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Power Laws in Preferential Attachment Graphs and Stein's Method for the Negative Binomial Distribution

Part of: Limit theorems Combinatorial probability Graph theory

Published online by Cambridge University Press:  04 January 2016

Nathan Ross
Nathan Ross*
Affiliation:
University of California, Berkeley
*
∗ Current address: Department of Mathematics and Statistics, Richard Berry Building, University of Melbourne, VIC 3010, Australia. Email address: nathan.ross@unimelb.edu.au
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Abstract

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For a family of linear preferential attachment graphs, we provide rates of convergence for the total variation distance between the degree of a randomly chosen vertex and an appropriate power law distribution as the number of vertices tends to ∞. Our proof uses a new formulation of Stein's method for the negative binomial distribution, which stems from a distributional transformation that has the negative binomial distributions as the only fixed points.

Keywords

Stein's methodnegative binomial distributionrandom graphpreferential attachmentdistributional transformationpower law

MSC classification

Primary: 05C80: Random graphs 60F05: Central limit and other weak theorems
Secondary: 60C05: Combinatorial probability
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 3 , September 2013 , pp. 876 - 893
Copyright
© Applied Probability Trust 

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