From: glevy@PRATT.EDU
Date: Sun Jan 01 2006 - 13:44:16 EST
Hi Paul Z,
I'll let Ian answer your question more. Here is part of the Wikipedia,
the free encyclopedia, entry for power law. I guess the more precise
way of stating Ian's claim is that income distribution follows a
power law probability distribution (see last section below).
In solidarity, Jerry
======================================
A power law relationship between two scalar quantities x and y is any such
that the relationship can be written as
k
y = ax
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
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:
a.. The Stefan-Boltzmann law
b.. The Gompertz Law of Mortality
c.. The Ramberg-Osgood stress-strain relationship
d.. The inverse-square law of Newtonian gravity
e.. Gamma correction relating light intensity with voltage
f.. Kleiber's law relating animal metabolism to size
g.. Behaviour near second-order phase transitions involving critical
exponents
h.. 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
i.. Proposed form of experience curve effects
j.. Scale-free networks, where the distribution of links is given by a
power law (in particular, the World Wide Web)
k.. The differential energy spectrum of cosmic-ray nuclei
Examples of power law probability distributions:
a.. The Pareto distribution
b.. Zipf's law
c.. 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.
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