## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 25 (2020), paper no. 13, 13 pp.

### Distributional analysis of the extra-clustering model with uniformly generated phylogenetic trees

Michael Fuchs, Chih-Hong Lee, and Ariel R. Paningbatan

#### Abstract

The extra-clustering model for the group formation process of social animals was introduced by Durand, Blum and François. The model uses the relatedness of the animals, which is described by phylogenetic trees. If these trees are drawn from the Yule-Harding model, it was analyzed in recent work. Here, we analyze it for the uniform model, which is the other widely-studied model on phylogenetics trees. More precisely, we derive moments and limit laws for the number of groups, the number of groups of fixed size and the largest group size. Our results show that, independent of the probability of extra-clustering, there is on average only a finite number of groups, one of which is very large whereas all others are small. This behavior considerably differs from the Yule-Harding case, where the finiteness of the number of groups is dependent on the extra-clustering probability.

#### Article information

**Source**

Electron. Commun. Probab., Volume 25 (2020), paper no. 13, 13 pp.

**Dates**

Received: 24 July 2019

Accepted: 21 January 2020

First available in Project Euclid: 8 February 2020

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1581130925

**Digital Object Identifier**

doi:10.1214/20-ECP291

**Subjects**

Primary: 05A16: Asymptotic enumeration 60F05: Central limit and other weak theorems 92B05: General biology and biomathematics

**Keywords**

animal grouping extra-cluster model phylogenetic tree PDA model number of groups largest group size moments limit laws negative binomial distribution convergence of moments

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Fuchs, Michael; Lee, Chih-Hong; Paningbatan, Ariel R. Distributional analysis of the extra-clustering model with uniformly generated phylogenetic trees. Electron. Commun. Probab. 25 (2020), paper no. 13, 13 pp. doi:10.1214/20-ECP291. https://projecteuclid.org/euclid.ecp/1581130925