase diagram that separates the region in which diffusion is possible from one in which diffusion is impossible, and (4) a transition in the structure of the networks at which the mean number of distinct sites visited vanishes, indicate completely different behavior for the computed quantities than those in SF networks with quenched disorder generated by simple random dilution.Comment: 18 pages including 8 figure

Analysis of Non-stationary Data for Heart-Rate Fluctuations in Terms of Drift and Diffusion Coefficients

We describe a method for analyzing the stochasticity in the non-stationary data for the beat-to-beat fluctuations in the heart rates of healthy subjects, as well as those with congestive heart failure. The method analyzes the returns time series of the data as a Markov process, and computes the Markov time scale, i.e., the time scale over which the data are a Markov process. We also construct an effective stochastic continuum equation for the return series. We show that the drift and diffusion coefficients, as well as the amplitude of the returns time series for healthy subjects are distinct from those with CHF. Thus, the method may potentially provide a diagnostic tool for distinguishing healthy subjects from those with congestive heart failure, as it can distinguish small differences between the data for the two classes of subjects in terms of well-defined and physically-motivated quantities.Comment: 6 pages, two columns, 6 figure

Geometrical Phase Transition on WO3_3 Surface

A topographical study on an ensemble of height profiles obtained from atomic force microscopy techniques on various independently grown samples of tungsten oxide WO$_3$ is presented by using ideas from percolation theory. We find that a continuous 'geometrical' phase transition occurs at a certain critical level-height $\delta_c$ below which an infinite island appears. By using the finite-size scaling analysis of three independent percolation observables i.e., percolation probability, percolation strength and the mean island-size, we compute some critical exponents which characterize the transition. Our results are compatible with those of long-range correlated percolation. This method can be generalized to a topographical classification of rough surface models.Comment: 3 pages, 4 figures, to appear in Applied Physics Letters (2010

Alternative criterion for two-dimensional wrapping percolation

Based on the differences between a spanning cluster and a wrapping cluster, an alternative criterion for testing wrapping percolation is provided for two-dimensional lattices. By following the Newman-Ziff method, the finite size scaling of estimates for percolation thresholds are given. The results are consistent with those from Machta's method.Comment: 4 pages, 2 figure

Gaussian model of explosive percolation in three and higher dimensions

The Gaussian model of discontinuous percolation, recently introduced by Ara\'ujo and Herrmann [Phys. Rev. Lett., 105, 035701 (2010)], is numerically investigated in three dimensions, disclosing a discontinuous transition. For the simple-cubic lattice, in the thermodynamic limit, we report a finite jump of the order parameter, $J=0.415 \pm 0.005$. The largest cluster at the threshold is compact, but its external perimeter is fractal with fractal dimension $d_A = 2.5 \pm 0.2$. The study is extended to hypercubic lattices up to six dimensions and to the mean-field limit (infinite dimension). We find that, in all considered dimensions, the percolation transition is discontinuous. The value of the jump in the order parameter, the maximum of the second moment, and the percolation threshold are analyzed, revealing interesting features of the transition and corroborating its discontinuous nature in all considered dimensions. We also show that the fractal dimension of the external perimeter, for any dimension, is consistent with the one from bridge percolation and establish a lower bound for the percolation threshold of discontinuous models with finite number of clusters at the threshold

Perspectives for Monte Carlo simulations on the CNN Universal Machine

Possibilities for performing stochastic simulations on the analog and fully parallelized Cellular Neural Network Universal Machine (CNN-UM) are investigated. By using a chaotic cellular automaton perturbed with the natural noise of the CNN-UM chip, a realistic binary random number generator is built. As a specific example for Monte Carlo type simulations, we use this random number generator and a CNN template to study the classical site-percolation problem on the ACE16K chip. The study reveals that the analog and parallel architecture of the CNN-UM is very appropriate for stochastic simulations on lattice models. The natural trend for increasing the number of cells and local memories on the CNN-UM chip will definitely favor in the near future the CNN-UM architecture for such problems.Comment: 14 pages, 6 figure

Statistical Properties of the Interbeat Interval Cascade in Human Subjects

Statistical properties of interbeat intervals cascade are evaluated by considering the joint probability distribution $P(\Delta x_2,\tau_2;\Delta x_1,\tau_1)$ for two interbeat increments $\Delta x_1$ and $\Delta x_2$ of different time scales $\tau_1$ and $\tau_2$. We present evidence that the conditional probability distribution $P(\Delta x_2,\tau_2|\Delta x_1,\tau_1)$ may obey a Chapman-Kolmogorov equation. The corresponding Kramers-Moyal (KM) coefficients are evaluated. It is shown that while the first and second KM coefficients, i.e., the drift and diffusion coefficients, take on well-defined and significant values, the higher-order coefficients in the KM expansion are very small. As a result, the joint probability distributions of the increments in the interbeat intervals obey a Fokker-Planck equation. The method provides a novel technique for distinguishing the two classes of subjects in terms of the drift and diffusion coefficients, which behave differently for two classes of the subjects, namely, healthy subjects and those with congestive heart failure.Comment: 5 pages, 6 figure