Power law

A power law relationship between two scalar quantities x and y is any such that the relationship can be written as

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:y = ax^k,!

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

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

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:log(y) = klog(x) + log(a),!

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which has the same form as the equation for a line

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:y = mx+c,!

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Power laws are observed in many fields, including physics, biology, geography, sociology, economics, and war and terrorism. Power laws are among the most frequent scaling laws that describe the scaling invariance found in many natural phenomena.

Related Topics:
Physics - Biology - Geography - Sociology - Economics - War - Terrorism - Scaling invariance

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Examples of power law relationships:

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• The Stefan-Boltzmann law
• 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
• 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
• Proposed form of experience curve effects

Examples of power law probability distributions:

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• The Pareto distribution
• Zipf's law

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.

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Power law: See also ►