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.