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Scale-free and power law distributions via fixed points and convergence of (thinning and conditioning) transformations


 
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1. Title Title of document Scale-free and power law distributions via fixed points and convergence of (thinning and conditioning) transformations
 
2. Creator Author's name, affiliation, country Richard Arratia; University of Southern California; United States
 
2. Creator Author's name, affiliation, country Thomas M. Liggett; UCLA; United States
 
2. Creator Author's name, affiliation, country Malcolm J. Williamson; Center For Communications Research; United States
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) thinning, power-law, scale-free, degree distribution, Pareto distribution
 
3. Subject Subject classification Primary 60B10; Secondary 05C82
 
4. Description Abstract

In discrete contexts such as the degree distribution for a graph, scale-free has traditionally been defined to be power-law. We propose a reasonable interpretation of scale-free, namely, invariance under the  transformation of $p$-thinning, followed by conditioning on being positive.

For each $\beta \in (1,2)$, we show that there is a unique distribution which is a fixed point of this transformation; the distribution is power-law-$\beta$, and different from the usual Yule-Simon power law-$\beta$ that arises in preferential attachment models.

In addition to characterizing these fixed points, we prove convergence results for iterates of the transformation.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s)
 
7. Date (YYYY-MM-DD) 2014-06-27
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/2923
 
10. Identifier Digital Object Identifier 10.1214/ECP.v19-2923
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 19
 
12. Language English=en en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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