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| [[Image:Long tail.svg|thumb|300px|right|An example power-law graph, being used to demonstrate ranking of popularity. To the right is the [[long tail]], and to the left are the few that dominate (also known as the [[Pareto principle|80–20 rule]]).]]
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| In [[statistics]], a '''power law''' is a functional relationship between two quantities, where one quantity varies as a [[Exponentiation|power]] of another. For instance, the number of cities having a certain population size is found to vary as a power of the size of the population. Empirical power-law distributions hold only approximately or over a limited range.
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| ==Empirical examples of power laws==
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| The distributions of a wide variety of physical, biological, and man-made phenomena approximately follow a power law over a wide range of magnitudes: these include the sizes of [[earthquake]]s, craters on the [[moon]] and of [[solar flare]]s,<ref name= Newman/> the foraging pattern of various species,<ref name=Humphries>{{cite journal | author = Humphries NE, Queiroz N, Dyer JR, Pade NG, Musyl MK, Schaefer KM, Fuller DW, Brunnschweiler JM, Doyle TK, Houghton JD, Hays GC, Jones CS, Noble LR, Wearmouth VJ, Southall EJ, Sims DW| year = 2010| title = Environmental context explains Lévy and Brownian movement patterns of marine predators | url = | journal = Nature | volume = 465 | issue = 7301| pages = 1066–1069| doi = 10.1038/nature09116 | pmid = 20531470 |bibcode = 2010Natur.465.1066H }}</ref> the sizes of activity patterns of neuronal populations,<ref name=Klaus>{{cite journal | author = Klaus A, Yu S, Plenz D | year = 2011 | title = Statistical Analyses Support Power Law Distributions Found in Neuronal Avalanches | url = http://www.plosone.org/article/info%3Adoi%2F10.1371%2Fjournal.pone.0019779 | journal = PLoS ONE | volume = 6 | issue = 5| pages = e19779 | doi = 10.1371/journal.pone.0019779 | editor1-last = Zochowski | editor1-first = Michal | pmid = 21720544 | pmc = 3102672|bibcode = 2011PLoSO...619779K }}</ref> the frequencies of [[word]]s in most languages, frequencies of [[family name]]s, the [[species richness]] in [[clades]] of organisms,<ref>{{cite book | |
| |editor1-last=Albert|editor1-first=J. S.
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| |editor2-first=R. E.|editor2-last=Reis
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| |year=2011
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| |title=Historical Biogeography of Neotropical Freshwater Fishes
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| |publisher=University of California Press
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| |location=Berkeley
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| |url=http://www.ucpress.edu/book.php?isbn=9780520268685
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| }}</ref> the sizes of [[power outage]]s, wars, and many other quantities.{{sfn|Clauset|Shalizi|Newman|2009}} Few empirical distributions fit a power law for all their values, but rather follow a power law in the tail.
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| [[Acoustic attenuation]] follows frequency power-laws within wide frequency bands for many complex media. [[allometric scaling| Allometric scaling laws ]] for relationships between biological variables are among the best known power-law functions in nature.
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| In [[criminology]] a ''criminal career'' is the number of crimes committed by an individual within a given time frame. N(x) = the number of individuals, N, with x charges follows a power distribution. Disaggregating career by crime type, ''s'' ranges from 2 to 3, typically higher the more serious the crime, which is a testimony to proportional police efforts. Charges form only a subset (avg. 20%) of total crimes committed. Knowledge of the power law for this tip of the iceberg can be use to simulate the underlying totality of crimes committed. E.g., for burglary with a clearance rate of 10% the apparent ''s'' based on charges was estimated as 2.3 whereas the true value was more likely to be 1.7, implying that crimes are much more concentrated on perpetrators than it would appear based on charges.
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| ==Properties of power laws==
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| ===Scale invariance===
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| One attribute of power laws is their [[scale invariance#Scale invariance of functions and self-similarity|scale invariance]]. Given a relation <math>f(x) = ax^k</math>, scaling the argument <math>x</math> by a constant factor <math>c</math> causes only a proportionate scaling of the function itself. That is,
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| :<math>f(c x) = a(c x)^k = c^k f(x) \propto f(x).\!</math>
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| That is, scaling by a constant <math>c</math> simply multiplies the original power-law relation by the constant <math>c^k</math>. Thus, it follows that all power laws with a particular scaling exponent are equivalent up to constant factors, since each is simply a scaled version of the others. This behavior is what produces the linear relationship when logarithms are taken of both <math>f(x)</math> and <math>x</math>, and the straight-line on the log-log plot is often called the ''signature'' of a power law. With real data, such straightness is a necessary, but not sufficient, condition for the data following a power-law relation. In fact, there are many ways to generate finite amounts of data that mimic this signature behavior, but, in their asymptotic limit, are not true power laws (e.g., if the generating process of some data follows a [[Log-normal distribution]]). Thus, accurately fitting and validating power-law models is an active area of research in statistics.
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| ===Universality===
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| The equivalence of power laws with a particular scaling exponent can have a deeper origin in the dynamical processes that generate the power-law relation. In physics, for example, [[phase transition]]s in thermodynamic systems are associated with the emergence of power-law distributions of certain quantities, whose exponents are referred to as the [[critical exponent]]s of the system. Diverse systems with the same critical exponents—that is, which display identical scaling behaviour as they approach [[critical point (thermodynamics)|criticality]]—can be shown, via [[renormalization group]] theory, to share the same fundamental dynamics. For instance, the behavior of water and CO<sub>2</sub> at their boiling points fall in the same universality class because they have identical critical exponents. In fact, almost all material phase transitions are described by a small set of universality classes. Similar observations have been made, though not as comprehensively, for various [[self-organized criticality|self-organized critical]] systems, where the critical point of the system is an [[attractor]]. Formally, this sharing of dynamics is referred to as [[universality (dynamical systems)|universality]], and systems with precisely the same critical exponents are said to belong to the same [[renormalization group#Relevant and irrelevant operators, universality classes|universality class]].
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| <!--
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| COMMENT: Rather than spin-glasses I'd like a concrete reference of two distinct systems that share the same universality class. I'd also like a reference about the how-common-is-universality issue. -->
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| <!--RESPONSE: Discussion of water and CO2 satisfies first request. Can someone else satisfy the second part? -->
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| <!--COMMENT: Note that scale-invariance is not necessarily observed for power-law-''tailed'' equations. For example, the [[Lévy distribution]] does not display the above property.-->
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| <!--RESPONSE: Any function that asymptotically follows a power law relation is scale invariant, by the definition given in this article. -->
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| ==Power-law functions==
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| Scientific interest in power-law relations stems partly from the ease with which certain general classes of mechanisms generate them.{{sfn|Sornette|2006}} The demonstration of a power-law relation in some data can point to specific kinds of mechanisms that might underlie the natural phenomenon in question, and can indicate a deep connection with other, seemingly unrelated systems;{{sfn|Simon|1955}} see also [[#Universality|universality]] above. The ubiquity of power-law relations in physics is partly due to [[dimensional analysis|dimensional constraints]], while in [[complex systems]], power laws are often thought to be signatures of hierarchy or of specific [[stochastic processes]]. A few notable examples of power laws are the [[Gutenberg–Richter law]] for earthquake sizes, [[Pareto principle|Pareto's law]] of income distribution, structural self-similarity of [[fractals]], and [[allometric law|scaling laws in biological systems]]. Research on the origins of power-law relations, and efforts to observe and validate them in the real world, is an active topic of research in many fields of science, including [[physics]], [[computer science]], [[linguistics]], [[geophysics]], [[neuroscience]], [[sociology]], [[economics]] and more.
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| However much of the recent interest in power laws comes from the study of [[probability distributions]]: The distributions of a wide variety of quantities seem to follow the power-law form, at least in their upper tail (large events). The behavior of these large events connects these quantities to the study of [[extreme value theory|theory of large deviations]] (also called [[extreme value theory]]), which considers the frequency of extremely rare events like [[stock market crash]]es and large [[natural disaster]]s. It is primarily in the study of statistical distributions that the name "power law" is used; in other areas, such as physics and engineering, a power-law functional form with a single term and a positive integer exponent is typically regarded as a [[polynomial]] function.
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| In empirical contexts, an approximation to a power-law <math>o(x^k)</math> often includes a deviation term <math>\varepsilon</math>, which can represent uncertainty in the observed values (perhaps measurement or sampling errors) or provide a simple way for observations to deviate from the power-law function (perhaps for [[stochastic process|stochastic]] reasons):
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| :<math>y = ax^k + \varepsilon.\!</math>
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| Mathematically, a strict power law cannot be a probability distribution, but a distribution that is a truncated [[power function]] is possible: <math>p(x) = C x^{-\alpha}</math> for <math>x > x_\text{min}</math> where the exponent <math>\alpha</math> is greater than 1 (otherwise the tail has infinite area), the minimum value <math>x_\text{min}</math> is needed otherwise the distribution has infinite area as ''x'' approaches 0, and the constant ''C'' is a scaling factor to ensure that the total area is 1, as required by a probability distribution. More often one uses an asymptotic power law – one that is only true in the limit; see [[#Power-law probability distributions|power-law probability distributions]] below for details. Typically the exponent falls in the range <math>2 < \alpha < 3</math>, though not always.{{sfn|Clauset|Shalizi|Newman|2009|page=2}}
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| ===Examples of power-law functions===
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| *The frequency-dependency of [[acoustic attenuation]] in complex media
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| *The [[Stevens' power law]] of psychophysics
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| *The [[Stefan–Boltzmann law]]
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| *The input-voltage–output-current curves of [[field-effect transistor]]s and [[vacuum tubes]] approximate a [[Electronic amplifier#Square-law|square-law]] relationship, a factor in "[[tube sound]]".
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| *A 3/2-power law can be found in the [[Current–voltage characteristic|plate characteristic curves]] of [[triode]]s.
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| *The [[inverse-square law]]s of [[Newtonian gravity]] and [[electrostatics]], as evidenced by the [[gravitational potential]] and [[Electrostatic potential]], respectively.
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| *Model of [[van der Waals force]]
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| *Force and potential in [[simple harmonic motion]]
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| *[[Kepler's third law]]
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| * The [[initial mass function]] of stars
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| * The [[M-sigma relation]]
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| *[[Gamma correction]] relating light intensity with voltage
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| * The two-thirds power law, relating speed to curvature in the human [[motor system]].
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| *[[Kleiber's law]] relating animal metabolism to size, and [[allometric law]]s in general
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| * The [[Taylor's law]] relating mean population size and variance of populations sizes in ecology
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| *[[Phase transition#Critical exponents and universality classes|Behaviour near second-order phase transitions]] involving [[critical exponent]]s
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| *Proposed form of [[Experience curve effects#The experience curve|experience curve effects]]
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| *The differential energy spectrum of [[cosmic-ray]] nuclei
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| *[[Square-cube law]] (ratio of surface area to volume)
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| *[[Fractal]]s
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| *The [[Pareto principle]] also called the "80–20 rule"
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| *[[Zipf's law]] in corpus analysis and population distributions amongst others, where frequency of an item or event is inversely proportional to its frequency rank (i.e. the second most frequent item/event occurring half as often the most frequent item and so on).
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| *The [[safe operating area]] relating to maximum simultaneous current and voltage in power semiconductors.
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| ===Variants===
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| ====Square law====
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| '''Square law''' refers to a power law with exponent of 2.
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| ====Inverse-Square law====
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| [[Inverse-square law]] refers to a power law with exponent of -2.
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| ====Broken power law====
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| A broken power law is defined with a threshold:<ref>{{cite paper |url=http://iopscience.iop.org/1538-4357/640/1/L5/fulltext/20296.text.html |title=Afterglow Light Curves and Broken Power Laws: A Statistical Study |accessdate=2013-07-07}}</ref>
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| :<math>f(x) \propto x^{\alpha_1}</math> for <math>x<x_\text{th},</math>
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| :<math>f(x) \propto x^{\alpha_1-\alpha_2}_\text{th}x^{\alpha_2}\text{ for } x>x_\text{th}</math>.
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| ====Power law with exponential cutoff====
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| A power law with an exponential cutoff is simply a power law multiplied by an exponential function:<ref>{{cite paper |url=http://arxiv.org/pdf/0706.1062.pdf |title=POWER-LAW DISTRIBUTIONS IN EMPIRICAL DATA |accessdate=2013-07-07}}</ref>
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| :<math>f(x) \propto x^{\alpha}e^{\beta x}.</math>
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| ====Curved power law====
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| :<math>f(x) \propto x^{\alpha + \beta x}</math><ref>{{cite web |url=http://www.mpe.mpg.de/xray/wave/rosat/doc/users-guide/node-files/node188.php |title=Curved-power law |accessdate=2013-07-07}}</ref>
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| ==Power-law probability distributions==
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| In a looser sense, a power-law [[probability distribution]] is a distribution whose density function (or mass function in the discrete case) has the form
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| :<math>p(x) \propto L(x) x^{-\alpha}</math>
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| where <math>\alpha > 1</math>, and <math>L(x)</math> is a [[slowly varying function]], which is any function that satisfies <math>\lim_{x\rightarrow\infty} L(t\,x) / L(x) = 1</math> with <math>t</math> constant and <math>t > 0</math>. This property of <math>L(x)</math> follows directly from the requirement that <math>p(x)</math> be asymptotically scale invariant; thus, the form of <math>L(x)</math> only controls the shape and finite extent of the lower tail. For instance, if <math>L(x)</math> is the constant function, then we have a power law that holds for all values of <math>x</math>. In many cases, it is convenient to assume a lower bound <math>x_{\mathrm{min}}</math> from which the law holds. Combining these two cases, and where <math>x</math> is a continuous variable, the power law has the form
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| :<math>p(x) = \frac{\alpha-1}{x_\min} \left(\frac{x}{x_\min}\right)^{-\alpha},</math>
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| where the pre-factor to <math>\frac{\alpha-1}{x_\min}</math> is the [[normalizing constant]]. We can now consider several properties of this distribution. For instance, its [[Moment (mathematics)|moments]] are given by
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| :<math>\langle x^{m} \rangle = \int_{x_\min}^\infty x^{m} p(x) \,\mathrm{d}x = \frac{\alpha-1}{\alpha-1-m}x_\min^m</math>
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| which is only well defined for <math>m < \alpha -1</math>. That is, all moments <math>m \geq \alpha - 1</math> diverge: when <math>\alpha<2</math>, the average and all higher-order moments are infinite; when <math>2<\alpha<3</math>, the mean exists, but the variance and higher-order moments are infinite, etc. For finite-size samples drawn from such distribution, this behavior implies that the [[central moment]] estimators (like the mean and the variance) for diverging moments will never converge – as more data is accumulated, they continue to grow. These power-law probability distributions are also called [[Pareto distribution|Pareto-type distributions]], distributions with Pareto tails, or distributions with regularly varying tails.
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| Another kind of power-law distribution, which does not satisfy the general form above, is the power law with an exponential cutoff{{clarify|reason=who counts this as a power law?|date=May 2012}}
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| :<math>p(x) \propto L(x) x^{-\alpha} \mathrm{e}^{-\lambda x}.</math>
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| In this distribution, the exponential decay term <math>\mathrm{e}^{-\lambda x}</math> eventually overwhelms the power-law behavior at very large values of <math>x</math>. This distribution does not scale and is thus not asymptotically a power law; however, it does approximately scale over a finite region before the cutoff. (Note that the pure form above is a subset of this family, with <math>\lambda=0</math>.) This distribution is a common alternative to the asymptotic power-law distribution because it naturally captures finite-size effects. For instance, although the [[Gutenberg–Richter law]] is commonly cited as an example of a power-law distribution, the distribution of earthquake magnitudes cannot scale as a power law in the limit <math>x\rightarrow\infty</math> because there is a finite amount of energy in the Earth's crust and thus there must be some maximum size to an earthquake. As the scaling behavior approaches this size, it must taper off.
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| The [[Tweedie distributions]] are a family of statistical models characterized by [[Closure (mathematics)|closure]] under additive and reproductive convolution as well as under scale transformation. Consequently these models all express a power-law relationship between the variance and the mean. These models have a fundamental role as foci of mathematical [[Limit (mathematics)|convergence]] similar to the role that the [[normal distribution]] has as a focus in the [[central limit theorem]]. This convergence effect explains why the variance-to-mean power law manifests so widely in natural processes, as with [[Taylor's law]] in ecology and with fluctuation scaling<ref name=Kendal2011a>Kendal WS & Jørgensen B (2011) Taylor's power law and fluctuation scaling explained by a central-limit-like convergence. ''Phys. Rev. E'' 83,066115</ref> in physics. It can also be shown that this variance-to-mean power law, when demonstrated by the [[Tweedie distributions|method of expanding bins]], implies the presence of 1/''f'' noise and that 1/''f'' noise can arise as a consequence of this [[Tweedie distributions|Tweedie convergence effect]].<ref name=Kendal2011b>Kendal WS & Jørgensen BR (2011) Tweedie convergence: a mathematical basis for Taylor's power law, 1/''f'' noise and multifractality. ''Phys. Rev E'' 84, 066120</ref>
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| ===Graphical methods for identification===
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| Although more sophisticated and robust methods have been proposed, the most frequently used graphical methods of identifying power-law probability distributions using random samples are Pareto quantile-quantile plots (or Pareto [[Q-Q plot]]s),{{citation needed|date=May 2012}} mean residual life plots<ref>Beirlant, J., Teugels, J. L., Vynckier, P. (1996a) ''Practical Analysis of Extreme Values'', Leuven: Leuven University Press</ref><ref>Coles, S. (2001) ''An introduction to statistical modeling of extreme values''. Springer-Verlag, London.</ref> and [[log-log plot]]s. Another, more robust graphical method uses bundles of residual quantile functions.<ref name=Diaz>{{cite journal | last1 = Diaz |first1=F. J. | year = 1999 | title = Identifying Tail Behavior by Means of Residual Quantile Functions | url = | journal = Journal of Computational and Graphical Statistics | volume = 8 | issue = 3| pages = 493–509 | doi = 10.2307/1390871 }}</ref> (Please keep in mind that power-law distributions are also called Pareto-type distributions.) It is assumed here that a random sample is obtained from a probability distribution, and that we want to know if the tail of the distribution follows a power law (in other words, we want to know if the distribution has a "Pareto tail"). Here, the random sample is called "the data".
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| Pareto Q-Q plots compare the [[quantile]]s of the log-transformed data to the corresponding quantiles of an exponential distribution with mean 1 (or to the quantiles of a standard Pareto distribution) by plotting the former versus the latter. If the resultant scatterplot suggests that the plotted points " asymptotically converge" to a straight line, then a power-law distribution should be suspected. A limitation of Pareto Q-Q plots is that they behave poorly when the tail index <math>\alpha</math> (also called Pareto index) is close to 0, because Pareto Q-Q plots are not designed to identify distributions with slowly varying tails.<ref name=Diaz/>
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| On the other hand, in its version for identifying power-law probability distributions, the mean residual life plot consists of first log-transforming the data, and then plotting the average of those log-transformed data that are higher than the ''i''-th order statistic versus the ''i''-th order statistic, for ''i'' = 1, ..., ''n'', where n is the size of the random sample. If the resultant scatterplot suggests that the plotted points tend to "stabilize" about a horizontal straight line, then a power-law distribution should be suspected. Since the mean residual life plot is very sensitive to outliers (it is not robust), it usually produces plots that are difficult to interpret; for this reason, such plots are usually called Hill horror plots <ref>Resnick, S. I. (1997) "Heavy Tail Modeling and Teletraffic Data", ''The Annals of Statistics'', 25, 1805–1869.</ref>
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| [[Log-log plot]]s are an alternative way of graphically examining the tail of a distribution using a random sample. This method consists of plotting the logarithm of an estimator of the probability that a particular number of the distribution occurs versus the logarithm of that particular number. Usually, this estimator is the proportion of times that the number occurs in the data set. If the points in the plot tend to "converge" to a straight line for large numbers in the x axis, then the researcher concludes that the distribution has a power-law tail. Examples of the application of these types of plot have been published.<ref>{{cite journal | last1 = Jeong|first1= H|last2= Tombor|first2= B. Albert|last3= Oltvai|first3= Z.N.|last4= Barabasi|first4= A.-L. | year = 2000 | title = The large-scale organization of metabolic networks | url = | journal = Nature | volume = 407 | issue = 6804| pages = 651–654 | doi = 10.1038/35036627 | pmid = 11034217 |arxiv = cond-mat/0010278 |bibcode = 2000Natur.407..651J }}</ref> A disadvantage of these plots is that, in order for them to provide reliable results, they require huge amounts of data. In addition, they are appropriate only for discrete (or grouped) data.
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| Another graphical method for the identification of power-law probability distributions using random samples has been proposed.<ref name=Diaz/> This methodology consists of plotting a ''bundle for the log-transformed sample''. Originally proposed as a tool to explore the existence of moments and the moment generation function using random samples, the bundle methodology is based on residual [[quantile function]]s (RQFs), also called residual percentile functions,<ref>Arnold, B. C., Brockett, P. L. (1983) "When does the βth percentile residual life function determine the distribution?", ''Operations Research'' 31 (2), 391–396.</ref><ref>Joe, H., Proschan, F. (1984) "Percentile residual life functions", ''Operations Research'' 32 (3), 668–678.</ref><ref>Joe, H. (1985), "Characterizations of life distributions from percentile residual lifetimes", ''Ann. Inst. Statist. Math.'' 37, Part A, 165–172.</ref><ref>Csorgo, S., Viharos, L. (1992) "Confidence bands for percentile residual lifetimes", ''Journal of Statistical Planning and Inference'' 30, 327–337.</ref><ref>Schmittlein, D. C., Morrison, D. G. (1981), "The median residual lifetime: A characterization theorem and an application", ''Operations Research'' 29 (2), 392–399.</ref><ref>Morrison, D. G., Schmittlein, D. C. (1980) "Jobs, strikes, and wars: Probability models for duration", ''Organizational Behavior and Human Performance'' 25, 224–251.</ref><ref>Gerchak, Y. (1984) "Decreasing failure rates and related issues in the social sciences", ''Operations Research'' 32 (3), 537–546.</ref> which provide a full characterization of the tail behavior of many well-known probability distributions, including power-law distributions, distributions with other types of heavy tails, and even non-heavy-tailed distributions. Bundle plots do not have the disadvantages of Pareto Q-Q plots, mean residual life plots and log-log plots mentioned above (they are robust to outliers, allow visually identifying power laws with small values of <math>\alpha</math>, and do not demand the collection of much data).{{citation needed|date=May 2012}} In addition, other types of tail behavior can be identified using bundle plots.
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| ===Plotting power-law distributions===
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| In general, power-law distributions are plotted on [[log-log graph|doubly logarithmic axes]], which emphasizes the upper tail region. The most convenient way to do this is via the (complementary) [[cumulative distribution function#Complementary cumulative distribution function (tail distribution)|cumulative distribution]] (cdf), <math>P(x) = \mathrm{Pr}(X > x)</math>,
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| :<math>P(x) = \Pr(X > x) = C \int_x^\infty p(X)\,\mathrm{d}X = \frac{\alpha-1}{x_\min^{-\alpha+1}} \int_x^\infty X^{-\alpha}\,\mathrm{d}X = \left(\frac{x}{x_\min} \right)^{-\alpha+1}.</math>
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| Note that the cdf is also a power-law function, but with a smaller scaling exponent. For data, an equivalent form of the cdf is the rank-frequency approach, in which we first sort the <math>n</math> observed values in ascending order, and plot them against the vector <math>\left[1,\frac{n-1}{n},\frac{n-2}{n},\dots,\frac{1}{n}\right]</math>.
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| Although it can be convenient to log-bin the data, or otherwise smooth the probability density (mass) function directly, these methods introduce an implicit bias in the representation of the data, and thus should be avoided.{{Citation needed|date=January 2011}} The cdf, on the other hand, introduces no bias in the data and preserves the linear signature on doubly logarithmic axes.
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| ===Estimating the exponent from empirical data===
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| There are many ways of estimating the value of the scaling exponent for a power-law tail, however not all of them yield [[Maximum likelihood#Asymptotics|unbiased and consistent answers]]. Some of the most reliable techniques are often based on the method of [[maximum likelihood estimation|maximum likelihood]]. Alternative methods are often based on making a linear regression on either the log-log probability, the log-log cumulative distribution function, or on log-binned data, but these approaches should be avoided as they can all lead to highly biased estimates of the scaling exponent.{{sfn|Clauset|Shalizi|Newman|2009}}
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| ====Maximum likelihood====
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| For real-valued, [[independent and identically distributed]] data, we fit a power-law distribution of the form
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| : <math>p(x) = \frac{\alpha-1}{x_\min} \left(\frac{x}{x_\min}\right)^{-\alpha}</math>
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| to the data <math>x\geq x_\min</math>, where the coefficient <math>\frac{\alpha-1}{x_\min}</math> is included to ensure that the distribution is [[Normalizing constant|normalized]]. Given a choice for <math>x_\min</math>, a simple derivation by this method yields the estimator equation
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| :<math>\hat{\alpha} = 1 + n \left[ \sum_{i=1}^n \ln \frac{x_i}{x_\min} \right]^{-1}</math>
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| where <math>\{x_i\}</math> are the <math>n</math> data points <math>x_{i}\geq x_\min</math>.<ref name=Newman/><ref name=Hall/> This estimator exhibits a small finite sample-size bias of order <math>O(n^{-1})</math>, which is small when ''n'' > 100. Further, the uncertainty{{clarify|what aspect of uncertainty does equation relate to|date=June 2012}} in the estimation can be derived from the maximum likelihood argument, and has the form <math>\sigma = \frac{\alpha-1}{\sqrt{n}}</math>. This estimator is equivalent to the popular{{citation needed|date=June 2012}} [[Hill estimator]] from [[quantitative finance]] and [[extreme value theory]].{{citation needed|date=June 2012}}
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| For a set of ''n'' integer-valued data points <math>\{x_i\}</math>, again where each <math>x_i\geq x_\min</math>, the maximum likelihood exponent is the solution to the transcendental equation
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| : <math>\frac{\zeta'(\hat\alpha,x_\min)}{\zeta(\hat{\alpha},x_\min)} = -\frac{1}{n} \sum_{i=1}^n \ln \frac{x_i}{x_\min} </math> | |
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| where <math>\zeta(\alpha,x_{\mathrm{min}})</math> is the [[Riemann zeta function#Generalizations|incomplete zeta function]]. The uncertainty in this estimate follows the same formula as for the continuous equation. However, the two equations for <math>\hat{\alpha}</math> are not equivalent, and the continuous version should not be applied to discrete data, nor vice versa.
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| Further, both of these estimators require the choice of <math>x_\min</math>. For functions with a non-trivial <math>L(x)</math> function, choosing <math>x_\min</math> too small produces a significant bias in <math>\hat\alpha</math>, while choosing it too large increases the uncertainty in <math>\hat{\alpha}</math>, and reduces the [[statistical power]] of our model. In general, the best choice of <math>x_\min</math> depends strongly on the particular form of the lower tail, represented by <math>L(x)</math> above.
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| More about these methods, and the conditions under which they can be used, can be found in {{harv|Clauset|Shalizi|Newman|2009}}. Further, this comprehensive review article provides [http://www.santafe.edu/~aaronc/powerlaws/ usable code] (Matlab, R and C++) for estimation and testing routines for power-law distributions.
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| ====Kolmogorov–Smirnov estimation====
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| Another method for the estimation of the power-law exponent, which does not assume [[independent and identically distributed]] (iid) data, uses the minimization of the [[Kolmogorov–Smirnov statistic]], <math>D</math>, between the cumulative distribution functions of the data and the power law:
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| : <math>\hat{\alpha} = \underset{\alpha}{\operatorname{arg\,min}} \, D_\alpha </math>
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| with
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| : <math> D_\alpha = \max_x | P_\mathrm{emp}(x) - P_\alpha(x) | </math>
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| where <math>P_\mathrm{emp}(x)</math> and <math>P_\alpha(x)</math> denote the cdfs of the data and the power law with exponent <math>\alpha</math>, respectively. As this method does not assume iid data, it provides an alternative way to determine the power-law exponent for data sets in which the temporal correlation can not be ignored.<ref name=Klaus/>
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| ====Two-point fitting method====
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| This criterion{{clarify|reason=need to give sme detail of method|date=May 2012}} can be applied for the estimation of power-law exponent in the case of scale free distributions and provides a more convergent estimate than the maximum likelihood method.<ref name=Guerriero>{{Cite journal
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| | first1 = V.|last1= Guerriero
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| | year = 2012
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| | title = Power Law Distribution: Method of Multi-scale Inferential Statistics
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| | journal = Journal of Modern Mathematics Frontier (JMMF) | url =http://www.sjmmf.org/paperInfo.aspx?ID=886
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| | volume = 1
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| | pages = 21–28}}</ref> It has been applied to study probability distributions of fracture apertures.<ref name=Guerriero/> In some contexts the probability distribution is described, not by the [[cumulative distribution function]], by the [[cumulative frequency analysis|cumulative frequency]] of a property ''X'', defined as the number of elements per meter (or area unit, second etc.) for which ''X'' > ''x'' applies, where ''x'' is a variable real number. As an example,<ref name=Guerriero/> the cumulative distribution of the fracture aperture, ''X'', for a sample of ''N'' elements is defined as 'the number of fractures per meter having aperture greater than ''x'' . Use of cumulative frequency has some advantages, e.g. it allows one to put on the same diagram data gathered from sample lines of different lengths at different scales (e.g. from outcrop and from microscope).
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| ==== R-program pseudocode ====
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| pwrdist = function(u,main="") {
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| #u is vector of event counts, eg. how many crimes was a given perpetrator charged for by the police
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| fx=table(u);i=as.numeric(names(fx));
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| y=(1:max(i))*0;y[i]=fx
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| m0=glm(y~log(1:max(i)),family=quasipoisson());print(summary(m0))
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| sub=paste("s=",round(m0$coef[2],2),"lambda=",sum(u),"/",length(u))
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| plot(i,fx,log="xy",main=main,,xlab="x",sub=sub,ylab="counts");grid()
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| points(1:max(i),(fitted(m0)),type="l")
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| m0
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| }
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| ===Examples of power-law distributions===
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| ;Power law in the supercritical state of matter
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| In recent years, a significant effort has been devoted to investigation of various properties of [[supercritical fluids]]. This has been an exciting field with a long history since 1822 when Baron Charles Cagniard de la Tour discovered supercritical fluids while conducting experiments involving the discontinuities of the sound in a sealed cannon barrel filled with various fluids at high temperature. More recently, supercritical fluids have started to be deployed in several important applications, ranging from the extraction of floral fragrance from flowers to applications in food science such as creating decaffeinated coffee, functional food ingredients, pharmaceuticals, cosmetics, polymers, powders, bio- and functional materials, nano-systems, natural products, biotechnology, fossil and bio-fuels, microelectronics, energy and environment.
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| Apart from comparing theory and experiments, studying the supercritical scaling exponents is interesting in the wider context of scaling behaviour of physical properties. In the area of [[phase transitions]], the scaling behaviour idea has been a crucial element in the subject of liquids and other systems from the nineteenth century onwards. [[Dima Bolmatov]] with colleagues predicted relationship between supercritical exponents of [[heat capacity]] and [[viscosity]] and derived a power law for the supercritical state.<ref>{{Cite journal|doi=10.1038/ncomms3331 |title=Thermodynamic behaviour of supercritical matter|year=2013|last1=Bolmatov|first1=D.|last2=Brazhkin|first2=V. V.|last3=Trachenko|first3=K.|journal=Nature Communications|volume=4|arxiv = 1303.3153v3 }}</ref>
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| ;distributions
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| *[[Pareto distribution]] (continuous)
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| *[[Zeta distribution]] (discrete)
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| *[[Yule–Simon distribution]] (discrete)
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| *[[Student's t-distribution]] (continuous), of which the [[Cauchy distribution]] is a special case
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| ;[[empirical statistical laws]]
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| *[[Zipf's law]] and its generalization, the [[Zipf–Mandelbrot law]] (discrete)
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| **[[Lotka's law]]
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| ;models
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| *The [[scale-free network]] model{{citation needed|date=June 2012}}
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| *[[Pink noise]]
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| ;applications
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| *Neuronal avalanches<ref name=Klaus/>
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| *The law of stream numbers, and the law of stream lengths ([[Robert E. Horton|Horton]]'s laws describing river systems){{citation needed|date=June 2012}}
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| *Populations of cities ([[Gibrat's law]]){{citation needed|date=June 2012}}
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| *[[Bibliogram]]s, and frequencies of words in a text ([[Zipf's law]]){{citation needed|date=June 2012}}
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| *[[Bibliometrics|Bibliometric measures]], such as the frequency of publications by authors in a given field ([[Lotka's law]]){{citation needed|date=June 2012}}
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| *[[90–9–1 principle]] on [[wiki]]s (also referred to as the [[1% Rule (Internet culture)|1% Rule]]){{citation needed|date=June 2012}}
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| *Richardson's Law for the severity of violent conflicts (wars and terrorism){Lewis Fry Richardson, The Statistics of Deadly Quarrels, 1950}
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| <!--
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| COMMENT: The above list was carefully chosen to include only those real-world examples for which there is strong statistical support. The list below was carefully chosen to include those for which there is less statistical support. Please do not edit without first consulting the scientific literature. -->
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| A great many power-law distributions have been conjectured in recent years. For instance, power laws are thought to characterize the behavior of the upper tails for the popularity of websites, the degree distribution of the [[webgraph]], describing the [[hyperlink]] structure of the [[world-wide web]], the net worth of individuals, the number of species per genus, the popularity of [[Given name#Popularity distribution of given names|given names]], [[Gutenberg–Richter law]] of [[earthquake]] magnitudes, the size of financial returns, and many others.{{citation needed|date=May 2012}} However, much debate remains as to which of these tails are actually power-law distributed and which are not. For instance, it is commonly accepted now{{citation needed|date=May 2012}} that the famous [[Gutenberg–Richter law]] decays more rapidly than a pure power-law tail because of a finite exponential cutoff in the upper tail.
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| [[Log-normal distribution]]s are often mistaken for power-law distributions.{{sfn|Mitzenmacher|2004}} For example, [[Gibrat's law]] about proportional growth processes can actually produce limiting distributions that are lognormal, although their log-log plots look linear. An explanation of this is that although the logarithm of the [[Log-normal distribution#Probability density function|lognormal density function]] is quadratic in {{math|log(<var>x</var>)}}, yielding a "bowed" shape in a log-log plot, if the quadratic term is small relative to the linear term then the result can appear almost linear. Therefore a log-log plot that is slightly "bowed" downwards can reflect a log-normal distribution – not a power law.
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| ==Validating power laws==
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| Although power-law relations are attractive for many theoretical reasons, demonstrating that data do indeed follow a power-law relation requires more than simply fitting a particular model to the data. In general, many alternative functional forms can appear to follow a power-law form for some extent.{{sfn|Laherrère|Sornette|1998}} Also, researchers usually have to face the problem of deciding whether or not a real-world probability distribution follows a power law. As a solution to this problem, Diaz<ref name=Diaz/> proposed a graphical methodology based on random samples that allow visually discerning between different types of tail behavior. This methodology uses bundles of residual quantile functions, also called percentile residual life functions, which characterize many different types of distribution tails, including both heavy and non-heavy tails.
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| One method to validate a power-law relation tests many orthogonal predictions of a particular generative mechanism against data. Simply fitting a power-law relation to a particular kind of data is not considered a rational approach. As such, the validation of power-law claims remains a very active field of research in many areas of modern science.{{sfn|Clauset|Shalizi|Newman|2009}}
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| ==Note==
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| Power law distributions have a well defined [[mean]] only if the exponent exceeds 2 and have a finite [[variance]] only when the exponent exceeds 3. To apply the [[central limit theorem]] (and related statistics) to a power law distributed variable it is first necessary to ensure that the distribution has an exponent that exceeds 3.
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| ==See also==
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| {{col-begin}}
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| {{col-break}}
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| *[[Acoustic attenuation]]
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| *[[Empirical relationship]]
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| *[[Fat tail]]
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| *[[Finite-time singularity]]
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| *[[Fractional calculus]]
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| *[[Fractional dynamics]]
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| *[[Heavy-tailed distribution]]s
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| *[[Hyperbolic growth]]
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| *[[Lévy flight]]
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| *[[Long Tail]]
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| {{col-break}}
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| *[[Power law fluid]]
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| *[[Simon model]]
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| *[[Stable distribution]]
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| *[[Stevens' power law]]
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| *[[Wealth condensation]]
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| *[[Allometric law]]
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| *[[Webgraph]]
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| {{col-break}}
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| {{col-end}}
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| ==Notes==
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| {{reflist|refs=
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| <ref name=Hall>{{Cite journal
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| | author = Hall, P. | year = 1982
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| | title = On Some Simple Estimates of an Exponent of Regular Variation
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| | journal = [[Journal of the Royal Statistical Society, Series B]]
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| | volume = 44 | issue = 1 | pages = 37–42
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| | jstor = 2984706
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| }}</ref>
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| <ref name=Newman>{{Cite journal
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| | author = Newman, M. E. J.
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| | year = 2005
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| | title = Power laws, Pareto distributions and Zipf's law
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| | journal = [[Contemporary Physics]]
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| | volume = 46 | issue = 5 | pages = 323–351
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| | arxiv =cond-mat/0412004 | doi = 10.1080/00107510500052444
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| |bibcode = 2005ConPh..46..323N }}</ref>
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| }}
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| ==Bibliography==
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| * Bak, Per (1997) ''How nature works'', Oxford University Press ISBN 0 19 850164 1
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| *{{cite doi|10.1137/070710111}}
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| *{{cite doi|10.1007/s100510050276}}
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| *{{cite doi|10.1080/15427951.2004.10129088}}
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| * Alexander Saichev, Yannick Malevergne and Didier Sornette (2009) ''Theory of Zipf's law and beyond'', Lecture Notes in Economics and Mathematical Systems, Volume 632, Springer (November 2009), ISBN 978-3-642-02945-5
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| *{{cite doi|10.2307/2333389}}
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| *{{cite isbn|9783540308829}}
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| * Mark Buchanan (2000) ''Ubiquity'', Wiedenfield & Nicholson ISBN 0-297-64376-2
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| * Stumpf, M.P.H. and Porter, M.A. "Critical Truths about Power Laws" ''Science'' '''2012''', 335, 665-6
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| ==External links==
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| *[http://www.nslij-genetics.org/wli/zipf/ Zipf's law]
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| *[http://www.hpl.hp.com/research/idl/papers/ranking/ranking.html Zipf, Power-laws, and Pareto - a ranking tutorial]
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| *[http://simscience.org/crackling/Advanced/Earthquakes/GutenbergRichter.html Gutenberg–Richter Law]
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| *[http://www.physicalgeography.net/fundamentals/10ab.html Stream Morphometry and Horton's Laws]
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| *[[Clay Shirky]] on [https://www.youtube.com/watch?v=sPQViNNOAkw Institutions & Collaboration: Power law in relation to the internet-based social networks]
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| *[[Clay Shirky]] on [http://www.shirky.com/writings/herecomeseverybody/powerlaw_weblog.html Power Laws, Weblogs, and Inequality]
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| *[http://www.fooledbyrandomness.com/fortune.pdf "How the Finance Gurus Get Risk All Wrong"] by Benoit Mandelbrot & Nassim Nicholas Taleb. ''Fortune'', July 11, 2005.
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| *[http://www.newyorker.com/fact/content/articles/060213fa_fact "Million-dollar Murray":] power-law distributions in homelessness and other social problems; by [[Malcolm Gladwell]]. ''The New Yorker'', February 13, 2006.
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| *Benoit Mandelbrot & Richard Hudson: <cite>The Misbehaviour of Markets (2004) </cite>
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| *Philip Ball: [http://www.agrfoto.com/philipball/criticalmass.php Critical Mass: How one thing leads to another] (2005)
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| *[http://econophysics.blogspot.com/2006/07/tyranny-of-power-law-and-why-we-should.html ''Tyranny of the Power Law''] from [http://econophysics.blogspot.com The Econophysics Blog]
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| *[http://www.cscs.umich.edu/~crshalizi/weblog/491.html ''So You Think You Have a Power Law — Well Isn't That Special?''] from [http://www.cscs.umich.edu/~crshalizi/weblog/ Three-Toed Sloth], the blog of [[Cosma Shalizi]], Professor of Statistics at Carnegie-Mellon University.
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| *[http://www.mathworks.com/matlabcentral/fileexchange/27176-log-binning-of-data Simple MATLAB script] which bins data to illustrate power-law distributions (if any) in the data.
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| *[http://web-graph.org The Erdős Webgraph Server] visualizes the distribution of the degrees of the webgraph on the [http://web-graph.org/index.php/download download page].
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| {{DEFAULTSORT:Power Law}}
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| [[Category:Exponentials]]
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| [[Category:Power laws|*]]
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| [[Category:Tails of probability distributions]]
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| [[Category:Statistical models]]
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| [[nl:Machtsfunctie]]
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