Translator Disclaimer
March 2016 The degree profile and weight in Apollonian networks and k-trees
Panpan Zhang, Hosam Mahmoud
Author Affiliations +
Adv. in Appl. Probab. 48(1): 163-175 (March 2016).


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.


Download Citation

Panpan Zhang. Hosam Mahmoud. "The degree profile and weight in Apollonian networks and k-trees." Adv. in Appl. Probab. 48 (1) 163 - 175, March 2016.


Published: March 2016
First available in Project Euclid: 8 March 2016

zbMATH: 1336.05124
MathSciNet: MR3473572

Primary: 05082 , 90B15
Secondary: 60C05

Keywords: degree profile , martingale , network , phase transition , Pólya urn , random graph , Random structure , self-similarity , stochastic recurrence

Rights: Copyright © 2016 Applied Probability Trust


This article is only available to subscribers.
It is not available for individual sale.

Vol.48 • No. 1 • March 2016
Back to Top