Advances in Applied Probability

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

Panpan Zhang and Hosam Mahmoud

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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.

Article information

Adv. in Appl. Probab., Volume 48, Number 1 (2016), 163-175.

First available in Project Euclid: 8 March 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05082 90B15: Network models, stochastic
Secondary: 60C05: Combinatorial probability

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


Zhang, Panpan; Mahmoud, Hosam. The degree profile and weight in Apollonian networks and k -trees. Adv. in Appl. Probab. 48 (2016), no. 1, 163--175.

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