# Power law

Not to be confused with Force (law).
For other uses, see Power.
An example power-law graph, being used to demonstrate ranking of popularity. To the right is the long tail, and to the left are the few that dominate (also known as the 80–20 rule).

In statistics, a power law is a functional relationship between two quantities, where one quantity varies as a power of another. For instance, the number of cities having a certain population size is found to vary as a power of the size of the population. Empirical power-law distributions hold only approximately or over a limited range.

## Empirical examples of power laws

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 earthquakes, craters on the moon and of solar flares,[1] the foraging pattern of various species,[2] the sizes of activity patterns of neuronal populations,[3] the frequencies of words in most languages, frequencies of family names, the species richness in clades of organisms,[4] the sizes of power outages, wars, criminal charges per convict, and many other quantities.[5] 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 of power laws

### Scale invariance

One attribute of power laws is their scale invariance. Given a relation $f(x) = ax^k$, scaling the argument $x$ by a constant factor $c$ causes only a proportionate scaling of the function itself. That is,

$f(c x) = a(c x)^k = c^k f(x) \propto f(x).\!$

That is, scaling by a constant $c$ simply multiplies the original power-law relation by the constant $c^k$. 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 is often called the signature of a power law. With real data, such straightness is a necessary, but not sufficient, condition for the data following a power-law relation. In fact, there are many ways to generate finite amounts of data that mimic this signature behavior, but, in their asymptotic limit, are not true power laws (e.g., if the generating process of some data follows a Log-normal distribution). Thus, accurately fitting and validating power-law models is an active area of research in statistics.

## Validating power laws

Although power-law relations are attractive for many theoretical reasons, demonstrating that data do indeed follow a power-law relation requires more than simply fitting a particular model to the data.[14] For example log-normal distributions are often mistaken for power-law distributions.[35] For example, Gibrat's law about proportional growth processes can actually produce limiting distributions that are lognormal, although their log-log plots look linear. An explanation of this is that although the logarithm of the lognormal density function is quadratic in log(x), yielding a "bowed" shape in a log-log plot, if the quadratic term is small relative to the linear term then the result can appear almost linear. Therefore a log-log plot that is slightly "bowed" downwards can reflect a log-normal distribution – not a power law. In general, many alternative functional forms can appear to follow a power-law form for some extent.[36] 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[23] 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.

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.[5]

## Notes

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2. ^ Humphries NE, Queiroz N, Dyer JR, Pade NG, Musyl MK, Schaefer KM, Fuller DW, Brunnschweiler JM, Doyle TK, Houghton JD, Hays GC, Jones CS, Noble LR, Wearmouth VJ, Southall EJ, Sims DW (2010). "Environmental context explains Lévy and Brownian movement patterns of marine predators". Nature 465 (7301): 1066–1069. Bibcode:2010Natur.465.1066H. doi:10.1038/nature09116. PMID 20531470.
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5. ^ Newman M. Power laws, Pareto distributions and Zipf’s law. Contemporary Phys 2005, 46, 323
6. ^ a b 9na CEPAL Charlas Sobre Sistemas Complejos Sociales (CCSSCS): Leyes de potencias, http://www.youtube.com/watch?v=4uDSEs86xCI
7. ^ G. Neukum and B. A. Ivanov, Crater size distributions and impact probabilities on Earth from lunar, terrestrial-planet, and asteroid cratering data. In T. Gehrels (ed.), Hazards Due to Comets and Asteroids, pp. 359–416, University of Arizona Press, Tucson, AZ (1994).
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