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Joint degree distributions of preferential attachment random graphs

Part of: Limit theorems Special processes Graph theory Combinatorial probability

Published online by Cambridge University Press:  26 June 2017

Erol Peköz ,
Adrian Röllin  and
Nathan Ross
Erol Peköz*
Affiliation:
Boston University
Adrian Röllin*
Affiliation:
National University of Singapore
Nathan Ross*
Affiliation:
University of Melbourne
*
* Postal address: Questrom School of Business, Boston University, 595 Commonwealth Avenue, Room 607, Boston, MA 02215, USA. Email address: pekoz@bu.edu
** Postal address: Department of Statistics and Applied Probability, National University of Singapore, 6 Science Drive 2, Singapore 117546, Singapore. Email address: adrian.roellin@nus.edu.sg
*** Postal address: School of Mathematics and Statistics, University of Melbourne, Richard Berry Building, VIC 3010, Australia. Email address: nathan.ross@unimelb.edu.au
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Abstract

We study the joint degree counts in linear preferential attachment random graphs and find a simple representation for the limit distribution in infinite sequence space. We show weak convergence with respect to the p-norm topology for appropriate p and also provide optimal rates of convergence of the finite-dimensional distributions. The results hold for models with any general initial seed graph and any fixed number of initial outgoing edges per vertex; we generate nontree graphs using both a lumping and a sequential rule. Convergence of the order statistics and optimal rates of convergence to the maximum of the degrees is also established.

Keywords

Preferential attachment random graphmulticolor Pólya urndistributional approximation

MSC classification

Primary: 60F05: Central limit and other weak theorems 60C05: Combinatorial probability
Secondary: 05C80: Random graphs 60K99: None of the above, but in this section
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 2 , June 2017 , pp. 368 - 387
Copyright
Copyright © Applied Probability Trust 2017 

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