## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 21 (2016), paper no. 3, 38 pp.

### Preferential attachment with fitness: unfolding the condensate

#### Abstract

Preferential attachment models with fitness are a substantial extension of the classical preferential attachment model, where vertices have an independent fitness that has a linear impact on its attractiveness in the network formation. As observed by Bianconi and Barabási [4] such network models show different phases. In the condensation phase a small number of exceptionally fit vertices collects a finite fraction of all new links and hence forms a condensate. In this article, we analyse the formation of the condensate for a variant of the model with deterministic normalisation. We consider the regime where the fitness distribution is bounded and has polynomial tail behaviour in its upper end. The central result is a law of large numbers for an appropriately scaled version of the condensate. It follows that a $\Gamma$-distributed shape is formed and, in particular, that the number of vertices contributing to the condensate rises to infinity with increasing network size, in probability.

#### Article information

**Source**

Electron. J. Probab., Volume 21 (2016), paper no. 3, 38 pp.

**Dates**

Received: 16 September 2014

Accepted: 29 August 2015

First available in Project Euclid: 3 February 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1454514663

**Digital Object Identifier**

doi:10.1214/16-EJP3801

**Mathematical Reviews number (MathSciNet)**

MR3485345

**Zentralblatt MATH identifier**

1338.05245

**Subjects**

Primary: 05C80: Random graphs [See also 60B20]

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

**Keywords**

Barabási-Albert model preferential attachment fitness condensation

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Dereich, Steffen. Preferential attachment with fitness: unfolding the condensate. Electron. J. Probab. 21 (2016), paper no. 3, 38 pp. doi:10.1214/16-EJP3801. https://projecteuclid.org/euclid.ejp/1454514663