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2014 Scale-free and power law distributions via fixed points and convergence of (thinning and conditioning) transformations
Richard Arratia, Thomas Liggett, Malcolm Williamson
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Electron. Commun. Probab. 19: 1-10 (2014). DOI: 10.1214/ECP.v19-2923

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 $, we show that there is a unique distribution which is a fixed point of this transformation; the distribution is power-law-$\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.

Citation

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Richard Arratia. Thomas Liggett. Malcolm Williamson. "Scale-free and power law distributions via fixed points and convergence of (thinning and conditioning) transformations." Electron. Commun. Probab. 19 1 - 10, 2014. https://doi.org/10.1214/ECP.v19-2923

Information

Accepted: 27 June 2014; Published: 2014
First available in Project Euclid: 7 June 2016

zbMATH: 1320.60010
MathSciNet: MR3225870
Digital Object Identifier: 10.1214/ECP.v19-2923

Subjects:
Primary: 60B10
Secondary: 05C82

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