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The degree profile and weight in Apollonian networks and k-trees

Part of: Operations research and management science Combinatorial probability

Published online by Cambridge University Press:  24 March 2016

Panpan Zhang  and
Hosam Mahmoud
Panpan Zhang*
Affiliation:
The George Washington University
Hosam Mahmoud*
Affiliation:
The George Washington University
*
* Postal address: Department of Statistics, The George Washington University, 801 22nd St. NW, Washington, DC 20052, USA.
* Postal address: Department of Statistics, The George Washington University, 801 22nd St. NW, Washington, DC 20052, USA.
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Abstract

We investigate the degree profile and total weight in Apollonian networks. We study the distribution of the degrees of vertices as they age in the evolutionary process. Asymptotically, the (suitably-scaled) degree of a node with a fixed label has a Mittag-Leffler-like limit distribution. The degrees of nodes of later ages have different asymptotic distributions, influenced by the time of their appearance. The very late arrivals have a degenerate distribution. The result is obtained via triangular Pólya urns. Also, via the Bagchi–Pal urn, we show that the number of terminal nodes asymptotically follows a Gaussian law. We prove that the total weight of the network asymptotically follows a Gaussian law, obtained via martingale methods. Similar results carry over to the sister structure of the k-trees, with minor modification in the proof methods, done mutatis mutandis.

Keywords

Random structurenetworkrandom graphself-similaritydegree profilephase transitionstochastic recurrencePólya urnmartingale

MSC classification

Primary: 90B15: Network models, stochastic
Secondary: 60C05: Combinatorial probability
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 1 , March 2016 , pp. 163 - 175
Copyright
Copyright © Applied Probability Trust 2016 

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