Power law

A power law relationship between two scalar quantities x and y is any such that 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 suck can be seen as a straight line on a log-log graph since, taking logs of both sides, the above equation is equal to

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

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 notoriously easy to confuse without using robust statistical methods such as Bayesian model selection or statistical hypothesis testing. One rule of thumb, however, is if the distribution is straight on a log-log graph over 3 or more orders of magnitude.

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

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
• 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
• Scale-free networks, where the distribution of links is given by a power law (in particular, the World Wide Web)
• The differential energy spectrum of cosmic-ray nuclei

Examples of power law probability distributions:

• The Pareto distribution
• Zipf's law
• Weibull distribution

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.

See also

• allometric law
• bibliogram
• constructal law
• scale invariance
• self-similarity

External links

• 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
• A claim that the blogosphere obeys a powerlaw distribution
• Stream Morphometry and Horton's Laws
• Scale invariance in global terrorism
• From old wars to new wars and global terrorism