Power-law Hypothesis Research Articles

Distinguishing subsampled power laws from other heavy-tailed distributions.

Distinguishing power-law distributions from other heavy-tailed distributions is challenging, and this task is often further complicated by subsampling effects. In this work, we evaluate the performance of two commonly used methods for detecting power-law distributions-the maximum likelihood method of Clauset etal. and the extreme value method of Voitalov etal.-in distinguishing subsampled power laws from two other heavy-tailed distributions, the lognormal and the stretched exponential distributions. We focus on a random subsampling method commonly applied in network science and biological sciences. In this subsampling scheme, we are ultimately interested in the frequency distribution of elements with a certain number of constituent parts-for example, species with k individuals or nodes with k connections-and each part is selected to the subsample with an equal probability. We investigate how well the results obtained from low-subsampling-depth subsamples generalize to the original distribution. Our results show that the power-law exponent of the original distribution can be estimated fairly accurately from subsamples, but classifying the distribution correctly is more challenging. The maximum likelihood method falsely rejects the power-law hypothesis for a large fraction of subsamples from power-law distributions. While the extreme value method correctly recognizes subsampled power-law distributions with all tested subsampling depths, its capacity to distinguish power laws from the heavy-tailed alternatives is limited. However, these false positives tend to result not from the subsampling itself but from the estimators' inability to classify the original sample correctly. In fact, we show that the extreme value method can sometimes be expected to perform better on subsamples than on the original samples from the lognormal and the stretched exponential distributions, while the contrary is true for the main tests included in the maximum likelihood method.

Testing the power-law hypothesis of the interconflict interval

War is an extreme form of collective human behaviour characterized by coordinated violence. We show that this nature of war is substantiated in the temporal patterns of conflict occurrence that obey power law. The focal metric is the interconflict interval (ICI), the interval between the end of a conflict in a dyad (i.e. a pair of states) and the start of the subsequent conflict in the same dyad. Using elaborate statistical tests, we confirmed that ICI samples compiled from the history of interstate conflicts from 1816 to 2014 followed a power-law distribution. We then demonstrate that the power-law properties of ICIs can be explained by a hypothetical model assuming an information-theoretic formulation of the Clausewitz thesis on war: the use of force is a means of interstate communication. Our findings help us to understand the nature of wars between regular states, the significance of which has increased since the Russian invasion of Ukraine in 2022.

Theoretical framework for the emergent floe size distribution in the marginal ice zone: the case for log-normality.

Sea ice is not horizontally homogeneous on large scales. Its morphology is inherently discrete and made of individual floes. In recent years, sea ice models have incorporated this horizontal heterogeneity. The modelling framework considers an evolution equation for the probability density function of the floe size distribution (FSD) with forcing terms that represent the effects of several physical processes. Despite the modelling effort, a key question remains: What is the FSD emerging from the collection of all forcing processes? Field observations have long suggested that the FSD follows a power law, but this result has not been reproduced by models or laboratory experiments. The theoretical framework for FSD dynamics in response to physical forcings is presented. Wave-induced breakup is further examined with an emphasis on how it affects the FSD. Recent modelling results suggesting the consistent emergence of a log-normal distribution as a result of that process are further discussed. Log-normality is also found in a dataset of floe sizes, which was originally analysed under the power law hypothesis. A simple stochastic process of FSD dynamics, based on random fragmentation theory, is further shown to predict log-normality. We therefore conjecture that, in some situations, the emergent FSD follows a log-normal distribution. This article is part of the theme issue 'Theory, modelling and observations of marginal ice zone dynamics: multidisciplinary perspectives and outlooks'.

PageRank asymptotics on directed preferential attachment networks

We characterize the tail behavior of the distribution of the PageRank of a uniformly chosen vertex in a directed preferential attachment graph and show that it decays as a power law with an explicit exponent that is described in terms of the model parameters. Interestingly, this power law is heavier than the tail of the limiting in-degree distribution, which goes against the commonly accepted power law hypothesis. This deviation from the power law hypothesis points at the structural differences between the inbound neighborhoods of typical vertices in a preferential attachment graph versus those in static random graph models where the power law hypothesis has been proven to hold (e.g., directed configuration models and inhomogeneous random digraphs). In addition to characterizing the PageRank distribution of a typical vertex, we also characterize the explicit growth rate of the PageRank of the oldest vertex as the network size grows.

Improved Characterization of the Astrophysical Muon–neutrino Flux with 9.5 Years of IceCube Data

Abstract We present a measurement of the high-energy astrophysical muon–neutrino flux with the IceCube Neutrino Observatory. The measurement uses a high-purity selection of 650k neutrino-induced muon tracks from the northern celestial hemisphere, corresponding to 9.5 yr of experimental data. With respect to previous publications, the measurement is improved by the increased size of the event sample and the extended model testing beyond simple power-law hypotheses. An updated treatment of systematic uncertainties and atmospheric background fluxes has been implemented based on recent models. The best-fit single power-law parameterization for the astrophysical energy spectrum results in a normalization of ϕ @ 100 TeV ν μ + ν ¯ μ = 1.44 − 0.26 + 0.25 × 10 − 18 GeV − 1 cm − 2 s − 1 sr − 1 and a spectral index γ SPL = 2.37 − 0.09 + 0.09 , constrained in the energy range from 15 TeV to 5 PeV. The model tests include a single power law with a spectral cutoff at high energies, a log-parabola model, several source-class-specific flux predictions from the literature, and a model-independent spectral unfolding. The data are consistent with a single power-law hypothesis, however, spectra with softening above one PeV are statistically more favorable at a two-sigma level.

The Generalized Gielis Geometric Equation and Its Application

Many natural shapes exhibit surprising symmetry and can be described by the Gielis equation, which has several classical geometric equations (for example, the circle, ellipse and superellipse) as special cases. However, the original Gielis equation cannot reflect some diverse shapes due to limitations of its power-law hypothesis. In the present study, we propose a generalized version by introducing a link function. Thus, the original Gielis equation can be deemed to be a special case of the generalized Gielis equation (GGE) with a power-law link function. The link function can be based on the morphological features of different objects so that the GGE is more flexible in fitting the data of the shape than its original version. The GGE is shown to be valid in depicting the shapes of some starfish and plant leaves.

Local weak convergence for PageRank

Page Rank is a well-known algorithm for measuring centrality in networks. It was originally proposed by Google for ranking pages in the World Wide Web. One of the intriguing empirical properties of PageRank is the socalled 'power-law hypothesis': in a scale-free network, the PageRank scores follow a power law with the same exponent as the (in-)degrees. To date, this hypothesis has been confirmed empirically and in several specific random graphs models. In contrast, this paper does not focus on one random graph model but investigates the existence of an asymptotic PageRank distribution, when the graph size goes to infinity, using local weak convergence. This may help to identify general network structures in which the power-law hypothesis holds. We start from the definition of local weak convergence for sequences of (random) undirected graphs, and extend this notion to directed graphs. To this end, we define an exploration process in the directed setting that keeps track of in- and out-degrees of vertices. Then we use this to prove the existence of an asymptotic PageRank distribution. As a result, the limiting distribution of PageRank can be computed directly as a function of the limiting object. We apply our results to the directed configuration model and continuous-time branching processes trees, as well as to preferential attachment models.

Estimating the sea ice floe size distribution using satellite altimetry: theory, climatology, and model comparison

Abstract. In sea-ice-covered areas, the sea ice floe size distribution (FSD) plays an important role in many processes affecting the coupled sea–ice–ocean–atmosphere system. Observations of the FSD are sparse – traditionally taken via a painstaking analysis of ice surface photography – and the seasonal and inter-annual evolution of floe size regionally and globally is largely unknown. Frequently, measured FSDs are assessed using a single number, the scaling exponent of the closest power-law fit to the observed floe size data, although in the absence of adequate datasets there have been limited tests of this “power-law hypothesis”. Here we derive and explain a mathematical technique for deriving statistics of the sea ice FSD from polar-orbiting altimeters, satellites with sub-daily return times to polar regions with high along-track resolutions. Applied to the CryoSat-2 radar altimetric record, covering the period from 2010 to 2018, and incorporating 11 million individual floe samples, we produce the first pan-Arctic climatology and seasonal cycle of sea ice floe size statistics. We then perform the first pan-Arctic test of the power-law hypothesis, finding limited support in the range of floe sizes typically analyzed in photographic observational studies. We compare the seasonal variability in observed floe size to fully coupled climate model simulations including a prognostic floe size and thickness distribution and coupled wave model, finding good agreement in regions where modeled ocean surface waves cause sea ice fracture.

A tale of two tails: Do Power Law and Lognormal models fit firm-size distributions in the mid-Victorian era?

The paper explores the frequency and size distributions of firm-size in a novel dataset for the mid-Victorian era from a recent extraction of the England and Wales population censuses of 1851, 1861, 1871, and 1881. The paper contrasts the hypothesis of the Power Laws against the Lognormal model for the tails of the distributions using maximum likelihood estimation, log likelihood ratio, clipped sample coefficient of variation UMPU-Wilks test, Kolmogorov–Smirnov statistic, among other state-of-the-art statistical methods. Our results show that the Power Law hypothesis is accepted for the size distribution for the years 1851 and 1861, while 1871 is marginally non-significant, but for 1881 the test is inconclusive. The paper discusses the process that generates these distributions citing recent literature that shows how after adding an i.i.d. noise to the Gibrat’s multiplicative model one can recreate a Power Law behaviour. Overall, the paper provides, describes and statistically tests for the very first time a unique historical dataset confirming that the tails of the distributions at least for 1851 and 1861 follow a Pareto model and that the Lognormal model is firmly rejected.

Complex network description of the ionosphere

Abstract. Complex networks have emerged as an essential approach of geoscience to generate novel insights into the nature of geophysical systems. To investigate the dynamic processes in the ionosphere, a directed complex network is constructed, based on a probabilistic graph of the vertical total electron content (VTEC) from 2012. The results of the power-law hypothesis test show that both the out-degree and in-degree distribution of the ionospheric network are not scale-free. Thus, the distribution of the interactions in the ionosphere is homogenous. None of the geospatial positions play an eminently important role in the propagation of the dynamic ionospheric processes. The spatial analysis of the ionospheric network shows that the interconnections principally exist between adjacent geographical locations, indicating that the propagation of the dynamic processes primarily depends on the geospatial distance in the ionosphere. Moreover, the joint distribution of the edge distances with respect to longitude and latitude directions shows that the dynamic processes travel further along the longitude than along the latitude in the ionosphere. The analysis of “small-world-ness” indicates that the ionospheric network possesses the small-world property, which can make the ionosphere stable and efficient in the propagation of dynamic processes.

The effect of fire on an abstract forest ecosystem: An agent based study

Our model considers a new element in forest fire modeling, namely the dynamics of a forest animal, intimately linked to the trees. We show that animals and trees react differently to different types of fire. A high probability of fire initiation results in several small fires, which do not allow for a large fuel accumulation and thus the destruction of many trees by fire, but is found to be generally devastating to the animal population at the same time. On the other hand, a low fire initiation probability allows for the accumulation of higher quantities of fuel, which in turn results in larger fires, more devastating to the trees than to the animals. Thus, we suggest that optimal fire management should take into account the relation between fire initiation and its different effects on animals and trees. Further, wildfires are often considered as prime examples for power-law-like frequency distributions, yet there is no agreement on the mechanisms responsible for the observed patterns. Our model suggests that instead of a single unified distribution, a superposition of at least two different distributions can be detected and this suggests multiform mechanisms acting on different scales. None of the discovered distributions are compatible with the power-law hypothesis.

City size distribution across the OECD: Does the definition of cities matter?

This study provides new comparative evidence on city size distribution in OECD (the Organisation for Economic Co-operation and Development) countries, by using consistently defined functional urban areas (FUAs). FUAs are identified by an algorithm based on population density at grid level and commuting patterns and are thought to better approximate economic agglomerations and their internal spatial organisation. Results show that Zipf's law provides a better description of city size distribution when cities are measured in terms of FUAs rather than using traditional administrative definitions. In addition, Zipf's law describes well city size distribution both at country level and wider spatial scales, that is, by continent and for the whole OECD. Finally, the power law hypothesis – of which Zipf's law is a particular case – was not rejected in most of the countries.

Power laws in citation distributions: evidence from Scopus.

Modeling distributions of citations to scientific papers is crucial for understanding how science develops. However, there is a considerable empirical controversy on which statistical model fits the citation distributions best. This paper is concerned with rigorous empirical detection of power-law behaviour in the distribution of citations received by the most highly cited scientific papers. We have used a large, novel data set on citations to scientific papers published between 1998 and 2002 drawn from Scopus. The power-law model is compared with a number of alternative models using a likelihood ratio test. We have found that the power-law hypothesis is rejected for around half of the Scopus fields of science. For these fields of science, the Yule, power-law with exponential cut-off and log-normal distributions seem to fit the data better than the pure power-law model. On the other hand, when the power-law hypothesis is not rejected, it is usually empirically indistinguishable from most of the alternative models. The pure power-law model seems to be the best model only for the most highly cited papers in “Physics and Astronomy”. Overall, our results seem to support theories implying that the most highly cited scientific papers follow the Yule, power-law with exponential cut-off or log-normal distribution. Our findings suggest also that power laws in citation distributions, when present, account only for a very small fraction of the published papers (less than 1 % for most of science fields) and that the power-law scaling parameter (exponent) is substantially higher (from around 3.2 to around 4.7) than found in the older literature.

Econometric Balancing of a Social Accounting Matrix Under a Power-law Hypothesis

Shannon-Kullback-Leibler cross-entropy (SKLCE) is particularly useful when ergodic system inverse problems require a solution. Though empirical application using the Shanon-Gibbs approach has recently met with notable success, it suffers from its ergodicity, constraining all micro-states of the system to appear with identical odds. The present document aims at extending applications of a non-extensive cross-entropy model (NECE) for balancing an input output stochastic system. The model then postulates that economic activity is characterized by long run complex behavioural interactions between economic agents and/or economic sectors. Applying scaling property of a Power-law we present a model which successfully balances a Polish national social accounting matrix (SAM) expected to exhibit Warlasian general equilibrium features. The Rao-Cramer-Kullback inferential information indexes are proposed. We note that increasing relative weight on the disturbance component of the dual criterion function leads to higher values of the q-Tsallis complexity index while smaller disturbance weights produce q values closer to unity, the case of Gaussian distribution. The great advantage of the approach presented over rival techniques is its allowing for the generalisation of Gaussian law enabled by its capability of including heavy tall distributions. The approach also constitutes a powerful instrument for the assessment of complexity in the analysed statistical system thanks to the q-Tsallis parameter.

Pinning down the cosmic ray source mechanism with new IceCube data

Very recently the IceCube Collaboration has reported an observation of 28 neutrino candidates with energies between 50 TeV and 2 PeV, constituting a $4.1\ensuremath{\sigma}$ excess compared to the atmospheric background. In this article we investigate the compatibility between the data and a hypothesized unbroken power-law neutrino spectrum for various values of spectral index $\mathrm{\ensuremath{\Gamma}}\ensuremath{\ge}2$. We show that $\mathrm{\ensuremath{\Gamma}}\ensuremath{\sim}2.3$ is consistent at the $\ensuremath{\sim}1.5\ensuremath{\sigma}$ level with the observed events up to 2 PeV and to the null observation of events at higher energies. We then assume that the sources of this unbroken spectrum are Galactic, and deduce (i) an energy-transfer fraction from parent protons to pions (finding ${\ensuremath{\epsilon}}_{{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}}$ and ${\ensuremath{\epsilon}}_{\ensuremath{\pi}}$), and (ii) a way of discriminating among models which have been put forth to explain the ``knee'' and ``ankle'' features of the cosmic ray spectrum. Future IceCube data will test the unbroken power-law hypothesis and provide a multimessenger approach to explaining features of the cosmic ray spectrum, including the transition from Galactic to extragalactic dominance.

Power-law distributions in binned empirical data

Many man-made and natural phenomena, including the intensity of earthquakes, population of cities and size of international wars, are believed to follow power-law distributions. The accurate identification of power-law patterns has significant consequences for correctly understanding and modeling complex systems. However, statistical evidence for or against the power-law hypothesis is complicated by large fluctuations in the empirical distribution’s tail, and these are worsened when information is lost from binning the data. We adapt the statistically principled framework for testing the power-law hypothesis, developed by Clauset, Shalizi and Newman, to the case of binned data. This approach includes maximum-likelihood fitting, a hypothesis test based on the Kolmogorov–Smirnov goodness-of-fit statistic and likelihood ratio tests for comparing against alternative explanations. We evaluate the effectiveness of these methods on synthetic binned data with known structure, quantify the loss of statistical power due to binning, and apply the methods to twelve real-world binned data sets with heavy-tailed patterns.

Fitting and goodness-of-fit test of non-truncated and truncated power-law distributions

Recently, Clauset, Shalizi, and Newman have proposed a systematic method to find over which range (if any) a certain distribution behaves as a power law. However, their method has been found to fail, in the sense that true (simulated) power-law tails are not recognized as such in some instances, and then the power-law hypothesis is rejected. Moreover, the method does not work well when extended to power-law distributions with an upper truncation. We explain in detail a similar but alternative procedure, valid for truncated as well as for non-truncated power-law distributions, based in maximum likelihood estimation, the Kolmogorov-Smirnov goodness-of-fit test, and Monte Carlo simulations. An overview of the main concepts as well as a recipe for their practical implementation is provided. The performance of our method is put to test on several empirical data which were previously analyzed with less systematic approaches. We find the functioning of the method very satisfactory.

New tsunami runup relationships based on long wave experiments

Tsunami are propagating waves characterized by long wavelengths and large amplitudes close to the shore. They may also have a profile characterised by a large trough preceding the positive wave. These waves are destructive, causing severe damage to structures and human casualties when they reach coastal areas (e.g., Indian Ocean, 2004; Japan, 2011). Runup is a local characteristic of a wave flow inland and this measure, easily identifiable in the field, is extensively used as an indicator of tsunami inundation and impact on the coast.In this paper, an innovative large scale experimental programme is applied to develop runup equations for long propagating waves. A pneumatic generator with a controlled valve system, capable of exchanging large volumes of water with the propagation flume is used. This novel generation system allows waves to be generated that have much larger wavelengths than those experimentally reproduced to date. Moreover, it enables leading depressed waves to be stably generated and analysed for the first time.To analyse the influence of wavelength and wave shape on runup, the experimental data was partitioned into groups classified by wave period T/Tb (where Tb is the travel time of a linear wave along the length of the beach) and wave shape (elevated or N-waves). In this paper, elevated waves refer to waves of translation having a single positive elevation above mean water level, and N-waves refer to waves of translation comprising both a negative elevation (below mean water level) and a positive elevation. Dimensional analysis (using experimentally determined wavelength, potential energy and wave amplitudes) was applied to identify correlated measures. A statistical analysis was used to determine a power law relationship between runup and measures of the waveform, and to test the significance of the power law hypothesis. The experimental results show how both the wavelength and wave shape influence the runup distance. For TTb<1, the runup is seen to scale as R∼a, while for TTb>1, it scales as R∼a.

And Models: A Self-Similar Approach

We study the models and their particular case, the so-called -models under the self-similarity hypothesis. In particular, we calculate the exact form that each quantity may take in order that field equations (FEs) admit self-similar solutions. The methods employed allow us to obtain general results that are valid not only for the FRW metric, but also for all the Bianchi types as well as for the Kantowski-Sachs model (under the self-similarity hypothesis and the power-law hypothesis for the scale factors).

The Power Laws of Violence against Women: Rescaling Research and Policies

BackgroundViolence against Women –despite its perpetuation over centuries and its omnipresence at all social levels– entered into social consciousness and the general agenda of Social Sciences only recently, mainly thanks to feminist research, campaigns, and general social awareness. The present article analyzes in a secondary analysis of German prevalence data on Violence against Women, whether the frequency and severity of Violence against Women can be described with power laws.Principal FindingsAlthough the investigated distributions all resemble power-law distributions, a rigorous statistical analysis accepts this hypothesis at a significance level of 0.1 only for 1 of 5 cases of the tested frequency distributions and with some restrictions for the severity of physical violence. Lowering the significance level to 0.01 leads to the acceptance of the power-law hypothesis in 2 of the 5 tested frequency distributions and as well for the severity of domestic violence. The rejections might be mainly due to the noise in the data, with biases caused by self-reporting, errors through rounding, desirability response bias, and selection bias.ConclusionFuture victimological surveys should be designed explicitly to avoid these deficiencies in the data to be able to clearly answer the question whether Violence against Women follows a power-law pattern. This finding would not only have statistical implications for the processing and presentation of the data, but also groundbreaking consequences on the general understanding of Violence against Women and policy modeling, as the skewed nature of the underlying distributions makes evident that Violence against Women is a highly disparate and unequal social problem. This opens new questions for interdisciplinary research, regarding the interplay between environmental, experimental, and social factors on victimization.