[1] Distributional analysis of river discharge time series is an important task in many areas of hydrological engineering, including optimal design of water storage and drainage networks, management of extreme events, risk assessment for water supply, and environmental flow management, among many others. Having diverging moments, heavy-tailed power law distributions have attracted widespread at...

In the study of the spectra of power-law graphs, there are basically two competing approaches. One is to prove analogues of Wigner's semicircle law, whereas the other predicts that the eigenvalues follow a power-law distribution. Although the semicircle law and the power law have nothing in common, we will show that both approaches are essentially correct if one considers the appropriate matric...

Herein we study energy exchange models of multiple interacting agents that conserve energy in each interaction. The models differ regarding the rules that regulate the energy exchange and boundary effects. We find a variety of stochastic behaviours that manifest energy equilibrium probability distributions of different types and interaction rules that yield not only the exponential distribution...

Extracting avalanche distributions from experimental microplasticity data can be hampered by limited time resolution. We compute the effects of low time resolution on avalanche size distributions and give quantitative criteria for diagnosing and circumventing problems associated with low time resolution. We show that traditional analysis of data obtained at low acquisition rates can lead to ava...

We summarize a book under publication with the above title written by the three present authors, on the theory of Zipf’s law, and more generally of power laws, driven by the mechanism of proportional growth. The preprint is available upon request from the authors. For clarity, consistence of language and conciseness, we discuss the origin and conditions of the validity of Zipf’s law using the t...

Pareto, Zipf and numerous subsequent investigators of inverse power distributions have often represented their findings as though their data conformed to a power law form for all ranges of the variable of interest. I refer to this ideal case as a strong inverse power law (SIPL). However, many of the examples used by Pareto and Zipf, as well as others who have followed them, have been truncated ...

Two models of binary fragmentation are introduced in which a time dependent transition size produces two regions of fragment sizes above and below the transition size. In the models we consider a fixed rate of fragmentation for the largest fragment and two different rates of fragmentation for the two regions of sizes above and below the transition size. The models are solved exactly in the long...

Citation distributions are crucial for the analysis and modeling of the activity of scientists. We investigated bibliometric data of papers published in journals of the American Physical Society, searching for the type of function which best describes the observed citation distributions. We used the goodness of fit with Kolmogorov-Smirnov statistics for three classes of functions: log-normal, s...

– We study how the presence of correlations in physical variables contributes to the form of probability distributions. We investigate a process with correlations in the variance generated by i) a Gaussian or ii) a truncated Lévy distribution. For both i) and ii), we find that due to the correlations in the variance, the process “dynamically” generates power law tails in the distributions, whos...

Cancer cells are characterized by having aberrant chromosomes. The number of aberrations and the specific chromosomes affected are correlated with tumor progression. We show that for breast, colorectal, and renal cell carcinomas the distribution of the number of such aberrations per tumor follow a power law distribution with an exponent close to unity. We present two stochastic models that in s...