In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, 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 often used synonymously. The set is called the domain of the functi ...
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'' va ...
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
Power most often refers to:
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power
* Power (social and political), the ability to influence people or events
** Abusive power
Power may a ...
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 solar phe ...
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 semantics, objective or pragmatics, practical semantics, meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of w ...
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 a ...
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 term, t ...
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 electricit ...
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 Acoustic attenuation is a measure of the energy loss of sound propagation in media. Most media have viscosity and are therefore not ideal media. When sound propagates in such media, there is always thermal consumption of energy caused by viscosity. ...
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 , scaling the argument by a constant factor causes only a proportionate scaling of the function itself. That is,
:
where 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 simply multiplies the original power-law relation by the constant . 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 and , 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 as ...
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 normal ...
). 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 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 ''arithme ...
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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 c ...
behavior.
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
(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
). 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''
is the minimum value for which the power law holds.