Electronic Journal of Probability

Preferential attachment with fitness: unfolding the condensate

Steffen Dereich

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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

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

Received: 16 September 2014
Accepted: 29 August 2015
First available in Project Euclid: 3 February 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20]
Secondary: 60C05: Combinatorial probability 90B15: Network models, stochastic

Barabási-Albert model preferential attachment fitness condensation

Creative Commons Attribution 4.0 International License.


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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