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In
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one quantity varies as a power of another. For instance, considering the area of a square in terms of the length of its side, if the length is doubled, the area is multiplied by a factor of four.


Empirical examples

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 craters on the
moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
and of
solar flare A solar flare is an intense localized eruption of electromagnetic radiation in the Sun's atmosphere. Flares occur in active regions and are often, but not always, accompanied by coronal mass ejections, solar particle events, and other sol ...
s, the foraging pattern of various species, the sizes of activity patterns of neuronal populations, the frequencies of
word A word is a basic element of language that carries an objective or practical meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of what a word is, there is no conse ...
s in most languages, frequencies of
family name In some cultures, a surname, family name, or last name is the portion of one's personal name that indicates one's family, tribe or community. Practices vary by culture. The family name may be placed at either the start of a person's full name, ...
s, the
species richness Species richness is the number of different species represented in an ecological community, landscape or region. Species richness is simply a count of species, and it does not take into account the abundances of the species or their relative ab ...
in
clades A clade (), also known as a monophyletic group or natural group, is a group of organisms that are monophyletic – that is, composed of a common ancestor and all its lineal descendants – on a phylogenetic tree. Rather than the English te ...
of organisms, the sizes of
power outage A power outage (also called a powercut, a power out, a power failure, a power blackout, a power loss, or a blackout) is the loss of the electrical power network supply to an end user. There are many causes of power failures in an electricity ...
s, volcanic eruptions, human judgments of stimulus intensity and many other quantities. Few empirical distributions fit a power law for all their values, but rather follow a power law in the tail. Acoustic attenuation follows frequency power-laws within wide frequency bands for many complex media. Allometric scaling laws for relationships between biological variables are among the best known power-law functions in nature.


Properties


Scale invariance

One attribute of power laws is their
scale invariance In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality. The technical term ...
. Given a relation f(x) = ax^, scaling the argument x by a constant factor c causes only a proportionate scaling of the function itself. That is, :f(c x) = a(c x)^ = c^ f( x ) \propto f(x),\! where \propto denotes
direct proportionality In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constan ...
. That is, scaling by a constant c simply multiplies the original power-law relation by the constant c^. 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 f(x) and x, and the straight-line on the
log–log plot In science and engineering, a log–log graph or log–log plot is a two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes. Power functions – relationships of the form y=ax^k – appear ...
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 In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a norma ...
). Thus, accurately fitting and validating power-law models is an active area of research in statistics; see below.


Lack of well-defined average value

A power-law x^ has a well-defined
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set. For a data set, the '' ar ...
over x \in ,\infty)_only_if__k_>_2_,_and_it_has_a_finite_variance_only_if_k_>3;_most_identified_power_laws_in_nature_have_exponents_such_that_the_mean_is_well-defined_but_the_variance_is_not,_implying_they_are_capable_of_black_swan_theory.html" "title="variance.html" ;"title=",\infty) only if k > 2 , and it has a finite variance">,\infty) only if k > 2 , and it has a finite variance only if k >3; most identified power laws in nature have exponents such that the mean is well-defined but the variance is not, implying they are capable of black swan theory">black swan The black swan (''Cygnus atratus'') is a large waterbird, a species of swan which breeds mainly in the southeast and southwest regions of Australia. Within Australia, the black swan is nomadic, with erratic migration patterns dependent upon ...
behavior. This can be seen in the following thought experiment: imagine a room with your friends and estimate the average monthly income in the room. Now imagine the world's richest person entering the room, with a monthly income of about 1 1,000,000,000, billion US$. What happens to the average income in the room? Income is distributed according to a power-law known as the
Pareto distribution The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto ( ), is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, ac ...
(for example, the net worth of Americans is distributed according to a power law with an exponent of 2). On the one hand, this makes it incorrect to apply traditional statistics that are based on
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
and
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...
(such as
regression analysis In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one ...
). On the other hand, this also allows for cost-efficient interventions. For example, given that car exhaust is distributed according to a power-law among cars (very few cars contribute to most contamination) it would be sufficient to eliminate those very few cars from the road to reduce total exhaust substantially. The median does exist, however: for a power law ''x'' –''k'', with exponent , it takes the value 21/(''k'' – 1)''x''min, where ''x''min is the minimum value for which the power law holds.


Universality

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 In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states ...
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 Critical or Critically may refer to: *Critical, or critical but stable, medical states **Critical, or intensive care medicine * Critical juncture, a discontinuous change studied in the social sciences. * Critical Software, a company specializing ...
s of the system. Diverse systems with the same critical exponents—that is, which display identical scaling behaviour as they approach criticality—can be shown, via
renormalization group In theoretical physics, the term renormalization group (RG) refers to a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in t ...
theory, to share the same fundamental dynamics. For instance, the behavior of water and CO2 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 critical systems, where the critical point of the system is an
attractor In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...
. Formally, this sharing of dynamics is referred to as universality, and systems with precisely the same critical exponents are said to belong to the same
universality class In statistical mechanics, a universality class is a collection of mathematical models which share a single scale invariant limit under the process of renormalization group flow. While the models within a class may differ dramatically at finite s ...
.


Power-law functions

Scientific interest in power-law relations stems partly from the ease with which certain general classes of mechanisms generate them. 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; see also universality above. The ubiquity of power-law relations in physics is partly due to dimensional constraints, while in
complex systems A complex system is a system composed of many components which may interact with each other. Examples of complex systems are Earth's global climate, organisms, the human brain, infrastructure such as power grid, transportation or communication sy ...
, power laws are often thought to be signatures of hierarchy or of specific
stochastic processes In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...
. A few notable examples of power laws are Pareto's law of income distribution, structural self-similarity of
fractals In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illus ...
, and 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 Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
,
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
,
linguistics Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Ling ...
,
geophysics Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' so ...
,
neuroscience Neuroscience is the science, scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions and disorders. It is a Multidisciplinary approach, multidisciplinary science that combines physiology, an ...
,
systematics Biological systematics is the study of the diversification of living forms, both past and present, and the relationships among living things through time. Relationships are visualized as evolutionary trees (synonyms: cladograms, phylogenetic t ...
,
sociology Sociology is a social science that focuses on society, human social behavior, patterns of social relationships, social interaction, and aspects of culture associated with everyday life. It uses various methods of empirical investigation an ...
,
economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...
and more. However, much of the recent interest in power laws comes from the study of
probability distributions In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
: 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 theory of large deviations (also called
extreme value theory Extreme value theory or extreme value analysis (EVA) is a branch of statistics dealing with the extreme deviations from the median of probability distributions. It seeks to assess, from a given ordered sample of a given random variable, the pr ...
), which considers the frequency of extremely rare events like
stock market crash A stock market crash is a sudden dramatic decline of stock prices across a major cross-section of a stock market, resulting in a significant loss of paper wealth. Crashes are driven by panic selling and underlying economic factors. They often foll ...
es and large
natural disaster A natural disaster is "the negative impact following an actual occurrence of natural hazard in the event that it significantly harms a community". A natural disaster can cause loss of life or damage property, and typically leaves some econ ...
s. It is primarily in the study of statistical distributions that the name "power law" is used. In empirical contexts, an approximation to a power-law o(x^k) often includes a deviation term \varepsilon, 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 Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...
reasons): :y = ax^k + \varepsilon.\! Mathematically, a strict power law cannot be a probability distribution, but a distribution that is a truncated power function is possible: p(x) = C x^ for x > x_\text where the exponent \alpha (Greek letter
alpha Alpha (uppercase , lowercase ; grc, ἄλφα, ''álpha'', or ell, άλφα, álfa) is the first letter of the Greek alphabet. In the system of Greek numerals, it has a value of one. Alpha is derived from the Phoenician letter aleph , whi ...
, not to be confused with scaling factor a used above) is greater than 1 (otherwise the tail has infinite area), the minimum value x_\text 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 below for details. Typically the exponent falls in the range 2 < \alpha < 3, though not always.


Examples

More than a hundred power-law distributions have been identified in physics (e.g. sandpile avalanches), biology (e.g. species extinction and body mass), and the social sciences (e.g. city sizes and income). Among them are:


Astronomy

*
Kepler's third law In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbi ...
* The
initial mass function In astronomy, the initial mass function (IMF) is an empirical function that describes the initial distribution of masses for a population of stars. The IMF is an output of the process of star formation. The IMF is often given as a probability d ...
of stars *The differential energy spectrum of cosmic-ray nuclei *The
M–sigma relation The M–sigma (or ''M''–''σ'') relation is an empirical correlation between the stellar velocity dispersion ''σ'' of a galaxy bulge and the mass M of the supermassive black hole at its center. The ''M''–''σ'' relation was first presented ...


Physics

*The Angstrom exponent in aerosol optics *The frequency-dependency of acoustic attenuation in complex media *The
Stefan–Boltzmann law The Stefan–Boltzmann law describes the power radiated from a black body in terms of its temperature. Specifically, the Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths ...
*The input-voltage–output-current curves of
field-effect transistor The field-effect transistor (FET) is a type of transistor that uses an electric field to control the flow of current in a semiconductor. FETs (JFETs or MOSFETs) are devices with three terminals: ''source'', ''gate'', and ''drain''. FETs co ...
s and
vacuum tubes A vacuum tube, electron tube, valve (British usage), or tube (North America), is a device that controls electric current flow in a high vacuum between electrodes to which an electric potential difference has been applied. The type known as a ...
approximate a square-law relationship, a factor in "
tube sound Tube sound (or valve sound) is the characteristic sound associated with a vacuum tube amplifier (valve amplifier in British English), a vacuum tube-based audio amplifier. At first, the concept of ''tube sound'' did not exist, because practically ...
". * Square–cube law (ratio of surface area to volume) *A 3/2-power law can be found in the plate characteristic curves of
triode A triode is an electronic amplifying vacuum tube (or ''valve'' in British English) consisting of three electrodes inside an evacuated glass envelope: a heated filament or cathode, a grid, and a plate (anode). Developed from Lee De Forest's ...
s. *The
inverse-square law In science, an inverse-square law is any scientific law stating that a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be unders ...
s of
Newtonian gravity Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distanc ...
and
electrostatics Electrostatics is a branch of physics that studies electric charges at rest ( static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for a ...
, as evidenced by the
gravitational potential In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric ...
and
Electrostatic potential Electrostatics is a branch of physics that studies electric charges at rest ( static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for a ...
, respectively. * Self-organized criticality with a critical point as an
attractor In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...
*Model of
van der Waals force In molecular physics, the van der Waals force is a distance-dependent interaction between atoms or molecules. Unlike ionic or covalent bonds, these attractions do not result from a chemical electronic bond; they are comparatively weak and ...
*Force and potential in
simple harmonic motion In mechanics and physics, simple harmonic motion (sometimes abbreviated ) is a special type of periodic motion of a body resulting from a dynamic equilibrium between an inertial force, proportional to the acceleration of the body away from the ...
*
Gamma correction Gamma correction or gamma is a nonlinear operation used to encode and decode luminance or tristimulus values in video or still image systems. Gamma correction is, in the simplest cases, defined by the following power-law expression: : V_\tex ...
relating light intensity with voltage * Behaviour near second-order phase transitions involving
critical exponent Critical or Critically may refer to: *Critical, or critical but stable, medical states **Critical, or intensive care medicine * Critical juncture, a discontinuous change studied in the social sciences. * Critical Software, a company specializing ...
s *The safe operating area relating to maximum simultaneous current and voltage in power semiconductors. *Supercritical
state of matter In physics, a state of matter is one of the distinct forms in which matter can exist. Four states of matter are observable in everyday life: solid, liquid, gas, and plasma. Many intermediate states are known to exist, such as liquid crystal, ...
and
supercritical fluids A supercritical fluid (SCF) is any substance at a temperature and pressure above its critical point, where distinct liquid and gas phases do not exist, but below the pressure required to compress it into a solid. It can effuse through porous so ...
, such as supercritical exponents of
heat capacity Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K). Heat cap ...
and
viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the int ...
. *The Curie–von Schweidler law in dielectric responses to step DC voltage input. * The damping force over speed relation in antiseismic dampers calculus * Folded solvent-exposed surface areas of centered
amino acids Amino acids are organic compounds that contain both amino and carboxylic acid functional groups. Although hundreds of amino acids exist in nature, by far the most important are the alpha-amino acids, which comprise proteins. Only 22 alpha am ...
in
protein structure Protein structure is the three-dimensional arrangement of atoms in an amino acid-chain molecule. Proteins are polymers specifically polypeptides formed from sequences of amino acids, the monomers of the polymer. A single amino acid monom ...
segments


Psychology

*
Stevens's power law Stevens' power law is an empirical relationship in psychophysics between an increased intensity or strength in a physical stimulus and the perceived magnitude increase in the sensation created by the stimulus. It is often considered to supersede ...
of psychophysics ( challenged with demonstrations that it may be logarithmic) * The power law of forgetting


Biology

* Kleiber's law relating animal metabolism to size, and
allometric law Allometry is the study of the relationship of body size to shape, anatomy, physiology and finally behaviour, first outlined by Otto Snell in 1892, by D'Arcy Thompson in 1917 in ''On Growth and Form'' and by Julian Huxley in 1932. Overview Allom ...
s in general * The two-thirds power law, relating speed to curvature in the human
motor system The motor system is the set of central and peripheral structures in the nervous system that support motor functions, i.e. movement. Peripheral structures may include skeletal muscles and neural connections with muscle tissues. Central structur ...
. * The Taylor's law relating mean population size and variance of populations sizes in ecology *Neuronal avalanches * The
species richness Species richness is the number of different species represented in an ecological community, landscape or region. Species richness is simply a count of species, and it does not take into account the abundances of the species or their relative ab ...
(number of species) in clades of freshwater fishes *The Harlow Knapp effect, where a subset of the
kinase In biochemistry, a kinase () is an enzyme that catalyzes the transfer of phosphate groups from high-energy, phosphate-donating molecules to specific substrates. This process is known as phosphorylation, where the high-energy ATP molecule don ...
s found in the human body compose a majority of published research *The size of forest patches globally follows a power law *The species-area relationship relating the number of species found in an area as a function of the size of the area


Meteorology

* The size of rain-shower cells, energy dissipation in cyclones, and the diameters of
dust devils A dust devil is a strong, well-formed, and relatively short-lived whirlwind. Its size ranges from small (half a metre wide and a few metres tall) to large (more than 10 m wide and more than 1 km tall). The primary vertical motion is ...
on Earth and Mars


General science

*
Exponential growth Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a ...
and random observation (or killing) *Progress through
exponential growth Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a ...
and exponential
diffusion of innovations Diffusion of innovations is a theory that seeks to explain how, why, and at what rate new ideas and technology spread. Everett Rogers, a professor of communication studies, popularized the theory in his book ''Diffusion of Innovations''; the boo ...
* Highly optimized tolerance *Proposed form of
experience curve effects In industry, models of the learning or experience curve effect express the relationship between experience producing a good and the efficiency of that production, specifically, efficiency gains that follow investment in the effort. The effect h ...
*
Pink noise Pink noise or noise is a signal or process with a frequency spectrum such that the power spectral density (power per frequency interval) is inversely proportional to the frequency of the signal. In pink noise, each octave interval (halving ...
*The law of stream numbers, and the law of stream lengths ( Horton's laws describing river systems) *Populations of cities ( Gibrat's law) * Bibliograms, and frequencies of words in a text ( Zipf's law) * 90–9–1 principle on
wiki A wiki ( ) is an online hypertext publication collaboratively edited and managed by its own audience, using a web browser. A typical wiki contains multiple pages for the subjects or scope of the project, and could be either open to the pub ...
s (also referred to as the 1% rule) *Richardson's Law for the severity of violent conflicts (wars and terrorism) *The relationship between a CPU's cache size and the number of cache misses follows the power law of cache misses. *The
spectral density The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies ...
of the weight matrices of
deep neural network Deep learning (also known as deep structured learning) is part of a broader family of machine learning methods based on artificial neural networks with representation learning. Learning can be supervised, semi-supervised or unsupervised. D ...
s


Mathematics

*
Fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as ill ...
s *
Pareto distribution The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto ( ), is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, ac ...
and the
Pareto principle The Pareto principle states that for many outcomes, roughly 80% of consequences come from 20% of causes (the "vital few"). Other names for this principle are the 80/20 rule, the law of the vital few, or the principle of factor sparsity. Manag ...
also called the "80–20 rule" * 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 occurs half as often as the most frequent item, the third most frequent item/event occurs one third as often as the most frequent item, and so on). * Zeta distribution (discrete) * Yule–Simon distribution (discrete) *
Student's t-distribution In probability and statistics, Student's ''t''-distribution (or simply the ''t''-distribution) is any member of a family of continuous probability distributions that arise when estimating the mean of a normally distributed population in situ ...
(continuous), of which the
Cauchy distribution The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...
is a special case * Lotka's law *The scale-free network model


Economics

* Population sizes of cities in a region or urban network, Zipf's law. *Distribution of artists by the average price of their artworks. *
Distribution of income In economics, income distribution covers how a country's total GDP is distributed amongst its population. Economic theory and economic policy have long seen income and its distribution as a central concern. Unequal distribution of income causes ec ...
in a market economy. *Distribution of degrees in banking networks.


Finance

* The mean absolute change of the logarithmic mid-prices * Number of tick counts over time * Size of the maximum price move * Average waiting time of a directional change * Average waiting time of an overshoot


Variants


Broken power law

A broken power law is a
piecewise function In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Pi ...
, consisting of two or more power laws, combined with a threshold. For example, with two power laws: :f(x) \propto x^ for x :f(x) \propto x^_\textx^\text x>x_\text.


Power law with exponential cutoff

A power law with an exponential cutoff is simply a power law multiplied by an exponential function: :f(x) \propto x^e^.


Curved power law

:f(x) \propto x^


Power-law probability distributions

In a looser sense, a power-law
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
is a distribution whose density function (or mass function in the discrete case) has the form, for large values of x, :P(X>x) \sim L(x) x^ where \alpha > 1, and L(x) is a slowly varying function, which is any function that satisfies \lim_ L(r\,x) / L(x) = 1 for any positive factor r. This property of L(x) follows directly from the requirement that p(x) be asymptotically scale invariant; thus, the form of L(x) only controls the shape and finite extent of the lower tail. For instance, if L(x) is the constant function, then we have a power law that holds for all values of x. In many cases, it is convenient to assume a lower bound x_ from which the law holds. Combining these two cases, and where x is a continuous variable, the power law has the form of the
Pareto distribution The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto ( ), is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, ac ...
:p(x) = \frac \left(\frac\right)^, where the pre-factor to \frac is the
normalizing constant The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. The normalizing constant is used to reduce any probability function to a probability density function with total probability of one. ...
. We can now consider several properties of this distribution. For instance, its moments are given by :\langle x^ \rangle = \int_^\infty x^ p(x) \,\mathrmx = \fracx_\min^m which is only well defined for m < \alpha -1. That is, all moments m \geq \alpha - 1 diverge: when \alpha\leq 2, the average and all higher-order moments are infinite; when 2<\alpha<3, 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 In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random ...
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-type distributions, distributions with Pareto tails, or distributions with regularly varying tails. A modification, which does not satisfy the general form above, with an exponential cutoff, is :p(x) \propto L(x) x^ \mathrm^. In this distribution, the exponential decay term \mathrm^ eventually overwhelms the power-law behavior at very large values of x. This distribution does not scale and is thus not asymptotically as a power law; however, it does approximately scale over a finite region before the cutoff. The pure form above is a subset of this family, with \lambda=0. This distribution is a common alternative to the asymptotic power-law distribution because it naturally captures finite-size effects. The
Tweedie distributions In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and the cl ...
are a family of statistical models characterized by 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 convergence similar to the role that the
normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
has as a focus in the
central limit theorem In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themsel ...
. 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 in physics. It can also be shown that this variance-to-mean power law, when demonstrated by the method of expanding bins, implies the presence of 1/''f'' noise and that 1/''f'' noise can arise as a consequence of this Tweedie convergence effect.


Graphical methods for identification

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 plots), mean residual life plots and
log–log plot In science and engineering, a log–log graph or log–log plot is a two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes. Power functions – relationships of the form y=ax^k – appear ...
s. Another, more robust graphical method uses bundles of residual quantile functions. (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". Pareto Q–Q plots compare the
quantile In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. There is one fewer quantile th ...
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 \alpha (also called Pareto index) is close to 0, because Pareto Q–Q plots are not designed to identify distributions with slowly varying tails. 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
Log–log plot In science and engineering, a log–log graph or log–log plot is a two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes. Power functions – relationships of the form y=ax^k – appear ...
s are an alternative way of graphically examining the tail of a distribution using a random sample. Caution has to be exercised however as a log–log plot is necessary but insufficient evidence for a power law relationship, as many non power-law distributions will appear as straight lines on a log–log plot. 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. 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. Another graphical method for the identification of power-law probability distributions using random samples has been proposed. 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 In probability and statistics, the quantile function, associated with a probability distribution of a random variable, specifies the value of the random variable such that the probability of the variable being less than or equal to that value equ ...
s (RQFs), also called residual percentile functions, 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 \alpha, and do not demand the collection of much data). In addition, other types of tail behavior can be identified using bundle plots.


Plotting power-law distributions

In general, power-law distributions are plotted on doubly logarithmic axes, which emphasizes the upper tail region. The most convenient way to do this is via the (complementary)
cumulative distribution In statistics, the frequency (or absolute frequency) of an event i is the number n_i of times the observation has occurred/recorded in an experiment or study. These frequencies are often depicted graphically or in tabular form. Types The cumul ...
(ccdf) that is, the survival function, P(x) = \mathrm(X > x), :P(x) = \Pr(X > x) = C \int_x^\infty p(X)\,\mathrmX = \frac \int_x^\infty X^\,\mathrmX = \left(\frac \right)^. 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 n observed values in ascending order, and plot them against the vector \left ,\frac,\frac,\dots,\frac\right/math>. 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. The survival function, on the other hand, is more robust to (but not without) such biases in the data and preserves the linear signature on doubly logarithmic axes. Though a survival function representation is favored over that of the pdf while fitting a power law to the data with the linear least square method, it is not devoid of mathematical inaccuracy. Thus, while estimating exponents of a power law distribution, maximum likelihood estimator is recommended.


Estimating the exponent from empirical data

There are many ways of estimating the value of the scaling exponent for a power-law tail, however not all of them yield unbiased and consistent answers. Some of the most reliable techniques are often based on the method of
maximum likelihood In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stat ...
. 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.


Maximum likelihood

For real-valued,
independent and identically distributed In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usual ...
data, we fit a power-law distribution of the form : p(x) = \frac \left(\frac\right)^ to the data x\geq x_\min, where the coefficient \frac is included to ensure that the distribution is normalized. Given a choice for x_\min, the log likelihood function becomes: :\mathcal(\alpha)=\log \prod _^n \frac \left(\frac\right)^ The maximum of this likelihood is found by differentiating with respect to parameter \alpha, setting the result equal to zero. Upon rearrangement, this yields the estimator equation: :\hat = 1 + n \left \sum_^n \ln \frac \right where \ are the n data points x_\geq x_\min. This estimator exhibits a small finite sample-size bias of order O(n^), which is small when ''n'' > 100. Further, the standard error of the estimate is \sigma = \frac + O(n^). This estimator is equivalent to the popular Hill estimator from
quantitative finance Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that require ...
and
extreme value theory Extreme value theory or extreme value analysis (EVA) is a branch of statistics dealing with the extreme deviations from the median of probability distributions. It seeks to assess, from a given ordered sample of a given random variable, the pr ...
. For a set of ''n'' integer-valued data points \, again where each x_i\geq x_\min, the maximum likelihood exponent is the solution to the transcendental equation : \frac = -\frac \sum_^n \ln \frac where \zeta(\alpha,x_) is the incomplete zeta function. The uncertainty in this estimate follows the same formula as for the continuous equation. However, the two equations for \hat are not equivalent, and the continuous version should not be applied to discrete data, nor vice versa. Further, both of these estimators require the choice of x_\min. For functions with a non-trivial L(x) function, choosing x_\min too small produces a significant bias in \hat\alpha, while choosing it too large increases the uncertainty in \hat, and reduces the
statistical power In statistics, the power of a binary hypothesis test is the probability that the test correctly rejects the null hypothesis (H_0) when a specific alternative hypothesis (H_1) is true. It is commonly denoted by 1-\beta, and represents the chances ...
of our model. In general, the best choice of x_\min depends strongly on the particular form of the lower tail, represented by L(x) above. More about these methods, and the conditions under which they can be used, can be found in . Further, this comprehensive review article provide
usable code
(Matlab, Python, R and C++) for estimation and testing routines for power-law distributions.


Kolmogorov–Smirnov estimation

Another method for the estimation of the power-law exponent, which does not assume
independent and identically distributed In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usual ...
(iid) data, uses the minimization of the Kolmogorov–Smirnov statistic, D, between the cumulative distribution functions of the data and the power law: : \hat = \underset \, D_\alpha with : D_\alpha = \max_x , P_\mathrm(x) - P_\alpha(x) , where P_\mathrm(x) and P_\alpha(x) denote the cdfs of the data and the power law with exponent \alpha, 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.


Two-point fitting method

This criterion 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. It has been applied to study probability distributions of fracture apertures. In some contexts the probability distribution is described, not by the
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Eve ...
, by the 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, 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).


Validating power laws

Although power-law relations are attractive for many theoretical reasons, demonstrating that data does indeed follow a power-law relation requires more than simply fitting a particular model to the data. This is important for understanding the mechanism that gives rise to the distribution: superficially similar distributions may arise for significantly different reasons, and different models yield different predictions, such as extrapolation. For example,
log-normal distribution In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a norma ...
s are often mistaken for power-law distributions: a data set drawn from a lognormal distribution will be approximately linear for large values (corresponding to the upper tail of the lognormal being close to a power law), but for small values the lognormal will drop off significantly (bowing down), corresponding to the lower tail of the lognormal being small (there are very few small values, rather than many small values in a power law). For example, Gibrat's law about proportional growth processes produce distributions that are lognormal, although their log–log plots look linear over a limited range. An explanation of this is that although the logarithm of the lognormal density function is quadratic in , 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, and the lognormal behavior is only visible when the quadratic term dominates, which may require significantly more data. Therefore, a log–log plot that is slightly "bowed" downwards can reflect a log-normal distribution – not a power law. In general, many alternative functional forms can appear to follow a power-law form for some extent. proposed plotting the empirical cumulative distribution function in the log-log domain and claimed that a candidate power-law should cover at least two orders of magnitude. 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 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. However, claimed the need for both a statistical and a theoretical background in order to support a power-law in the underlying mechanism driving the data generating process. 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.


See also

* Fat-tailed distribution *
Heavy-tailed distribution In probability theory, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded: that is, they have heavier tails than the exponential distribution. In many applications it is the right tail of the distrib ...
s *
Hyperbolic growth When a quantity grows towards a singularity under a finite variation (a "finite-time singularity") it is said to undergo hyperbolic growth. More precisely, the reciprocal function 1/x has a hyperbola as a graph, and has a singularity at 0, meani ...
*
Lévy flight A Lévy flight is a random walk in which the step-lengths have a Lévy distribution, a probability distribution that is heavy-tailed. When defined as a walk in a space of dimension greater than one, the steps made are in isotropic random directi ...
*
Long tail In statistics and business, a long tail of some distributions of numbers is the portion of the distribution having many occurrences far from the "head" or central part of the distribution. The distribution could involve popularities, random nu ...
*
Pareto distribution The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto ( ), is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, ac ...
* Power law fluid * Simon model *
Stable distribution In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be sta ...
*
Stevens's power law Stevens' power law is an empirical relationship in psychophysics between an increased intensity or strength in a physical stimulus and the perceived magnitude increase in the sensation created by the stimulus. It is often considered to supersede ...


References

Notes Bibliography * * * * * * * * * *


External links


Zipf, Power-laws, and Pareto – a ranking tutorial


by Benoit Mandelbrot & Nassim Nicholas Taleb. ''Fortune'', July 11, 2005.
"Million-dollar Murray"
power-law distributions in homelessness and other social problems; by
Malcolm Gladwell Malcolm Timothy Gladwell (born 3 September 1963) is an English-born Canadian journalist, author, and public speaker. He has been a staff writer for ''The New Yorker'' since 1996. He has published seven books: '' The Tipping Point: How Little ...
. ''The New Yorker'', February 13, 2006. *Benoit Mandelbrot & Richard Hudson: ''The Misbehaviour of Markets (2004)'' *Philip Ball
Critical Mass: How one thing leads to another
(2005)

fro
The Econophysics Blog''So You Think You Have a Power Law – Well Isn't That Special?''
fro
Three-Toed Sloth
the blog of
Cosma Shalizi Cosma Rohilla Shalizi (born February 28, 1974) is an associate professor in the Department of Statistics at Carnegie Mellon University in Pittsburgh. Life Cosma Rohilla Shalizi is of Tamil, Afghan and Italian heritage and was born in Boston, ...
, Professor of Statistics at Carnegie-Mellon University.
Simple MATLAB script
which bins data to illustrate power-law distributions (if any) in the data.
The Erdős Webgraph Server
visualizes the distribution of the degrees of the webgraph on th
download page
{{DEFAULTSORT:Power Law Exponentials Theory of probability distributions Statistical laws Articles with example R code