Power law

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Power laws are observed in many subject areas, including physics, biology, geography, sociology, economics, and linguistics. Power laws are among the most frequent scaling laws that describe the scale invariance found in many natural phenomena.

A power law relationship between two scalar quantities x and y is one where the relationship can be written as

$y = ax^k\,\!$

where a (the constant of proportionality) and k (the exponent of the power law) are constants.

Power laws can be seen as a straight line on a log-log graph since, taking logs of both sides, the above equation becomes

$\log(y) = k\log(x) + \log(a)\,\!$

which has the same form as the equation for a line

$y = mx+c\,\!$

Because both the power law and the log-normal distribution are asymptotic distributions, they can be easy to confuse without using robust statistical methods such as Bayesian model selection or statistical hypothesis testing. Indeed, a log-log plot of a log-normal distribution can often look nearly straight for specific ranges of x and y. One rule of thumb is the distribution conforms to a power law if it is straight on a log-log graph over 3 or more orders of magnitude.

Examples of power law relationships:

The Stefan-Boltzmann law
The Gompertz Law of Mortality
The Ramberg-Osgood stress-strain relationship
The inverse-square law of Newtonian gravity
Gamma correction relating light intensity with voltage
Kleiber's law relating animal metabolism to size
Behaviour near second-order phase transitions involving critical exponents
Proposed form of experience curve effects
The differential energy spectrum of cosmic-ray nuclei

Examples of power law probability distributions:

The Pareto distribution, for example, the distribution of wealth in capitalist economies
Zipf's law, for example, the frequency of unique words in large texts
Scale-free networks, where the distribution of links is given by a power law (in particular, the World Wide Web)
Frequency of events or effects of varying size in self-organized critical systems, e.g. Gutenberg-Richter Law of earthquake magnitudes and Horton's laws describing river systems

These appear to fit such disparate phenomena as the popularity of websites, the wealth of individuals, the popularity of given names, and the frequency of words in documents. Benoît Mandelbrot and Nassim Taleb have recently popularised the analysis of financial market volatility in terms of a power law distribution (as opposed to the traditional Gaussian distribution), and Aventis science prize-winning author Philip Ball has argued that the same power law relationships that are evident in phase transitions also apply to various manifestations of collective human behaviour.