A review of power laws in real life phenomena
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paper_title: True reason for Zipf's law in language

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Analysis of word frequency have historically used data that included English, French, or other language, data typically described by Zipf's law. Using data on traditional and modern Chinese literatures, we show here that Chinese character frequency stroked Zipf's law based on literature before Qin dynasty; however, it departed from Zipf's law based on literature after Qin dynasty. Combined with data about English dictionaries and Chinese dictionaries, we show that the true reason for Zipf's Law in language is that growth and preferential selection mechanism of word or character in given language.
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paper_title: Common ecology quantifies human insurgency

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Many collective human activities, including violence, have been shown to exhibit universal patterns1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19. The size distributions of casualties both in whole wars from 1816 to 1980 and terrorist attacks have separately been shown to follow approximate power-law distributions6, 7, 9, 10. However, the possibility of universal patterns ranging across wars in the size distribution or timing of within-conflict events has barely been explored. Here we show that the sizes and timing of violent events within different insurgent conflicts exhibit remarkable similarities. We propose a unified model of human insurgency that reproduces these commonalities, and explains conflict-specific variations quantitatively in terms of underlying rules of engagement. Our model treats each insurgent population as an ecology of dynamically evolving, self-organized groups following common decision-making processes. Our model is consistent with several recent hypotheses about modern insurgency18, 19, 20, is robust to many generalizations21, and establishes a quantitative connection between human insurgency, global terrorism10 and ecology13, 14, 15, 16, 17, 22, 23. Its similarity to financial market models24, 25, 26 provides a surprising link between violent and non-violent forms of human behaviour.
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paper_title: Vertical transmission of culture and the distribution of family names

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A stochastic model for the evolution of a growing population is proposed, in order to explain empirical power-law distributions in the frequency of family names as a function of the family size. Preliminary results show that the predicted exponents are in good agreement with real data. The evolution of family-name distributions is discussed in the frame of vertical transmission of cultural features.
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paper_title: Growth dynamics of the World-Wide Web

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,The exponential growth of the World-Wide Web has transformed it into an ecology of knowledge in which highly diverse information is linked in an extremely complex and arbitrary manner. But even so, as we show here, there is order hidden in the web. We find that web pages are distributed among sites according to a universal power law: many sites have only a few pages, whereas very few sites have hundreds of thousands of pages. This universal distribution can be explained by using a simple stochastic dynamical growth model. The existence of a power law in the growth of the web not only implies the lack of any length scale for the web, but also allows the expected number of sites of any given size to be determined without exhaustively crawling the web. The distribution of site sizes for crawls by Alexa and Infoseek is shown in Fig. 1. Both data sets display a power law over several orders of magnitude, so on a log‐log scale the distribution of the number of pages per site appears as a straight line. This distribution should not be confused with Zipf ’s like distributions 1,2 , where a power law arises from rank ordering the variables 3 .
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paper_title: Applying Pareto distribution theory to electrolytic powder production

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Abstract Well known in several scientific and technical areas, the Pareto distribution of probability theory is employed to describe the distribution of mass fraction of metal or metal oxide powders produced by electrolytic means, with respect to limited quantitative observations of their particle size. The estimation of Pareto parameters from experimental mean/median or mean–variance data, and via the maximum likelihood method is specifically demonstrated.
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paper_title: Towards a seascape typology. I. Zipf versus Pareto laws

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Abstract Two data analysis methods, referred to as the Zipf and Pareto methods, initially introduced in economics and linguistics two centuries ago and subsequently used in a wide range of fields (word frequency in languages and literature, human demographics, finance, city formation, genomics and physics), are described and proposed here as a potential tool to classify space–time patterns in marine ecology. The aim of this paper is, first, to present the theoretical bases of Zipf and Pareto laws, and to demonstrate that they are strictly equivalent. In that way, we provide a one-to-one correspondence between their characteristic exponents and argue that the choice of technique is a matter of convenience. Second, we argue that the appeal of this technique is that it is assumption-free for the distribution of the data and regularity of sampling interval, as well as being extremely easy to implement. Finally, in order to allow marine ecologists to identify and classify any structure in their data sets, we provide a step by step overview of the characteristic shapes expected for Zipf's law for the cases of randomness, power law behavior, power law behavior contaminated by internal and external noise, and competing power laws illustrated on the basis of typical ecological situations such as mixing processes involving non-interacting and interacting species, phytoplankton growth processes and differential grazing by zooplankton.
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paper_title: Scale dependence in plant biodiversity.

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The relationship between the number of species and the area sampled is one of the oldest and best-documented patterns in community ecology. Several theoretical models and field data from a wide range of plant and animal taxa suggest that the slope, z, of a graph of the logarithm of species richness against the logarithm of area is roughly constant, with z approximately 0.25. We collected replicated and randomized plant data at 11 spatial scales from 0.01 to 10(8) square meters in Great Britain which show that the slope of the log-log plot is not constant, but varies systematically with spatial scale, and from habitat to habitat at the same spatial scale. Values of z were low (0.1 to 0.2) at small scales (<100 square meters), high (0.4 to 0.5) at intermediate scales (1 hectare to 10 square kilometers), and low again (0.1 to 0.2) for the largest scale transitions (e.g., East Berks to all of Berkshire). Instead of one process determining changes in species richness across a wide range of scales, different processes might determine plant biodiversity at different spatial scales.
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paper_title: An Abelian model for rainfall

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Statistical analyses of long-term records of daily rain suggest that rain phenomena might be a manifestation of self-organized criticality. In this work the essential mechanisms of rain phenomena, the growth of droplets inside a cloud and the subsequent rainfall, are described by an Abelian sandpile model of self-organized criticality. Several simulations support the existence of scale invariance. The introduction of variations of the basic model, to provide a better description of the phenomena, does not alter the critical behavior.
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paper_title: Self-similarity in World Wide Web traffic: evidence and possible causes

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The notion of self-similarity has been shown to apply to wide-area and local-area network traffic. We show evidence that the subset of network traffic that is due to World Wide Web (WWW) transfers can show characteristics that are consistent with self-similarity, and we present a hypothesized explanation for that self-similarity. Using a set of traces of actual user executions of NCSA Mosaic, we examine the dependence structure of WWW traffic. First, we show evidence that WWW traffic exhibits behavior that is consistent with self-similar traffic models. Then we show that the self-similarity in such traffic can be explained based on the underlying distributions of WWW document sizes, the effects of caching and user preference in file transfer, the effect of user "think time", and the superimposition of many such transfers in a local-area network. To do this, we rely on empirically measured distributions both from client traces and from data independently collected at WWW servers.
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paper_title: Organization of growing random networks

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The organizational development of growing random networks is investigated. These growing networks are built by adding nodes successively, and linking each to an earlier node of degree k with an attachment probability A(k). When A(k) grows more slowly than linearly with k, the number of nodes with k links, N(k)(t), decays faster than a power law in k, while for A(k) growing faster than linearly in k, a single node emerges which connects to nearly all other nodes. When A(k) is asymptotically linear, N(k)(t) approximately tk(-nu), with nu dependent on details of the attachment probability, but in the range 2<nu<infinity. The combined age and degree distribution of nodes shows that old nodes typically have a large degree. There is also a significant correlation in the degrees of neighboring nodes, so that nodes of similar degree are more likely to be connected. The size distributions of the in and out components of the network with respect to a given node--namely, its "descendants" and "ancestors"-are also determined. The in component exhibits a robust s(-2) power-law tail, where s is the component size. The out component has a typical size of order ln t, and it provides basic insights into the genealogy of the network.
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paper_title: The Gompertz-Pareto Income Distribution

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This work analyzes the Gompertz-Pareto distribution (GPD) of personal income, formed by the combination of the Gompertz curve, representing the overwhelming majority of the economically less favorable part of the population of a country, and the Pareto power law, which describes its tiny richest part. Equations for the Lorenz curve, Gini coefficient and the percentage share of the Gompertzian part relative to the total income are all written in this distribution. We show that only three parameters, determined by linear data fitting, are required for its complete characterization. Consistency checks are carried out using income data of Brazil from 1981 to 2007 and they lead to the conclusion that the GPD is consistent and provides a coherent and simple analytical tool to describe personal income distribution data.
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paper_title: Power Laws in Cities Population, Financial Markets and Internet Sites (Scaling in Systems with a variable number of components) Aharon Blank

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We study a few dynamical systems composed of many components whose sizes evolve according to multiplicative stochastic rules. We compare them with respect to the emergence of power laws in the size distribution of their components. We show that the details specifying and enforcing the smallest size of the components are crucial as well as the rules for creating new components. In particular, a growing system with a fixed number of components and a fixed smallest component size does not converge to a power law. We present a new model with variable number of components that converges to a power law for a very wide range of parameters. In a very large subset of this range, one obtains for the exponent α the special value 1 specific for the city populations distribution. We discuss the conditions in which α can take different values. In the case of the stock market, the distribution of the investors’ wealth is related to the ratio between the new capital invested in stock and the rate of increase of the stock index.
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paper_title: Scaling And Correlation In Financial Time Series

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We discuss the results of three recent phenomenological studies focussed on understanding the distinctive statistical properties of financial time series – (i) The probability distribution of stock price fluctuations: Stock price fluctuations occur in all magnitudes, in analogy to earthquakes – from tiny fluctuations to very drastic events, such as the crash of 19 October 1987, sometimes referred to as “Black Monday”. The distribution of price fluctuations decays with a power-law tail well outside the Levy stable regime and describes fluctuations that differ by as much as 8 orders of magnitude. In addition, this distribution preserves its functional form for fluctuations on time scales that differ by 3 orders of magnitude, from 1 min up to approximately 10 days. (ii) Correlations in financial time series: While price fluctuations themselves have rapidly decaying correlations, the magnitude of fluctuations measured by either the absolute value or the square of the price fluctuations has correlations that decay as a power-law, persisting for several months. (iii) Volatility and trading activity: We quantify the relation between trading activity – measured by the number of transactions NΔt – and the price change GΔt for a given stock, over a time interval [t,t+Δt]. We find that NΔt displays long-range power-law correlations in time, which leads to the interpretation that the long-range correlations previously found for |GΔt| are connected to those of NΔt.
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paper_title: Mechanical model of an earthquake fault

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We examine the dynamic behavior of a simple mechanical model of an earthquake fault. This model, introduced originally by Burridge and Knopoff (Bull. Seismol. Soc. Am. 57, 341 (1967)), consists of an elastically coupled chain of masses in contact with a moving rough surface. Our version of the model retains the full Newtonian dynamics with inertial effects and contains no externally imposed stochasticity or spatial inhomogeneity. The only nonlinear feature is a velocity-weakening stick-slip friction force between the masses and the moving surface. This system is being driven persistently toward a slipping instability and, therefore, exhibits noisy sequences of earthquakelike events. We observe these events in numerical simulations and are able to predict many of their features analytically. Their size distributions are found numerically to be consistent with the Gutenberg-Richter law. Some aspects of the size distributions can be understood by scaling arguments.
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paper_title: Pareto law in a kinetic model of market with random saving propensity

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We have numerically simulated the ideal-gas models of trading markets, where each agent is identified with a gas molecule and each trading as an elastic or money-conserving two-body collision. Unlike in the ideal gas, we introduce (quenched) saving propensity of the agents, distributed widely between the agents (0⩽λ<1). The system remarkably self-organizes to a critical Pareto distribution of money P(m)∼m−(ν+1) with ν≃1. We analyze the robustness (universality) of the distribution in the model. We also argue that although the fractional saving ingredient is a bit unnatural one in the context of gas models, our model is the simplest so far, showing self-organized criticality, and combines two century-old distributions: Gibbs (1901) and Pareto (1897) distributions.
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paper_title: The Nature of Markets in the World Wide Web

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Much has been said about the possibility that in the information age, ease of entry and global access will lead to market characteristics with few inefficiencies. While several arguments have been articulated on the nature of markets and electronic commerce, little is known about their characteristics and performance. We have studied the statistics in the number of visitors to sites of the World Wide Web by examining AOL usage logs covering 120,000 sites. We found out that, among all sites as well as for specific categories, the distribution of visitors per site follows a universal power law similar to that found by Pareto for income distributions. This implies that a small number of sites commands the traffic of a large segment of the Web population, a phenomenon reminiscent of winner-take-all markets. We have also developed a dynamic theory of site popularity that takes into account the stochastic nature of user decisions to visit given sites as well as the fact that newer sites are appearing at an ever increasing rate. The model accounts for the observed power law behavior and naturally provides the amplification factor responsible for the increased performance of the top performers.
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paper_title: Frequency of occurrence of numbers in the World Wide Web

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The distribution of numbers in human documents is determined by a variety of diverse natural and human factors, whose relative significance can be evaluated by studying the numbers' frequency of occurrence. Although it has been studied since the 1880's, this subject remains poorly understood. Here, we obtain the detailed statistics of numbers in the World Wide Web, finding that their distribution is a heavy-tailed dependence which splits in a set of power-law ones. In particular, we find that the frequency of numbers associated to western calendar years shows an uneven behavior: 2004 represents a `singular critical' point, appearing with a strikingly high frequency; as we move away from it, the decreasing frequency allows us to compare the amounts of existing information on the past and on the future. Moreover, while powers of ten occur extremely often, allowing us to obtain statistics up to the huge 10^127, `non-round' numbers occur in a much more limited range, the variations of their frequencies being dramatically different from standard statistical fluctuations. These findings provide a view of the array of numbers used by humans as a highly non-equilibrium and inhomogeneous system, and shed a new light on an issue that, once fully investigated, could lead to a better understanding of many sociological and psychological phenomena.
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paper_title: A Brief History of Generative Models for Power Law and Lognormal Distributions

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Recently, I became interested in a current debate over whether file size distributions are best modelled by a power law distribution or a lognormal distribution. In trying to learn enough about these distributions to settle the question, I found a rich and long history, spanning many fields. Indeed, several recently proposed models from the computer science community have antecedents in work from decades ago. Here, I briefly survey some of this history, focusing on underlying generative models that lead to these distributions. One finding is that lognormal and power law distributions connect quite naturally, and hence, it is not surprising that lognormal distributions have arisen as a possible alternative to power law distributions across many fields.
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paper_title: The Concentration of Commercial Success in Popular Music: An Analysis of the Distribution of Gold Records

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This paper examines whether commercial success in the popular recorded-music industry, as measured by gold-record output, conforms to an empirical concentration. We find that Lotka's Law overestimates the number of artists with one gold record and underestimates the number of multiplegold-record performing artists. However, for all measures of “successful” records, theGeneralized Lotka's Law provides an excellent fit, which suggests that the number of performers producingn gold records in about 1/nc of those producing one gold record.
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paper_title: The Self Organizing Economy

paper_content:

Rethinking international trade (collected papers) the age of diminished expectations geography and trade currencies and crises target zones and currency bands trade with Japan peddling prosperity empirical studies of strategic trade policy.
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paper_title: Predicting the long tail of book sales : Unearthing the power-law exponent

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The concept of the long tail has recently been used to explain the phenomenon in e-commerce where the total volume of sales of the items in the tail is comparable to that of the most popular items. In the case of online book sales, the proportion of tail sales has been estimated using regression techniques on the assumption that the data obeys a power-law distribution. Here we propose a different technique for estimation based on a generative model of book sales that results in an asymptotic power-law distribution of sales, but which does not suffer from the problems related to power-law regression techniques. We show that the proportion of tail sales predicted is very sensitive to the estimated power-law exponent. In particular, if we assume that the power-law exponent of the cumulative distribution is closer to 1.1 rather than to 1.2 (estimates published in 2003, calculated using regression by two groups of researchers), then our computations suggest that the tail sales of Amazon.com, rather than being 40% as estimated by Brynjolfsson, Hu and Smith in 2003, are actually closer to 20%, the proportion estimated by its CEO.
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paper_title: Kinematics and dynamics of Pareto–Zipf's law and Gibrat's law

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We discuss Pareto–Zipf's law and Gibrat's law found in the high-end regions of personal income, company's income, and various measures of company size. The fact that these phenomenological laws coexist in wide range of data suggests some deep mathematical relations between them. In this paper we find just such relationships under the law of detailed-balance. This sets a basis for perturbative treatment of the economic change.
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paper_title: Power laws, Pareto distributions and Zipf's law

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When the probability of measuring a particular value of some quantity varies inversely as a power of that value, the quantity is said to follow a power law, also known variously as Zipf's law or the Pareto distribution. Power laws appear widely in physics, biology, earth and planetary sciences, economics and finance, computer science, demography and the social sciences. For instance, the distributions of the sizes of cities, earthquakes, forest fires, solar flares, moon craters and people's personal fortunes all appear to follow power laws. The origin of power-law behaviour has been a topic of debate in the scientific community for more than a century. Here we review some of the empirical evidence for the existence of power-law forms and the theories proposed to explain them.
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paper_title: The Pareto-positive stable distribution: A new descriptive model for city size data

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The Pareto-positive stable (PPS) distribution is introduced as a new model for describing city size data in a country. The PPS distribution provides a flexible model for fitting the entire range of a set of city size data, where zero and unimodality are possible, and the classical Pareto and Zipf distributions are included as a particular case. Expressions for the shape, moments and other descriptive probabilistic measures are given. Estimation methods are discussed and a simple graphical method for studying the adequacy of the data to model is given. Finally, we consider city size data for Spain for several different years. The new distribution is compared with three classical models: Pareto, lognormal and Tsallis distributions. In all the data sets considered, the PPS distribution outperforms the fits of these three previous distributions.
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paper_title: Consumer surplus in the digital economy: Estimating the value of increased product variety at online booksellers

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We present a framework and empirical estimates that quantify the economic impact of increased product variety made available through electronic markets. While efficiency gains from increased competition significantly enhance consumer surplus, for instance, by leading to lower average selling prices, our present research shows that increased product variety made available through electronic markets can be a significantly larger source of consumer surplus gains.One reason for increased product variety on the Internet is the ability of online retailers to catalog, recommend, and provide a large number of products for sale. For example, the number of book titles available at Amazon.com is more than 23 times larger than the number of books on the shelves of a typical Barnes & Noble superstore, and 57 times greater than the number of books stocked in a typical large independent bookstore.Our analysis indicates that the increased product variety of online bookstores enhanced consumer welfare by $731 million to $1.03 billion in the year 2000, which is between 7 and 10 times as large as the consumer welfare gain from increased competition and lower prices in this market. There may also be large welfare gains in other SKU-intensive consumer goods such as music, movies, consumer electronics, and computer software and hardware.
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paper_title: Crater Size Distributions and Impact Probabilities on Earth from Lunar, Terrestrial Planeta, and Asteroid Cratering Data

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The terrestrial planets, the Earth's Moon, Mars,Venus, Mercury and the asteroids Gaspra and Ida showstriking similarities in their production crater size distribution characteristics. The investigation of hte crater populations in Gaspra and Ida yield information on the crater size distribution in the source region of the bodieslargely responsibe for cratering the inner solar system planets. Comparison of these data especially with the lunar impact record, which is our most reliable data base, confirms the complex shape of the crater size distribution curve (standarddistibution), and lends strong support to the idea of a common population of bodies impacting the inner planets and largely stemming from the asteroid belt. The steepening of the production crater size distribution at sizes D>= 1 km is confirmed to be due to the characteristics of the primary impactor production size distribution and not to an admixture of objects from secondary cratering processes. The impact hazard for craters, or objects, is assessed for the Earth and the other terrestrial planets relative to the lunar case through application of the lunar production size-frequency distribution which is well known for the crater diameter range 10m>D>1000km. current or past impact rates can be calculated for any size of crater or projectile and are given for specific crater sizes and respective projektile sizes as well as for projectile energies. The impact hazard for projectiles, e.g., capableof forming 1-km craters on solid surfacesof the Earth (if the atomsphere were absent)is assessed for the present and results in a production rate of one crater every 1600 yr for the surface area of the whole Earth, or every 6000 yr for the area of the continents; for 100-km craters, e.g. the corresponding number is one event every 27 Myr for the whole Earth.
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paper_title: Changes in Web client access patterns: Characteristics and caching implications

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Understanding the nature of the workloads and system demands created by users of the World Wide Web is crucial to properly designing and provisioning Web services. Previous measurements of Web client workloads have been shown to exhibit a number of characteristic featuress however, it is not clear how those features may be changing with time. In this study we compare two measurements of Web client workloads separated in time by three years, both captured from the same computing facility at Boston University. The older dataset, obtained in 1995, is well known in the research literature and has been the basis for a wide variety of studies. The newer dataset was captured in 1998 and is comparable in size to the older dataset. The new dataset has the drawback that the collection of users measured may no longer be representative of general Web userss however, using it has the advantage that many comparisons can be drawn more clearly than would be possible using a new, different source of measurement. Our results fall into two categories. First we compare the statistical and distributional properties of Web requests across the two datasets. This serves to reinforce and deepen our understanding of the characteristic statistical properties of Web client requests. We find that the kinds of distributions that best describe document sizes have not changed between 1995 and 1998, although specific values of the distributional parameters are different. Second, we explore the question of how the observed differences in the properties of Web client requests, particularly the popularity and temporal locality properties, affect the potential for Web file caching in the network. We find that for the computing facility represented by our traces between 1995 and 1998, (1) the benefits of using size-based caching policies have diminisheds and (2) the potential for caching requested files in the network has declined.
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paper_title: Bones, Bombs and Break Points: The Geography of Economic Activity

paper_content:

We consider the distribution of economic activity within a country in light of three leading theories - increasing returns, random growth, and locational fundamentals. To do so, we examine the distribution of regional population in Japan from the Stone Age to the modern era. We also consider the Allied bombing of Japanese cities in WWII as a shock to relative city sizes. Our results support a hybrid theory in which locational fundamentals establish the spatial pattern of relative regional densities, but increasing returns may help to determine the degree of spatial differentiation. One implication of these results is that even large temporary shocks to urban areas have no long-run impact on city size.
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paper_title: Zipf's Law For Cities: An Explanation

paper_content:

Zipf ’s law is a very tight constraint on the class of admissible models of local growth. It says that for most countries the size distribution of cities strikingly fits a power law: the number of cities with populations greater than S is proportional to 1/S. Suppose that, at least in the upper tail, all cities follow some proportional growth process (this appears to be verified empirically). This automatically leads their distribution to converge to Zipf ’s law.
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paper_title: Zipf Law for Brazilian Cities

paper_content:

This work studies the Zipf law for cities in Brazil. Data from censuses of 1970, 1980, 1991 and 2000 were used to select a sample containing only cities with 30,000 inhabitants or more. The results show that the population distribution in Brazilian cities does follow a power-law similar to the ones found in other countries. Estimates of the power-law exponent were found to be 2.22±0.34 for the 1970 and 1980 censuses, and 2.26±0.11 for censuses of 1991 and 2000. More accurate results were obtained with the maximum likelihood estimator, showing an exponent equal to 2.41 for 1970 and 2.36 for the other 3 years.
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paper_title: Role of Intermittency in Urban Development: A Model of Large-Scale City Formation

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A stochastic model that incorporates the essential mechanisms supposed to govern city formation is numerically analyzed. The model generates intermittent spatiotemporal structures and predicts a power-law population distribution whose exponent is in excellent agreement with the universal exponent observed in real human demography. Preliminary results of cluster analysis of the model also coincide with actual data. We thus suggest that urban development at large scales could be driven by intermittency processes. [S0031-9007(97)03578-3]
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paper_title: The Self Organizing Economy

paper_content:

Rethinking international trade (collected papers) the age of diminished expectations geography and trade currencies and crises target zones and currency bands trade with Japan peddling prosperity empirical studies of strategic trade policy.
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paper_title: Is the Zipf law spurious in explaining city-size distributions?

paper_content:

The Zipf law, which states that that the rank associated with some size S is proportional to S to some negative power, is a regularity observed in natural and social sciences. One popular application of the Zipf law is the relationship between city sizes and their ranks. This paper examines the rank–size relationship through Monte Carlo simulations and two examples. We show that a good fit (indicated by a high R 2 value) can be found for many statistical distributions. The Zipf law’s good fit is a statistical phenomenon, and therefore, it does not require an economic theory that determines city-size distributions. D 2006 Elsevier B.V. All rights reserved.
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paper_title: The size distribution of Chinese cities

paper_content:

Abstract This paper uses urban data to investigate two important issues regarding city sizes in China, the relative growth of cities and the nature of the city size distribution. The manner in which cities of different sizes grow relative to each other is examined and, contrary to the common empirical finding that the relative size and rank of cities remains stable over time, it is found that the Economic Reforms and the One Child Policy since 1979 have delivered significant structural change in the Chinese urban system. The city size distribution remains stable before the reforms but exhibits a convergent growth pattern in the post-reform period. The theoretical literature on city sizes highlights a link between log normal and Pareto distributions for city sizes prompting the employment of Pearson goodness-of-fit tests to examine directly which theoretical distribution provides the best approximation to the empirical city size distribution. Contrary to the evidence for other countries, a log normal rather than Pareto specification turns out to be the preferred distribution.
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paper_title: A simple and efficient test for Zipf ’ s law * ́

paper_content:

Abstract This paper presents a simple and locally optimal test for Zipf’s law. Its use is illustrated in the case of the largest US metropolitan areas. An objection to the general relevance of that law is also presented.
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paper_title: A Century of Shocks: The Evolution of the German City Size Distribution 1925-1999

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The empirical literature on city size distributions has mainly focused on the USA. The first major contribution of this paper is to provide empirical evidence on the evolution and structure of the West-German city size distribution. Using a unique annual data set that covers most of the 20th century for 62 of West-Germany's largest cities, we look at the evolution of both the city size distribution as a whole and each city separately. The West-German case is of particular interest as it has undergone major shocks, most notably WWII. Our data set allows us to identify these shocks and provide evidence on the effects of these `quasi-natural experiments' on the city size distribution. The second major contribution of this paper is that we perform unit-root tests on individual German city sizes using a substantial number of observations to analyze the evolution of the individual cities that make up the German city size distribution. Our main findings are twofold. First, WWII has had a major and lasting impact on the city size distribution. Second, the overall city size distribution does not adhere to Zipf's Law. This second finding is largely based on the results of unit root tests for individual cities to test for Gibrat's Law, the latter being a requirement for Zipf's Law to hold for the overall city-size distribution. Together these two findings are consistent with theories emphasizing increasing returns to scale in city growth.
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paper_title: Laws of Population Growth

paper_content:

An important issue in the study of cities is defining a metropolitan area, as different definitions affect the statistical distribution of urban activity. A commonly employed method of defining a metropolitan area is the Metropolitan Statistical Areas (MSA), based on rules attempting to capture the notion of city as a functional economic region, and is constructed using experience. The MSA is time-consuming and is typically constructed only for a subset (few hundreds) of the most highly populated cities. Here, we introduce a new method to designate metropolitan areas, denoted the"City Clustering Algorithm"(CCA). The CCA is based on spatial distributions of the population at a fine geographic scale, defining a city beyond the scope of its administrative boundaries. We use the CCA to examine Gibrat's law of proportional growth, postulating that the mean and standard deviation of the growth rate of cities are constant, independent of city size. We find that the mean growth rate of a cluster utilizing the CCA exhibits deviations from Gibrat's law, and that the standard deviation decreases as a power-law with respect to the city size. The CCA allows for the study of the underlying process leading to these deviations, shown to arise from the existence of long-range spatial correlations in the population growth. These results have socio-political implications, such as those pertaining to the location of new economic development in cities of varied size.
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paper_title: Bones, Bombs and Break Points: The Geography of Economic Activity

paper_content:

We consider the distribution of economic activity within a country in light of three leading theories - increasing returns, random growth, and locational fundamentals. To do so, we examine the distribution of regional population in Japan from the Stone Age to the modern era. We also consider the Allied bombing of Japanese cities in WWII as a shock to relative city sizes. Our results support a hybrid theory in which locational fundamentals establish the spatial pattern of relative regional densities, but increasing returns may help to determine the degree of spatial differentiation. One implication of these results is that even large temporary shocks to urban areas have no long-run impact on city size.
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paper_title: The "DGX" distribution for mining massive, skewed data

paper_content:

Skewed distributions appear very often in practice. Unfortunately, the traditional Zipf distribution often fails to model them well. In this paper, we propose a new probability distribution, the Discrete Gaussian Exponential (DGX), to achieve excellent fits in a wide variety of settings; our new distribution includes the Zipf distribution as a special case. We present a statistically sound method for estimating the DGX parameters based on maximum likelihood estimation (MLE). We applied DGX to a wide variety of real world data sets, such as sales data from a large retailer chain, us-age data from ATT in all cases, DGX fits these distributions very well, with almost a 99% correlation coefficient in quantile-quantile plots. Our algorithm also scales very well because it requires only a single pass over the data. Finally, we illustrate the power of DGX as a new tool for data mining tasks, such as outlier detection.
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paper_title: The size distribution across all cities - Double Pareto lognormal strikes

paper_content:

Using un-truncated settlement size data from eight countries, we show that the "double Pareto lognormal" (DPLN) distribution provides a better fit to actual city sizes than the simple lognormal (LN) distribution. The DPLN has a lognormal body and features a power law in both the lower and the upper tail. It emerges in the steady-state of a stochastic urban growth process with random city formation. Our findings reconcile a recent debate on the Zipfian rank-size rule for city sizes.
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paper_title: Examining City Size Distributions using Urban Areas

paper_content:

An input/output port address selection circuit for a device that couples to a local bus of a computer operates in two modes. In a first mode, the I/O port address is maintained as the last I/O port address saved. In the second mode, the selection circuit changes the I/O port address and saves the change when the circuit is reset. A computer user resolves I/O port address conflicts by placing the selection circuit in the second mode then resetting the system until the conflict is resolved and the system works properly. The selection circuit is then placed and left in the first mode so that the non-conflicting address is reused. Non-volatile memory, an electrically erasable-programmable read-only memory, is used for saving an index which indicates the last I/O port address set. The non-volatile memory is changed when the selection circuit is operated in the second mode.
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paper_title: Possible generalization of Boltzmann-Gibbs statistics

paper_content:

With the use of a quantity normally scaled in multifractals, a generalized form is postulated for entropy, namelyS q ≡k [1 – ∑ i=1 W p i q ]/(q-1), whereq∈ℝ characterizes the generalization andpi are the probabilities associated withW (microscopic) configurations (W∈ℕ). The main properties associated with this entropy are established, particularly those corresponding to the microcanonical and canonical ensembles. The Boltzmann-Gibbs statistics is recovered as theq→1 limit.
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paper_title: The Pareto-positive stable distribution: A new descriptive model for city size data

paper_content:

The Pareto-positive stable (PPS) distribution is introduced as a new model for describing city size data in a country. The PPS distribution provides a flexible model for fitting the entire range of a set of city size data, where zero and unimodality are possible, and the classical Pareto and Zipf distributions are included as a particular case. Expressions for the shape, moments and other descriptive probabilistic measures are given. Estimation methods are discussed and a simple graphical method for studying the adequacy of the data to model is given. Finally, we consider city size data for Spain for several different years. The new distribution is compared with three classical models: Pareto, lognormal and Tsallis distributions. In all the data sets considered, the PPS distribution outperforms the fits of these three previous distributions.
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paper_title: The temporal evolution of the city size distribution

paper_content:

We present an application of a growth model for a system of cities. This computer model simulates the evolution of systems with measurable entities (e.g. city size), and takes into account the growth of the entities in terms of size and number. It includes a random multiplicative process for the growth of individual entities and for the creation of new ones. We use a new mathematical expression with a positive exponent α (which we call the ‘shape exponent’) and additional three parameters, to describe the dynamics of the systems’ size distributions through time. We compare the changes of a real system of cities and the model’s results quantitatively and qualitatively. Our findings suggest that there is a good agreement at the macro level between the model and the real data.
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paper_title: Statistical mechanics of money, income and wealth: a short survey

paper_content:

In this short paper, we overview and extend the results of our papers cond-mat/0001432, cond-mat/0008305, and cond-mat/0103544, where we use an analogy with statistical physics to describe probability distributions of money, income, and wealth in society. By making a detailed quantitative comparison with the available statistical data, we show that these distributions are described by simple exponential and power-law functions.
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paper_title: Pareto law in a kinetic model of market with random saving propensity

paper_content:

We have numerically simulated the ideal-gas models of trading markets, where each agent is identified with a gas molecule and each trading as an elastic or money-conserving two-body collision. Unlike in the ideal gas, we introduce (quenched) saving propensity of the agents, distributed widely between the agents (0⩽λ<1). The system remarkably self-organizes to a critical Pareto distribution of money P(m)∼m−(ν+1) with ν≃1. We analyze the robustness (universality) of the distribution in the model. We also argue that although the fractional saving ingredient is a bit unnatural one in the context of gas models, our model is the simplest so far, showing self-organized criticality, and combines two century-old distributions: Gibbs (1901) and Pareto (1897) distributions.
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paper_title: Scaling behavior of an economic index

paper_content:

A satellite apparatus is provided whereby a plurality of experiments may be monitored in a single satellite for a plurality of different organizations. The satellite has modular casings for placement on a frame and adapted to place experimental components therein with instrumentation facing a centrally located scanner which transmits data on the experiments to a receiving station.
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paper_title: New evidence for the power-law distribution of wealth

paper_content:

We present a non-conventional approach for studying the distribution of wealth in society. We analyze data from the 1996 Forbes 400 list of the richest people in the US. Our results confirm that wealth is distributed according to a power law. The measured exponent of the power-law is 1.36. As theoretically predicted, this value is in close agreement with the exponent of the Levy distribution of stock market fluctuations.
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paper_title: A study of forest fire danger prediction system in Japan

paper_content:

All over the world, forest fires cause severe damages to valuable natural environment and loss of human lives. The fire statistics showed that the number of Japanese forest fires remains high at 3,000 per year, most fires are caused by human activities and occur in the populated area, although the forest fires are of course mainly affected by weather conditions. Based on the analyzed fundamentals of the weather conditions together with the city related data in relation to the fire occurrence, we have developed a system to predict forest fire danger. The fire danger simulations were made, using not only previous weather conditions, but also population density data and others. The comparison between the map of predicted fire danger and the locations of real fire occurrence indicated almost reasonable agreement. In addition, CFD simulations were conducted to examine the fire progression behavior and the suppression.
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paper_title: The flaming sandpile: self-organized criticality and wildfires

paper_content:

A large series of wildfire records of the Regional Forest Service of Liguria (northern Italy) from 1986 to 1993 was examined for agreement with power-law behavior between frequency of occurrence and size of the burned area. The statistical analysis shows that the idea of self-organized criticality (SOC) applies well to explain wildfire occurrence on a regional basis.
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paper_title: Self-organized critical forest-fire model

paper_content:

A forest-fire model is introduced which contains a lightning probability f. This leads to a self-organized critical state in the limit f\ensuremath{\rightarrow}0 provided that the time scales of tree growth and burning down of forest clusters are separated. We derive scaling laws and calculate all critical exponents. The values of the critical exponents are confirmed by computer simulations. For a two-dimensional system, we show that the forest density in the critical state assumes its minimum possible value, i.e., that energy dissipation is maximum.
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paper_title: The Gompertz-Pareto Income Distribution

paper_content:

This work analyzes the Gompertz-Pareto distribution (GPD) of personal income, formed by the combination of the Gompertz curve, representing the overwhelming majority of the economically less favorable part of the population of a country, and the Pareto power law, which describes its tiny richest part. Equations for the Lorenz curve, Gini coefficient and the percentage share of the Gompertzian part relative to the total income are all written in this distribution. We show that only three parameters, determined by linear data fitting, are required for its complete characterization. Consistency checks are carried out using income data of Brazil from 1981 to 2007 and they lead to the conclusion that the GPD is consistent and provides a coherent and simple analytical tool to describe personal income distribution data.
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paper_title: Pareto law in a kinetic model of market with random saving propensity

paper_content:

We have numerically simulated the ideal-gas models of trading markets, where each agent is identified with a gas molecule and each trading as an elastic or money-conserving two-body collision. Unlike in the ideal gas, we introduce (quenched) saving propensity of the agents, distributed widely between the agents (0⩽λ<1). The system remarkably self-organizes to a critical Pareto distribution of money P(m)∼m−(ν+1) with ν≃1. We analyze the robustness (universality) of the distribution in the model. We also argue that although the fractional saving ingredient is a bit unnatural one in the context of gas models, our model is the simplest so far, showing self-organized criticality, and combines two century-old distributions: Gibbs (1901) and Pareto (1897) distributions.
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paper_title: Forest Fires: An Example of Self-Organized Critical Behavior

paper_content:

Despite the many complexities concerning their initiation and propagation, forest fires exhibit power-law frequency-area statistics over many orders of magnitude. A simple forest fire model, which is an example of self-organized criticality, exhibits similar behavior. One practical implication of this result is that the frequency-area distribution of small and medium fires can be used to quantify the risk of large fires, as is routinely done for earthquakes.
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paper_title: Kinematics and dynamics of Pareto–Zipf's law and Gibrat's law

paper_content:

We discuss Pareto–Zipf's law and Gibrat's law found in the high-end regions of personal income, company's income, and various measures of company size. The fact that these phenomenological laws coexist in wide range of data suggests some deep mathematical relations between them. In this paper we find just such relationships under the law of detailed-balance. This sets a basis for perturbative treatment of the economic change.
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paper_title: A forest-fire model and some thoughts on turbulence

paper_content:

Abstract In the context of a forest-fire model we demonstrate critical scaling behavior in a “turbulent” non-equilibrium system. Energy is injected uniformly, and dissipated on a fractal. Critical exponents are estimated by means of a Monte Carlo renormalization- group calculation.
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paper_title: Self-organized criticality of forest fires in China

paper_content:

Self-organized criticality (SOC) of forest fires in China from 1950 to 1989 is studied. The stability, scaleinvariant character of SOC and external effects on SOC of forest fires in China are analyzed in detail. Forest-fire cellular automata model is a typical model for the research of SOC. Based on the traditional forest-fire model, an improved model, in which effects of tree species, meteorological conditions and human efforts on forest fires are considered, is introduced. Actual forest fire data in China are compared with simulation results of the two models. It is shown that forest fire data in China have SOC behavior and simulation results of the improved model accord better with actual forest fire data than those of the traditional model.
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paper_title: Pareto law of the expenditure of a person in convenience stores

paper_content:

We study the statistical laws of the expenditure of a person in convenience stores by analyzing around 100 million receipts. The density function of expenditure exhibits a fat tail that follows a power law. Using the Lorenz curve, the Gini coefficient is estimated to be 0.70; this implies that loyal customers contribute significantly to a store’s sales. We observe the Pareto principle where both the top 25% and 2% of the customers account for 80% and 25% of the store’s sales, respectively.
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paper_title: On Stable Pareto Laws in a Hierarchical Model of Economy ∗

paper_content:

This study considers a model of the income distribution of agents whose pairwise interaction is asymmetric and price-invariant. Asymmetric transactions are typical for chain-trading groups who arrange their business such that commodities move from senior to junior partners and money moves in the opposite direction. The price-invariance of transactions means that the probability of a pairwise interaction is a function of the ratio of incomes, which is independent of the price scale or absolute income level. These two features characterize the hierarchical model. The income distribution in this class of models is a well-defined double-Pareto function, which possesses Pareto tails for the upper and lower incomes. For gross and net upper incomes, the model predicts definite values of the Pareto exponents, agross and anet, which are stable with respect to quantitative variation of the pair-interaction. The Pareto exponents are also stable with respect to the choice of a demand function within two classes of status-dependent behavior of agents: linear demand (agross=1, anet=2) and unlimited slowly varying demand (agross=anet=1). For the sigmoidal demand that describes limited returns, agross=anet=1+α, with some α>0 satisfying a transcendental equation. The low-income distribution may be singular or vanishing in the neighborhood of the minimal income; in any case, it is L1-integrable and its Pareto exponent is given explicitly.
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paper_title: Three types of power-law distribution of forest fires in Japan

paper_content:

Abstract Study on the forest-fire distribution is not only a phenomenological insight of forest ecological dynamics, but also essential for developing forest-fire danger rating method and system. In this paper, three types of power-law characteristics of forest fires are investigated. The first one is the distribution of fire areas. The frequency–area distribution is found obeying good power-law relation that is invariant with time. The second one is the distribution of fire intervals. It is indicated that the frequency–interval distribution of forest fires in Japan is also accord with power law. What is the most interesting is that the frequency–interval distribution is a power law with periodic change, different to that of frequency–size distribution. The third one is the distribution of fire probability against population density. There is positive-exponent power-law relation between fire probability and population density. The results discussed in this paper are expected to promote the forest-fire danger rating method and are helpful to understand the ecological action of forest fires.
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paper_title: Power-law distribution of city fires

paper_content:

This paper presents results of fire distributions in China and Switzerland. The frequency-size (loss) distribution and the rank-size distribution (or Zipf plot) of city fires are plotted to check their fit with a power-law relationship. The results indicate that fire data in both countries have good power-law distributions and the power-law relation is invariant with scale and time, i.e. fires in different places or in different periods will have the same distribution characteristics. Steady power-law distribution corresponds with self-organized criticality (SOC) and is expected to provide practical implications for fire protection.
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paper_title: Predicting the long tail of book sales : Unearthing the power-law exponent

paper_content:

The concept of the long tail has recently been used to explain the phenomenon in e-commerce where the total volume of sales of the items in the tail is comparable to that of the most popular items. In the case of online book sales, the proportion of tail sales has been estimated using regression techniques on the assumption that the data obeys a power-law distribution. Here we propose a different technique for estimation based on a generative model of book sales that results in an asymptotic power-law distribution of sales, but which does not suffer from the problems related to power-law regression techniques. We show that the proportion of tail sales predicted is very sensitive to the estimated power-law exponent. In particular, if we assume that the power-law exponent of the cumulative distribution is closer to 1.1 rather than to 1.2 (estimates published in 2003, calculated using regression by two groups of researchers), then our computations suggest that the tail sales of Amazon.com, rather than being 40% as estimated by Brynjolfsson, Hu and Smith in 2003, are actually closer to 20%, the proportion estimated by its CEO.
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paper_title: Power laws, Pareto distributions and Zipf's law

paper_content:

When the probability of measuring a particular value of some quantity varies inversely as a power of that value, the quantity is said to follow a power law, also known variously as Zipf's law or the Pareto distribution. Power laws appear widely in physics, biology, earth and planetary sciences, economics and finance, computer science, demography and the social sciences. For instance, the distributions of the sizes of cities, earthquakes, forest fires, solar flares, moon craters and people's personal fortunes all appear to follow power laws. The origin of power-law behaviour has been a topic of debate in the scientific community for more than a century. Here we review some of the empirical evidence for the existence of power-law forms and the theories proposed to explain them.
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paper_title: Consumer surplus in the digital economy: Estimating the value of increased product variety at online booksellers

paper_content:

We present a framework and empirical estimates that quantify the economic impact of increased product variety made available through electronic markets. While efficiency gains from increased competition significantly enhance consumer surplus, for instance, by leading to lower average selling prices, our present research shows that increased product variety made available through electronic markets can be a significantly larger source of consumer surplus gains.One reason for increased product variety on the Internet is the ability of online retailers to catalog, recommend, and provide a large number of products for sale. For example, the number of book titles available at Amazon.com is more than 23 times larger than the number of books on the shelves of a typical Barnes & Noble superstore, and 57 times greater than the number of books stocked in a typical large independent bookstore.Our analysis indicates that the increased product variety of online bookstores enhanced consumer welfare by $731 million to $1.03 billion in the year 2000, which is between 7 and 10 times as large as the consumer welfare gain from increased competition and lower prices in this market. There may also be large welfare gains in other SKU-intensive consumer goods such as music, movies, consumer electronics, and computer software and hardware.
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paper_title: Measuring Prices and Price Competition Online: Amazon.com and BarnesandNoble.com

paper_content:

Despite the interest in measuring price sensitivity of online consumers, most academic work on Internet commerce is hindered by a lack of data on quantity. In this paper we use publicly available data on the sales ranks of about 20,000 books to derive quantity proxies at the two leading online booksellers. Matching this information to prices, we can directly estimate the elasticities of demand facing both merchants as well as create a price index for online books. The results show significant price sensitivity at both merchants but demand at BarnesandNoble.com is much more price-elastic than is demand at Amazon.com. The data also allow us to estimate the magnitude of bias in the CPI due to the rise of Internet sales. Copyright Kluwer Academic Publishers 2003
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paper_title: Common ecology quantifies human insurgency

paper_content:

Many collective human activities, including violence, have been shown to exhibit universal patterns1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19. The size distributions of casualties both in whole wars from 1816 to 1980 and terrorist attacks have separately been shown to follow approximate power-law distributions6, 7, 9, 10. However, the possibility of universal patterns ranging across wars in the size distribution or timing of within-conflict events has barely been explored. Here we show that the sizes and timing of violent events within different insurgent conflicts exhibit remarkable similarities. We propose a unified model of human insurgency that reproduces these commonalities, and explains conflict-specific variations quantitatively in terms of underlying rules of engagement. Our model treats each insurgent population as an ecology of dynamically evolving, self-organized groups following common decision-making processes. Our model is consistent with several recent hypotheses about modern insurgency18, 19, 20, is robust to many generalizations21, and establishes a quantitative connection between human insurgency, global terrorism10 and ecology13, 14, 15, 16, 17, 22, 23. Its similarity to financial market models24, 25, 26 provides a surprising link between violent and non-violent forms of human behaviour.
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paper_title: Mechanical model of an earthquake fault

paper_content:

We examine the dynamic behavior of a simple mechanical model of an earthquake fault. This model, introduced originally by Burridge and Knopoff (Bull. Seismol. Soc. Am. 57, 341 (1967)), consists of an elastically coupled chain of masses in contact with a moving rough surface. Our version of the model retains the full Newtonian dynamics with inertial effects and contains no externally imposed stochasticity or spatial inhomogeneity. The only nonlinear feature is a velocity-weakening stick-slip friction force between the masses and the moving surface. This system is being driven persistently toward a slipping instability and, therefore, exhibits noisy sequences of earthquakelike events. We observe these events in numerical simulations and are able to predict many of their features analytically. Their size distributions are found numerically to be consistent with the Gutenberg-Richter law. Some aspects of the size distributions can be understood by scaling arguments.
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paper_title: Universal patterns underlying ongoing wars and terrorism

paper_content:

We report a remarkable universality in the patterns of violence arising in three high-profile ongoing wars, and in global terrorism. Our results suggest that these quite different conflict arenas currently feature a common type of enemy, i.e. the various insurgent forces are beginning to operate in a similar way regardless of their underlying ideologies, motivations and the terrain in which they operate. We provide a microscopic theory to explain our main observations. This theory treats the insurgent force as a generic, self-organizing system which is dynamically evolving through the continual coalescence and fragmentation of its constituent groups.
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paper_title: On the Frequency of Severe Terrorist Events

paper_content:

In the spirit of Lewis Richardsonâ€™s original study of the statistics of deadly conflicts, we study the frequency and severity of terrorist attacks worldwide since 1968. We show that these events are uniformly characterized by the phenomenon of â€œscale invariance,â€ that is, the frequency scales as an inverse power of the severity, P(x) I‘x-I±. We find that this property is a robust feature of terrorism, persisting when we control for economic development of the target country, the type of weapon used, and even for short time scales. Further, we show that the center of the distribution oscillates slightly with a period of roughly I„â‰ˆ 13 years, that there exist significant temporal correlations in the frequency of severe events, and that current models of event incidence cannot account for these variations or the scale invariance property of global terrorism. Finally, we describe a simple toy model for the generation of these statistics and briefly discuss its implications.
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paper_title: Bones, Bombs and Break Points: The Geography of Economic Activity

paper_content:

We consider the distribution of economic activity within a country in light of three leading theories - increasing returns, random growth, and locational fundamentals. To do so, we examine the distribution of regional population in Japan from the Stone Age to the modern era. We also consider the Allied bombing of Japanese cities in WWII as a shock to relative city sizes. Our results support a hybrid theory in which locational fundamentals establish the spatial pattern of relative regional densities, but increasing returns may help to determine the degree of spatial differentiation. One implication of these results is that even large temporary shocks to urban areas have no long-run impact on city size.
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paper_title: Semi-empirical power-law scaling of new infection rate to model epidemic dynamics with inhomogeneous mixing

paper_content:

The expected number of new infections per day per infectious person during an epidemic has been found to exhibit power-law scaling with respect to the susceptible fraction of the population. This is in contrast to the linear scaling assumed in traditional epidemiologic modeling. Based on simulated epidemic dynamics in synthetic populations representing Los Angeles, Chicago, and Portland, we find city-dependent scaling exponents in the range of 1.7–2.06. This scaling arises from variations in the strength, duration, and number of contacts per person. Implementation of power-law scaling of the new infection rate is quite simple for SIR, SEIR, and histogram-based epidemic models. Treatment of the effects of the social contact structure through this power-law formulation leads to significantly lower predictions of final epidemic size than the traditional linear formulation.
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paper_title: Use of power law models in detecting region of interest

paper_content:

In this paper, we shall address the issue of semantic extraction of different regions of interest. The proposed approach is based on statistical methods and models inspired from linguistic analysis. Here, the models used are Zipf law and inverse Zipf law. They are used to model the frequency of appearance of the patterns contained in images as power law distributions. The use of these models allows to characterize the structural complexity of image textures. This complexity measure indicates a perceptually salient region in the image. The image is first partitioned into sub-images that are to be compared in some sense. Zipf or inverse Zipf law are applied to these sub-images and they are classified according to the characteristics of the power law models involved. The classification method consists in representing the characteristics of the Zipf and inverse Zipf model of each sub-image by a point in a representation space in which a clustering process is performed. Our method allows detection of regions of interest which are consistent with human perception, inverse Zipf law is particularly significant. This method has good performances compared to more classical detection methods. Alternatively, a neural network can be used for the classification phase.
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Title: A Review of Power Laws in Real Life Phenomena

Section 1: Introduction
Description 1: Provide an overview of power law distributions, historical context, and their application across various fields.

Section 2: Power Laws for Cities
Description 2: Summarize research and findings on the application of power laws to the size distribution of cities.

Section 3: Economy and Power Laws
Description 3: Detail studies and theories related to power laws in economic contexts, including income and wealth distributions.

Section 4: Online Sales
Description 4: Discuss how power laws describe the distribution of sales in e-commerce, with a focus on book sales and related data.

Section 5: Wars and Terrorists Attacks
Description 5: Explore the application of power laws to the frequency and severity of wars and terrorist attacks.

Section 6: Other Examples of Power Law in Sciences
Description 6: Provide additional examples where power laws are observed, including RNA folding, gene expression, and epidemic modeling.

Section 7: Practical Examples of PLs
Description 7: Investigate the cumulative distributions of various real-world data sets to confirm power law behavior.

Section 8: A Global Perspective
Description 8: Compare and contrast the power law behaviors across different phenomena to provide a broader understanding.

Section 9: Conclusion
Description 9: Summarize the key points, controversies, and implications of power law research in real-life phenomena.