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In
statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, a power law is a
functional relationship In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously. The set is called ...
between two quantities, where a
relative change In any quantitative science, the terms relative change and relative difference are used to compare two quantities while taking into account the "sizes" of the things being compared, i.e. dividing by a ''standard'' or ''reference'' or ''starting' ...
in one quantity results in a relative change in the other quantity proportional to the change raised to a constant
exponent In mathematics, exponentiation, denoted , is an operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, i ...
: one quantity varies as a power of another. The change is independent of the initial size of those quantities. For instance, the area of a square has a power law relationship with the length of its side, since if the length is doubled, the area is multiplied by 2, while if the length is tripled, the area is multiplied by 3, and so on.


Empirical examples

The distributions of a wide variety of physical, biological, and human-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 Orbit of the Moon, orbits around Earth at Lunar distance, an average distance of (; about 30 times Earth diameter, Earth's diameter). The Moon rotation, rotates, with a rotation period (lunar ...
and of
solar flare A solar flare is a relatively intense, localized emission 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 ot ...
s, cloud sizes, 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 semantics, 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 consensus among linguist ...
s in most languages, frequencies of
family name In many societies, a surname, family name, or last name is the mostly hereditary portion of one's personal name that indicates one's family. It is typically combined with a given name to form the full name of a person, although several give ...
s, the
species richness Species richness is the number of different species represented in an community (ecology), ecological community, landscape or region. Species richness is simply a count of species, and it does not take into account the Abundance (ecology), abunda ...
in
clades In biology, a clade (), also known as a monophyletic group or natural group, is a group of organisms that is composed of a common ancestor and all of its descendants. Clades are the fundamental unit of cladistics, a modern approach to taxonomy ...
of organisms, the sizes of
power outage A power outage, also called a blackout, a power failure, a power blackout, a power loss, a power cut, or a power out is the complete loss of the electrical power network supply to an end user. There are many causes of power failures in an el ...
s, volcanic eruptions, human judgments of stimulus intensity and many other quantities. Empirical distributions can only fit a power law for a limited range of values, because a pure power law would allow for arbitrarily large or small values.
Acoustic attenuation In acoustics, acoustic attenuation is a measure of the energy loss of sound propagation through an acoustic transmission medium. Most media have viscosity and are therefore not ideal media. When sound propagates in such media, there is always th ...
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


Statistical incompleteness

The power-law model does not obey the treasured paradigm of statistical completeness. Especially probability bounds, the suspected cause of typical bending and/or flattening phenomena in the high- and low-frequency graphical segments, are parametrically absent in the standard model.


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. 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. Exponentiation#Power_functions, Power functions – relationshi ...
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. 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 A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
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">black swan The black swan (''Cygnus atratus'') is a large Anatidae, 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 ...
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, actuarial scien ...
(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 expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
and
standard deviation In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
(such as regression analysis). 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 physics, chemistry, and other related fields like biology, 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 Sta ...
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 exponents describe the behavior of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e. they do not depend on the details of the physical system, but only on some of its g ...
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 renormalization group (RG) is 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 the underlying p ...
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 c ...
. 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 sc ...
.


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 that may interact with one another. Examples of complex systems are Earth's global climate, organisms, the human brain, infrastructure such as power grid, transportation or communication s ...
, 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 in a probability space, where the index of the family often has the interpretation of time. Stoc ...
. 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 Shape, 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 scale ...
, scaling laws in biological systems, and scaling laws in cities. 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 scientific study of matter, its Elementary particle, 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 whi ...
,
computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...
,
linguistics Linguistics is the scientific study of language. The areas of linguistic analysis are syntax (rules governing the structure of sentences), semantics (meaning), Morphology (linguistics), morphology (structure of words), phonetics (speech sounds ...
,
geophysics Geophysics () is a subject of natural science concerned with the physical processes and Physical property, properties of Earth and its surrounding space environment, and the use of quantitative methods for their analysis. Geophysicists conduct i ...
,
neuroscience Neuroscience is the scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions, and its disorders. It is a multidisciplinary science that combines physiology, anatomy, molecular biology, ...
,
systematics 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: phylogenetic trees, phylogenies). Phy ...
,
sociology Sociology is the scientific study of human society that focuses on society, human social behavior, patterns of Interpersonal ties, social relationships, social interaction, and aspects of culture associated with everyday life. The term sociol ...
,
economics Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interac ...
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 a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample spac ...
: 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 the study of extremes in statistical distributions. It is widely used in many disciplines, such as structural engineering, finance, economics, earth sciences, traffic prediction, and Engin ...
), 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 fol ...
es and large
natural disaster A natural disaster is the very harmful impact on a society or community brought by natural phenomenon or Hazard#Natural hazard, hazard. Some examples of natural hazards include avalanches, droughts, earthquakes, floods, heat waves, landslides ...
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 reasons): :y = ax^k + \varepsilon.\! Mathematically, a strict power law cannot be a probability distribution, but a distribution that is a truncated
power function In mathematics, exponentiation, denoted , is an operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, i ...
is possible: p(x) = C x^ for x > x_\text where the exponent \alpha (Greek letter
alpha Alpha (uppercase , lowercase ) 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'' , whose name comes from the West Semitic word for ' ...
, 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:


Artificial Intelligence

*
Neural scaling law In machine learning, a neural scaling law is an empirical scaling law that describes how Neural network (machine learning), neural network performance changes as key factors are scaled up or down. These factors typically include the number of para ...


Astronomy

* Kepler's third law * 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 during star formation. IMF not only describes the formation and evolution of individual stars, it a ...
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 present ...
*
Solar flares A solar flare is a relatively intense, localized emission of electromagnetic radiation in the Stellar atmosphere, Sun's atmosphere. Flares occur in active regions and are often, but not always, accompanied by coronal mass ejections, solar partic ...


Biology

*
Kleiber's law Kleiber's law, named after Max Kleiber for his biology work in the early 1930s, states, after many observations that, for a vast number of animals, an animal's Basal Metabolic Rate scales to the power of the animal's mass. More precisely : posi ...
relating animal metabolism to size, and
allometric law Allometry (Ancient Greek "other", "measurement") is the study of the relationship of body size to shape, anatomy, physiology and behaviour, first outlined by Otto Snell in 1892, by D'Arcy Thompson in 1917 in ''On Growth and Form'' and by Juli ...
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 nervous system, central and peripheral nervous system, peripheral structures in the nervous system that support motor functions, i.e. movement. Peripheral structures may include skeletal muscles and Efferen ...
. * 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 community (ecology), ecological community, landscape or region. Species richness is simply a count of species, and it does not take into account the Abundance (ecology), abunda ...
(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


Chemistry

* Rate law


Climate science

* Sizes of cloud areas and perimeters, as viewed from space * The size of rain-shower cells * Energy dissipation in cyclones * Diameters of
dust devils A dust devil (also known regionally as a dirt devil) is a strong, well-formed, and relatively short-lived whirlwind. Its size ranges from small (18 in/half a metre wide and a few yards/metres tall) to large (more than 30 ft/10 m ...
on Earth and Mars


General science

* 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 ha ...
*
Pink noise Pink noise, noise, fractional noise or fractal noise is a signal (information theory), signal or process with a frequency spectrum such that the power spectral density (power per frequency interval) is inversely proportional to the frequenc ...
*The law of stream numbers, and the law of stream lengths ( Horton's laws describing river systems) *Populations of cities (
Gibrat's law Gibrat's law, sometimes called Gibrat's rule of proportionate growth or the law of proportionate effect, is a rule defined by Robert Gibrat (1904–1980) in 1931 stating that the proportional rate of growth of a firm is independent of its absolut ...
) * Bibliograms, and frequencies of words in a text (
Zipf's law Zipf's law (; ) is an empirical law stating that when a list of measured values is sorted in decreasing order, the value of the -th entry is often approximately inversely proportional to . The best known instance of Zipf's law applies to the ...
) * 90–9–1 principle on
wiki A wiki ( ) is a form of hypertext publication on the internet which is collaboratively edited and managed by its audience directly through a web browser. A typical wiki contains multiple pages that can either be edited by the public or l ...
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 In signal processing, the power spectrum S_(f) of a continuous time signal x(t) describes the distribution of power into frequency components f composing that signal. According to Fourier analysis, any physical signal can be decomposed into ...
of the weight matrices of
deep neural network Deep learning is a subset of machine learning that focuses on utilizing multilayered neural network (machine learning), neural networks to perform tasks such as Statistical classification, classification, Regression analysis, regression, and re ...
s *Associated with
exponential growth Exponential growth occurs when a quantity grows as an exponential function of time. The quantity grows at a rate directly proportional to its present size. For example, when it is 3 times as big as it is now, it will be growing 3 times as fast ...
: **Tails in statistical distributions for exponential growth processes with random observation (or killing) **Progress through exponential growth 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. The theory was popularized by Everett Rogers in his book ''Diffusion of Innovations'', first published in 1962. Rogers argue ...


Economics

* Population sizes of cities in a region or urban network,
Zipf's law Zipf's law (; ) is an empirical law stating that when a list of measured values is sorted in decreasing order, the value of the -th entry is often approximately inversely proportional to . The best known instance of Zipf's law applies to the ...
. *Distribution of artists by the average price of their artworks. *
Income distribution 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 e ...
in a market economy. *Distribution of degrees in banking networks. *Firm-size distributions. *Scaling laws of socio-economic quantities with respect to population size (see urban scaling).


Finance

* Returns for high-risk
venture capital Venture capital (VC) is a form of private equity financing provided by firms or funds to start-up company, startup, early-stage, and emerging companies, that have been deemed to have high growth potential or that have demonstrated high growth in ...
investments * The mean absolute change of the logarithmic mid-prices * Large price changes, volatility, and transaction volume on stock exchanges * Average waiting time of a directional change * Average waiting time of an overshoot


Mathematics

*
Fractal In mathematics, a fractal is a Shape, 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 scale ...
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, actuarial scien ...
and the Pareto principle also called the "80–20 rule" *
Zipf's law Zipf's law (; ) is an empirical law stating that when a list of measured values is sorted in decreasing order, the value of the -th entry is often approximately inversely proportional to . The best known instance of Zipf's law applies to the ...
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 theory and statistics, Student's  distribution (or simply the  distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
(continuous), of which the
Cauchy distribution The Cauchy distribution, named after Augustin-Louis 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) ...
is a special case *
Lotka's law Lotka's law, named after Alfred J. Lotka, is one of a variety of special applications of Zipf's law. It describes the frequency of publication by authors in any given field. Definition Let X be the number of publications, Y be the number of ...
*The
scale-free network A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. That is, the fraction ''P''(''k'') of nodes in the network having ''k'' connections to other nodes goes for large values of ''k'' as : P( ...
model


Physics

*The Angstrom exponent in aerosol optics *The frequency-dependency of
acoustic attenuation In acoustics, acoustic attenuation is a measure of the energy loss of sound propagation through an acoustic transmission medium. Most media have viscosity and are therefore not ideal media. When sound propagates in such media, there is always th ...
in complex media *The
Stefan–Boltzmann law The Stefan–Boltzmann law, also known as ''Stefan's law'', describes the intensity of the thermal radiation emitted by matter in terms of that matter's temperature. It is named for Josef Stefan, who empirically derived the relationship, and Lu ...
*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 current through a semiconductor. It comes in two types: junction FET (JFET) and metal-oxide-semiconductor FET (MOSFET). FETs have three termi ...
s and
vacuum tubes A vacuum tube, electron tube, thermionic 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 voltage, potential difference has been applied. It ...
approximate a square-law relationship, a factor in " tube sound". *
Square–cube law The square–cube law (or cube–square law) is a mathematical principle, applied in a variety of scientific fields, which describes the relationship between the volume and the surface area as a shape's size increases or decreases. It was first ...
(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 amplifier, amplifying vacuum tube (or ''thermionic valve'' in British English) consisting of three electrodes inside an evacuated glass envelope: a heated Electrical filament, filament or cathode, a control grid, grid ...
s. *The
inverse-square law In science, an inverse-square law is any scientific law stating that the observed "intensity" of a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental ca ...
s of
Newtonian gravity Newton's law of universal gravitation describes gravity as a force by stating 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 sq ...
and
electrostatics Electrostatics is a branch of physics that studies slow-moving or stationary electric charges. Since classical antiquity, classical times, it has been known that some materials, such as amber, attract lightweight particles after triboelectric e ...
, as evidenced by the
gravitational potential In classical mechanics, the gravitational potential is a scalar potential associating with each point in space the work (energy transferred) per unit mass that would be needed to move an object to that point from a fixed reference point in the ...
and
Electrostatic potential Electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as electric potential energy per unit of electric charge. More precisely, electric potential is the amount of work needed ...
, 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 c ...
*Model of
van der Waals force In molecular physics and chemistry, the van der Waals force (sometimes 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 elec ...
*Force and potential in
simple harmonic motion In mechanics and physics, simple harmonic motion (sometimes abbreviated as ) is a special type of periodic motion an object experiences by means of a restoring force whose magnitude is directly proportional to the distance of the object from ...
*
Gamma correction Gamma correction or gamma is a Nonlinearity, nonlinear operation used to encode and decode Relative luminance, luminance or CIE 1931 color space#Tristimulus values, tristimulus values in video or still image systems. Gamma correction is, in the s ...
relating light intensity with voltage * Behaviour near second-order phase transitions involving
critical exponent Critical exponents describe the behavior of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e. they do not depend on the details of the physical system, but only on some of its g ...
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 (physics), plasma. Different states are distinguished by the ways the ...
and supercritical fluids, 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 capacity is a ...
and
viscosity Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
. *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 over 500 amino acids exist in nature, by far the most important are the Proteinogenic amino acid, 22 α-amino acids incorporated into p ...
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, which are the monomers of the polymer. A single amino acid ...
segments


Political Science

* Cube root law of assembly sizes


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 supersed ...
of psychophysics ( challenged with demonstrations that it may be logarithmic) * The power law of forgetting


Variants


Broken power law

A broken power law is a piecewise function, 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.


Smoothly broken power law

The pieces of a broken power law can be smoothly spliced together to construct a smoothly broken power law. There are different possible ways to splice together power laws. One example is the following:\ln \left(\frac + a\right) = c_0 \ln \left(\frac\right) + \sum_^n \frac \ln \left(1 + \left(\frac\right)^\right)where 0 < x_0 < x_1 < \cdots < x_n. When the function is plotted as a log-log plot with horizontal axis being \ln x and vertical axis being \ln(y/y_0 + a), the plot is composed of n+1 linear segments with slopes c_0, c_1, ..., c_n, separated at x = x_1, ..., x_n, smoothly spliced together. The size of f_i determines the sharpness of splicing between segments i-1, i.


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 a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
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, actuarial scien ...
:p(x) = \frac \left(\frac\right)^, where the pre-factor to \frac is the
normalizing constant In probability theory, a normalizing constant or normalizing factor is used to reduce any probability function to a probability density function with total probability of one. For example, a Gaussian function can be normalized into a probabilit ...
. We can now consider several properties of this distribution. For instance, its moments are given by :\mathbb \left(X^ \right) = \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 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
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similar to the role that the
normal distribution In probability theory and 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 ...
has as a focus in the
central limit theorem In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. 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 plot In statistics, a Q–Q plot (quantile–quantile plot) is a probability plot, a List of graphical methods, graphical method for comparing two probability distributions by plotting their ''quantiles'' against each other. A point on the plot ...
s), 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. Exponentiation#Power_functions, Power functions – relationshi ...
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

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 t ...
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.


Mean residual life plots

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 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. Exponentiation#Power_functions, Power functions – relationshi ...
s are an alternative way of graphically examining the tail of a distribution using a random sample. Taking the logarithm of a power law of the form f(x) = ax^ results in: :\begin \log(f(x)) &= \log(ax^) \\ &= \log(a) + \log(x^k) \\ &= \log(a) + k \cdot \log(x), \end which forms a straight line with slope k on a log-log scale. 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.


Bundle plots

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 is a function Q: ,1\mapsto \mathbb which maps some probability x \in ,1/math> of a random variable v to the value of the variable y such that P(v\leq y) = x according to its probability distr ...
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 (ccdf) that is, the
survival function The survival function is a function that gives the probability that a patient, device, or other object of interest will survive past a certain time. The survival function is also known as the survivor function or reliability function. The term ...
, 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 stati ...
. 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,
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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 in the financial field. In general, there exist two separate branches of finance that requ ...
and
extreme value theory Extreme value theory or extreme value analysis (EVA) is the study of extremes in statistical distributions. It is widely used in many disciplines, such as structural engineering, finance, economics, earth sciences, traffic prediction, and Engin ...
. 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 frequentist statistics, power is the probability of detecting a given effect (if that effect actually exists) using a given test in a given context. In typical use, it is a function of the specific test that is used (including the choice of tes ...
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
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(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. Ever ...
, 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 normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...
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 Gibrat's law, sometimes called Gibrat's rule of proportionate growth or the law of proportionate effect, is a rule defined by Robert Gibrat (1904–1980) in 1931 stating that the proportional rate of growth of a firm is independent of its absolut ...
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 A fat-tailed distribution is a probability distribution that exhibits a large skewness or kurtosis, relative to that of either a normal distribution or an exponential distribution. In common usage, the terms fat-tailed and Heavy-tailed distribut ...
*
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. Roughly speaking, “heavy-tailed” means the distribu ...
s * Hyperbolic growth * Lévy flight *
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 n ...
* Low-degree saturation *
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, actuarial scien ...
* 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 st ...
*
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 supersed ...


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 a Canadian journalist, author, and public speaker. He has been a staff writer for ''The New Yorker'' since 1996. He has published eight books. He is also the host of the podcast ''Revisionist ...
. ''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)

from The Econophysics Blog *
So You Think You Have a Power Law – Well Isn't That Special?
' from Three-Toed Sloth, the blog of Cosma Shalizi, 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