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Investigating Power Laws with Mathematica

Fiona Maclachlan
Organization: Manhattan College
Department: Economics & Finance
URL: http://home.manhattan.edu/~fiona.maclachlan/

2006 Wolfram Technology Conference
Conference location

Champaign IL

Power-law distributions are found in a broad range of disciplines. In the late nineteenth century, Vilfredo Pareto identified a power law for the distribution of income. More recently, power laws have been discovered in the degree distributions of socially constructed networks like the World Wide Web, and have been associated with phenomena characterized by preferential attachment. They have also been linked with the idea of self-organized criticality and have been observed in the size distributions of many natural phenomena, such as sandpiles and earthquakes. Recent empirical studies of economic data have turned up power-law behavior in the return distribution of financial assets, and in the size distributions of firms and market shares. The latter work has been picked up in the marketing literature and has even found its way into popular business books like The Long Tail.

This presentation illustrates ways in which Mathematica can be used to investigate issues related to power laws. I show how the nonlinear regression capability can be used to investigate the goodness of fit of a power-law distribution to a data set, and to compare the fit with that of other similarly shaped distributions, such as the lognormal. In addition, I show how simple computational models can be constructed to investigate the extent to which power laws arise from randomness in an underlying process. For example, the hypothesis that Zipf’s law in word frequency would be generated by monkeys typing can be investigated with a simulation, as can a model of market shares based on the idea of random division.

*Business and Economics

power laws, economics, finance
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PowerLaws.nb (2.4 MB) - Mathematica Notebook [for Mathematica 5.2]