## Electronic Journal of Probability

### Preferential attachment with fitness: unfolding the condensate

Steffen Dereich

#### 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
Accepted: 29 August 2015
First available in Project Euclid: 3 February 2016

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

#### 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

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