From: Paul Cockshott (wpc@DCS.GLA.AC.UK)
Date: Mon Jan 02 2006 - 16:01:45 EST
I am attaching an explantory note that I recently wrote on the difference between power law and Boltzmann Gibbs distributions. The Wikepedia account accords with my understanding of it. -----Original Message----- From: OPE-L [mailto:OPE-L@SUS.CSUCHICO.EDU] On Behalf Of Paul Zarembka Sent: 01 January 2006 20:48 To: OPE-L@SUS.CSUCHICO.EDU Subject: Re: [OPE-L] Absolutes in Marxian Theory? I have experience that Wikipedia leaves a lot to desired in terms of being a responsible source. Maybe it's OK for this, maybe not. Paul Z. ************************************************************************ RESEARCH IN POLITICAL ECONOMY, Paul Zarembka, editor, Elsevier Science ********************* http://ourworld.compuserve.com/homepages/PZarembka On Sun, 1 Jan 2006 glevy@PRATT.EDU wrote: > 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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