Power law

A power law relationship between two scalar quantities x and y is any such that 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 is equal to

which has the same form as the equation for a line

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

Examples of power law relationships:

• 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
• Horton's laws describing river systems

Examples of power law probability distributions:

• The Pareto distribution and Zipf's law, that appear to fit such disparate phenomena as the popularity of websites, the wealth of individuals, and the frequency of words in documents.

See also

• allometric law
• constructal law

External links

• Zipf, Power-laws, and Pareto - a ranking tutorial
• A claim that the blogosphere obeys a powerlaw distribution
• Zipf Law, Zipf Distribution: An Introduction