Low prevalence, quasi-stationarity and power-law distribution in a model of spreading

Understanding how contagions (information, infections, etc) are spread on complex networks is important both from practical as well as theoretical point of view. Considerable work has been done in this regard in the past decade or so. However, most models are limited in their scope and as a result only capture general features of spreading phenomena. Here, we propose and study a model of spreading which takes into account the strength or quality of contagions as well as the local (probabilistic) dynamics occurring at various nodes. Transmission occurs only after the quality-based fitness of the contagion has been evaluated by the local agent. The model exhibits quality-dependent exponential time scales at early times leading to a slowly evolving quasi-stationary state. Low prevalence is seen for a wide range of contagion quality for arbitrary large networks. We also investigate the activity of nodes and find a power-law distribution with a robust exponent independent of network topology. Our results are consistent with recent empirical observations.Comment: 7 pages, 8 figures. (Submitted

Markov Properties of Electrical Discharge Current Fluctuations in Plasma

Using the Markovian method, we study the stochastic nature of electrical discharge current fluctuations in the Helium plasma. Sinusoidal trends are extracted from the data set by the Fourier-Detrended Fluctuation analysis and consequently cleaned data is retrieved. We determine the Markov time scale of the detrended data set by using likelihood analysis. We also estimate the Kramers-Moyal's coefficients of the discharge current fluctuations and derive the corresponding Fokker-Planck equation. In addition, the obtained Langevin equation enables us to reconstruct discharge time series with similar statistical properties compared with the observed in the experiment. We also provide an exact decomposition of temporal correlation function by using Kramers-Moyal's coefficients. We show that for the stationary time series, the two point temporal correlation function has an exponential decaying behavior with a characteristic correlation time scale. Our results confirm that, there is no definite relation between correlation and Markov time scales. However both of them behave as monotonic increasing function of discharge current intensity. Finally to complete our analysis, the multifractal behavior of reconstructed time series using its Keramers-Moyal's coefficients and original data set are investigated. Extended self similarity analysis demonstrates that fluctuations in our experimental setup deviates from Kolmogorov (K41) theory for fully developed turbulence regime.Comment: 25 pages, 9 figures and 4 tables. V3: Added comments, references, figures and major correction

Canonical horizontal visibility graphs are uniquely determined by their degree sequence

Horizontal visibility graphs (HVGs) are graphs constructed in correspondence with number sequences that have been introduced and explored recently in the context of graph-theoretical time series analysis. In most of the cases simple measures based on the degree sequence (or functionals of these such as entropies over degree and joint degree distributions) appear to be highly informative features for automatic classification and provide nontrivial information on the associated dynam- ical process, working even better than more sophisticated topological metrics. It is thus an open question why these seemingly simple measures capture so much information. Here we prove that, under suitable conditions, there exist a bijection between the adjacency matrix of an HVG and its degree sequence, and we give an explicit construction of such bijection. As a consequence, under these conditions HVGs are unigraphs and the degree sequence fully encapsulates all the information of these graphs, thereby giving a plausible reason for its apparently unreasonable effectiveness