Rain,power laws,and advection

Localized rain events have been found to follow power-law size and duration distributions over several decades, suggesting parallels between precipitation and seismic activity [O. Peters, Phys. Rev. Lett. 88, 018701 (2002)]]. Similar power laws are generated by treating rain as a passive tracer undergoing advection in a velocity field generated by a two-dimensional system of point vortices.     

Comparison of Pareto and tapered Pareto distributions for environmental phenomena

The Pareto distribution is often used to describe environmental phenomena such as the sizes of earthquakes or wildfires, or the interevent times or distances between such environmental disturbances. Because it is heavy-tailed, the Pareto distribution, or power-law distribution as it is occasionally called, suggests that a higher frequency of extremely large values occur compared to other, more familiar distributions such as the normal, exponential, or uniform. However, an alternative distribution called the tapered Pareto has been shown in some cases to fit as well or better to data than the Pareto distribution, and the tapered Pareto distribution is not heavy-tailed, suggesting a far lower frequency of extreme events. Even with rather large datasets, it is often quite difficult to distinguish which of these distributions is preferable, as they only differ markedly in the extreme upper tail where few, if any, observations are recorded. This article reviews the evidence and arguments related to these two competing distributions, especially in the context of earthquakes and wildfires.     

Mobile user forecast and power-law acceleration invariance of scale-free networks

This paper studies and predicts the number growth of China's mobile users by using the power-law regression. We find that the number growth of the mobile users follows a power law. Motivated by the data on the evolution of the mobile users, we consider scenarios of self-organization of accelerating growth networks into scale-free structures and propose a directed network model, in which the nodes grow following a power-law acceleration. The expressions for the transient and the stationary average degree distributions are obtained by using the Poisson process. This result shows that the model generates appropriate power-law connectivity distributions. Therefore, we find a power-law acceleration invariance of the scale-free networks. The numerical simulations of the models agree with the analytical results well.     

Fractional power-law spatial dispersion in electrodynamics

Electric fields in non-local media with power-law spatial dispersion are discussed. Equations involving a fractional Laplacian in the Riesz form that describe the electric fields in such non-local media are studied. The generalizations of Coulomb’s law and Debye’s screening for power-law non-local media are characterized. We consider simple models with anomalous behavior of plasma-like media with power-law spatial dispersions. The suggested fractional differential models for these plasma-like media are discussed to describe non-local properties of power-law type.     

Effects of introduction of new resources and fragmentation of existing resources on limiting wealth distribution in asset exchange models

Pareto law, which states that wealth distribution in societies has a power-law tail, has been the subject of intensive investigations in the statistical physics community. Several models have been employed to explain this behavior. However, most of the agent based models assume the conservation of number of agents and wealth. Both these assumptions are unrealistic. In this paper, we study the limiting wealth distribution when one or both of these assumptions are not valid. Given the universality of the law, we have tried to study the wealth distribution from the asset exchange models point of view. We consider models in which (a) new agents enter the market at a constant rate (b) richer agents fragment with higher probability introducing newer agents in the system (c) both fragmentation and entry of new agents is taking place. While models (a) and (c) do not conserve total wealth or number of agents, model (b) conserves total wealth. All these models lead to a power-law tail in the wealth distribution pointing to the possibility that more generalized asset exchange models could help us to explain the emergence of a power-law tail in wealth distribution.     

Scale-free behavior of the Internet global performance

Measurements and data analysis have proved very effective in the study of the Internet's physical fabric and have shown heterogeneities and statistical fluctuations extending over several orders of magnitude. Here we focus on the relationship between the Round-Trip-Time (RTT) and the geographical distance. We define dimensionless variables that contain information on the quality of Internet connections finding that their probability distributions are characterized by a slow power-law decay signalling the presence of scale-free features. These results point out the extreme heterogeneity of Internet delay since the transmission speed between different points of the network exhibits very large fluctuations. The associated scaling exponents appear to have fairly stable values in different data sets and thus define an invariant characteristic of the Internet that might be used in the future as a benchmark of the overall state of “health” of the Internet. Received 25 January 2003 Published online 7 May 2003     

Runaway events dominate the heavy tail of citation distributions

Statistical distributions with heavy tails are ubiquitous in natural and social phenomena. Since the entries in heavy tail have unproportional significance, the knowledge of its exact shape is very important. Citations of scientific papers form one of the best-known heavy tail distributions. Even in this case there is a considerable debate whether citation distribution follows the log-normal or power-law fit. The goal of our study is to solve this debate by measuring citation distribution for a very large and homogeneous data. We measured citation distribution for 418, 438 Physics papers published in 1980–1989 and cited by 2008. While the log-normal fit deviates too strong from the data, the discrete power-law function with the exponent γ = 3.15 does better and fits 99.955% of the data. However, the extreme tail of the distribution deviates upward even from the power-law fit and exhibits a dramatic “runaway” behavior. The onset of the runaway regime is revealed macroscopically as the paper garners 1000-1500 citations, however the microscopic measurements of autocorrelation in citation rates are able to predict this behavior in advance.     

Flat-head power-law, size-independent clustering, and scaling of coevolutionary scale-free networks

Scale-free topology and high clustering coexist in some real networks, and keep invariant for growing sizes of the systems. Previous models could hardly give out size-independent clustering with selforganized mechanism when succeeded in producing power-law degree distributions. Always ignored, some empirical statistic results display flat-head power-law behaviors. We modify our recent coevolutionary model to explain such phenomena with the inert property of nodes to retain small portion of unfavorable links in self-organized rewiring process. Flat-head power-law and size-independent clustering are induced as the new characteristics by this modification. In addition, a new scaling relation is found as the result of interplay between node state growth and adaptive variation of connections.     

Precursory phenomena associated with large avalanches in the long-range connective sandpile (LRCS) model

Reduction in b-values before a large earthquake is a very popular topic for discussion. This study proposes an alternative sandpile model being able to demonstrate reduction in scaling exponents before large events through adaptable long-range connections. The distant connection between two separated cells was introduced in the sandpile model. We found that our modified long-range connective sandpile (LRCS) system repeatedly approaches and retreats from a critical state. When a large avalanche occurs in the LRCS model, accumulated energy dramatically dissipates and the system simultaneously retreats from criticality. The system quickly approaches the critical state accompanied by the increase in the slopes of the power-law frequency-size distributions of events. Afterwards, and most interestingly, the power-law slope declines before the next large event. The precursory b-value reduction before large earthquakes observed from earthquake catalogues closely mimics the evolution in power-law slopes for the frequency-size distributions of events derived in the LRCS models. Our paper, thus, provides a new explanation for declined b-values before large earthquakes.     

Fluctuations in Wikipedia access-rate and edit-event data

Internet-based social networks often reflect extreme events in nature and society by drastic increases in user activity. We study and compare the dynamics of the two major complex processes necessary for information spread via the online encyclopedia ‘Wikipedia’, i.e., article editing (information upload) and article access (information viewing) based on article edit-event time series and (hourly) user access-rate time series for all articles. Daily and weekly activity patterns occur in addition to fluctuations and bursting activity. The bursts (i.e., significant increases in activity for an extended period of time) are characterized by a power-law distribution of durations of increases and decreases. For describing the recurrence and clustering of bursts we investigate the statistics of the return intervals between them. We find stretched exponential distributions of return intervals in access-rate time series, while edit-event time series yield simple exponential distributions. To characterize the fluctuation behavior we apply detrended fluctuation analysis (DFA), finding that most article access-rate time series are characterized by strong long-term correlations with fluctuation exponents α≈0.9$\alpha \approx 0.9$. The results indicate significant differences in the dynamics of information upload and access and help in understanding the complex process of collecting, processing, validating, and distributing information in self-organized social networks.     

Scaling Properties of Field-Induced Superdiffusion in Continuous Time Random Walks

We consider a broad class of Continuous Time Random Walks (CTRW) with large fluctuations effects in space and time distributions: a random walk with trapping, describing subdiffusion in disordered and glassy materials, and a Lévy walk process, often used to model superdiffusive effects in inhomogeneous materials. We derive the scaling form of the probability distributions and the asymptotic properties of all its moments in the presence of a field by two powerful techniques, based on matching conditions and on the estimate of the contribution of rare events to power-law tails in a field.     

Inertial ranges for turbulent solutions of complex Ginzburg-Landau equations

We consider the spatially periodic, complex Ginzburg-Landau (CGL) equation in regimes close to that of a critical or supercritical focusing non-linear Schrödinger (NLS) equation, which is known to have solutions that exhibit self-similar blow-up. We use the NLS blow-up solutions as a template to develop a theory of how nearly self-similar intermittent burst events can create a power-law inertial range in the time-averaged wave-number spectrum of CGL solutions. Numerical experiments in one dimension with a quintic (critical) and septant (supercritical) non-linearity show a that power-law inertial range emerges which differs from that predicted by the theory. However, as one approaches the NLS limit in the supercritical case, a second power-law inertial range is seen to emerge that agrees with the theory.     

Tuning degree distributions: Departing from scale-free networks

Scale-free networks are characterized by a degree distribution with power-law behavior. Although scale-free networks have been shown to arise in many areas, ranging from the World Wide Web to transportation or social networks, degree distributions of other observed networks often differ from the power-law type. Data based investigations require modifications of the typical scale-free network.We present an algorithm that generates networks in which the shape of the degree distribution is tunable by modifying the preferential attachment step of the Barabási-Albert construction algorithm. The shape of the distribution is represented by dispersion measures such as the variance and the skewness, both of which are highly correlated with the maximal degree of the network and, therefore, adequately represents the influence of superspreaders or hubs. By combining our algorithm with work of Holme and Kim, we show how to generate networks with a variety of degree distributions and clustering coefficients.     

Velocity statistics distinguish quantum turbulence from classical turbulence

By analyzing trajectories of solid hydrogen tracers, we find that the distributions of velocity in decaying quantum turbulence in superfluid 4He are strongly non-Gaussian with 1/v(3) power-law tails. These features differ from the near-Gaussian statistics of homogenous and isotropic turbulence of classical fluids. We examine the dynamics of many events of reconnection between quantized vortices and show by simple scaling arguments that they produce the observed power-law tails.     

The critical properties of the agent-based model with environmental-economic interactions

The steady-state and nonequilibrium properties of the model of environmental-economic interactions are studied. The interacting heterogeneous agents are simulated on the platform of the emission dynamics of cellular automaton. The diffusive emissions are produced by the factory agents, and the local pollution is monitored by the randomly walking (mobile) sensors. When the threshold concentration is exceeded, a feedback signal is transmitted from the sensor to the nearest factory that affects its actual production rate. The model predicts the discontinuous phase transition between safe and catastrophic ecology. Right at the critical line, the broad-scale power-law distributions of emission rates have been identified. The power-law fluctuations are triggered by the screening effect of factories and by the time delay between the environment contamination and its detection. The system shows the typical signs of the self-organized critical systems, such as power-law distributions and scaling laws. The text was submitted by the authors in English.     

Local Events and Dynamics on Weighted Complex Networks

We examine the weighted networks grown and evolved by local events, such as the addition of new vertices and links and we show that depending on frequency of the events, a generalized power-law distribution of strength can emerge. Continuum theory is used to predict the scaling function as well as the exponents, which is in good agreement with the numerical simulation results. Depending on event frequency, power-law distributions of degree and weight can also be expected. Probability saturation phenomena for small strength and degree in many real world networks can be reproduced. Particularly, the non-trivial clustering coefficient, assortativity coefficient and degree-strength correlation in our model are all consistent with empirical evidences.     

On the probability distribution of stock returns in the Mike-Farmer model

Recently, Mike and Farmer have constructed a very powerful and realistic behavioral model to mimick the dynamic process of stock price formation based on the empirical regularities of order placement and cancelation in a purely order-driven market, which can successfully reproduce the whole distribution of returns, not only the well-known power-law tails, together with several other important stylized facts. There are three key ingredients in the Mike-Farmer (MF) model: the long memory of order signs characterized by the Hurst index H_s, the distribution of relative order prices x in reference to the same best price described by a Student distribution (or Tsallis’ q-Gaussian), and the dynamics of order cancelation. They showed that different values of the Hurst index H_s and the freedom degree α_x of the Student distribution can always produce power-law tails in the return distribution f_r(r) with different tail exponent α_r. In this paper, we study the origin of the power-law tails of the return distribution f_r(r) in the MF model, based on extensive simulations with different combinations of the left part L(x) for x < 0 and the right part R(x) for x > 0 of f_x(x). We find that power-law tails appear only when L(x) has a power-law tail, no matter R(x) has a power-law tail or not. In addition, we find that the distributions of returns in the MF model at different timescales can be well modeled by the Student distributions, whose tail exponents are close to the well-known cubic law and increase with the timescale.     

Fractional-order formulation of power-law and exponential distributions

We present a novel approach to data analysis using fractional order calculus. In principle the approach can be applied to any distribution and shows remarkable improvement even if the parameters of a particular distribution have been optimised to achieve the best fit to data. The method is demonstrated for two important distributions that are used in data analysis, namely, power-law and exponential distributions. We show that the approach can allow composite distributions to be constructed for improved accuracy and robustness in the characterisation of data sets.     

Preservation of long range temporal correlations under extreme random dilution

Many natural time series exhibit long range temporal correlations that may be characterized by power-law scaling exponents. However, in many cases, the time series have uneven time intervals due to, for example, missing data points, noisy data, and outliers. Here we study the effect of randomly missing data points on the power-law scaling exponents of time series that are long range temporally correlated. The Fourier transform and detrended fluctuation analysis (DFA) techniques are used for scaling exponent estimation. We find that even under extreme dilution of more than 50%, the value of the scaling exponent remains almost unaffected. Random dilution is also applied on heart interbeat interval time series. It is found that dilution of 70%-80% of the data points leads to a reduction of only 8% in the scaling exponent; it is also found that it is possible to discriminate between healthy and heart failure subjects even under extreme dilution of more than 90%.     

Dragon-Kings in rock fracturing: Insights gained from rock fracture tests in the laboratory

In order to shed some lights to the “dragon-kings” concept, this paper re-examines experimental results on rock fracture tests in the laboratory, obtained from acoustic emission monitoring. The fracture of intact rocks as well as rocks containing natural structures (joints, faults, foliations) under constant stress rate loading or creep conditions is generally characterized by typical stages with different underlying physics. The primary phase reflects the initial rupture of pre-existing microcrack population in the sample or in the fault zone. Sub-critical growth dominates the secondary phase. The third phases termed nucleation phase corresponds to the initiation and accelerated growth of the ultimate fracture. The secondary and nucleation phases in both intact rock and faulted rock show power-law (of time-to-failure) increasing event rate and moment release. Samples containing planar structures such as foliations and faults demonstrate very similar features to natural earthquakes including: 1) small number of immediate foreshocks by which fault nucleation zones could be mapped; 2) the critical nucleation zone size is normally a fraction of the sample dimension; 3) a lot of aftershocks concentrated on the fault ruptured during the main event; 4) stress drop due to the main rupture is of the order from a few tens to a few hundreds MPa; 5) b-value drops during foreshocks and recovers during the aftershocks. All these results agree with the suggestion that laboratory measurements require no scaling but can be applied directly to the Earth to represent local fault behavior. The ultimate failure of the sample, or fracture of major asperities on the fault surface, normally lead to extreme events, i.e., dragon-kings, which has a magnitude significantly greater than that expected by the Gutenberg-Richter power-law relation in the magnitude-frequency distribution for either foreshocks or aftershocks. There are at least two mechanisms that may lead to dragon-kings: 1) The power-law increasing event rate and moment release; and 2) Hierarchical fracturing behavior resulting from hierarchical inhomogeneities in the sample. In the 1st mechanism, the final failure corresponds to the end point of the progressive occurrence of events and thus the resulted dragon-king event can be interpreted as a superposition of many small events. While for the 2nd mechanism an event of extreme size is the result of fracture growth stepping from a lower hierarchy into a higher hierarchy on fault surface having asperities characterized by hierarchical distribution (of size or strength) rather than simple fractal distribution. In both mechanisms the underlying physics is that fracture in rocks is hard to stop beyond certain threshold corresponding to the critical nucleation zone size.