## Advances in Applied Probability

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

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

Panpan Zhang and Hosam Mahmoud

#### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 8 March 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.aap/1457466160

**Mathematical Reviews number (MathSciNet)**

MR3473572

**Zentralblatt MATH identifier**

1336.05124

**Subjects**

Primary: 05082 90B15: Network models, stochastic

Secondary: 60C05: Combinatorial probability

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aap/1457466160