## Journal of Applied Probability

- J. Appl. Probab.
- Volume 53, Number 2 (2016), 434-447.

### Degrees in random self-similar bipolar networks

#### Abstract

We investigate several aspects of a self-similar evolutionary process that builds a random bipolar network from building blocks that are themselves small bipolar networks. We characterize admissible outdegrees in the history of the evolution. We obtain the limit distribution of the polar degrees (when suitably scaled) characterized by its sequence of moments. We also obtain the asymptotic joint multivariate normal distribution of the number of nodes of small admissible outdegrees. Five possible substructures arise, and each has its own parameters (mean vector and covariance matrix) in the multivariate distribution. Several results are obtained by mapping bipolar networks into Pólya urns.

#### Article information

**Source**

J. Appl. Probab., Volume 53, Number 2 (2016), 434-447.

**Dates**

First available in Project Euclid: 17 June 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.jap/1466172865

**Mathematical Reviews number (MathSciNet)**

MR3514289

**Zentralblatt MATH identifier**

1342.05156

**Subjects**

Primary: 05082 90B15: Network models, stochastic

Secondary: 60C05: Combinatorial probability 60F05: Central limit and other weak theorems

**Keywords**

Random structure network self-similarity random graph degree stochastic recurrence Pólya urn multivariate normal distribution

#### Citation

Chen, Chen; Mahmoud, Hosam. Degrees in random self-similar bipolar networks. J. Appl. Probab. 53 (2016), no. 2, 434--447. https://projecteuclid.org/euclid.jap/1466172865