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# Power law

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 Title: Power law Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Power law

An example power-law graph, being used to demonstrate ranking of popularity. To the right is the long tail, and to the left are the few that dominate (also known as the 80–20 rule).

In statistics, a power law is a functional relationship between two quantities, where one quantity varies as a power of another. For instance, the number of cities having a certain population size is found to vary as a power of the size of the population. Empirical power-law distributions hold only approximately or over a limited range.

## Contents

• Empirical examples of power laws 1
• Properties of power laws 2
• Scale invariance 2.1
• No average 2.2
• Universality 2.3
• Power-law functions 3
• Examples of power-law functions 3.1
• Variants 3.2
• Broken power law 3.2.1
• Power law with exponential cutoff 3.2.2
• Curved power law 3.2.3
• Power-law probability distributions 4
• Graphical methods for identification 4.1
• Plotting power-law distributions 4.2
• Estimating the exponent from empirical data 4.3
• Maximum likelihood 4.3.1
• Kolmogorov–Smirnov estimation 4.3.2
• Two-point fitting method 4.3.3
• R function 4.3.4
• Validating power laws 5
• Notes 7
• Bibliography 8
• External links 9

## Empirical examples of power laws

The distributions of a wide variety of physical, biological, and man-made phenomena approximately follow a power law over a wide range of magnitudes: these include the sizes of

• Zipf's law
• Zipf, Power-laws, and Pareto – a ranking tutorial
• Gutenberg–Richter Law
• Stream Morphometry and Horton's Laws
• Clay Shirky on Institutions & Collaboration: Power law in relation to the internet-based social networks
• Clay Shirky on Power Laws, Weblogs, and Inequality
• "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, February 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
• So You Think You Have a Power Law – Well Isn't That Special? from Three-Toed Sloth, the blog of Cosma Shalizi, Professor of Statistics at Carnegie-Mellon University.
• Simple MATLAB script which bins data to illustrate power-law distributions (if any) in the data.
• The Erdős Webgraph Server visualizes the distribution of the degrees of the webgraph on the download page.