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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
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
which has the same form as the equation for a line
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
- Kleiber's law relating animal metabolism to size
- Behaviour near second-order phase transitions involving critical exponents
- Proposed form of experience curve effects
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
- Frequency of events or effects of varying size in self-organized critical systems, e.g. Gutenberg-Richter Law of earthquake magnitudes and Robert E. Horton|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.
- Allometric law
- Constructal law
- Inverse-square law
- Scale invariance
- Square-cube law
- Weibull distribution
- Goldstein, M. L., Morris, S. A. and Yen, G. G. (2004). Problems with fitting to the power-law distribution. European Physical Journal B 41: 255–258.
- Mitzenmacher, M. (2003). A brief history of generative models for power law and lognormal distributions. Internet Mathematics 1: 226–251.
- Newman, M. E. J. (2005). Power laws, Pareto distributions and Zipf's law. Contemporary Physics 46: 323–351.
- Zipf's law
- Power laws, Pareto distributions and Zipf's law
- Zipf, Power-laws, and Pareto - a ranking tutorial
- Zipf Law, Zipf Distribution: An Introduction
- Gutenberg-Richter Law
- Stream Morphometry and Horton's Laws
- A claim that the blogosphere obeys a powerlaw distribution
- "How the Finance Gurus Get Risk All Wrong" 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. The New Yorker, Feb 13, 2006.
- Benoit Mandelbrot & Richard Hudson: The Misbehaviour of Markets (2004)
- Philip Ball: Critical Mass: How one thing leads to another (2005)
- Tyranny of the Power Law from The Econophysics Blog
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