  The preferential attachment network with fitness is a dynamic random graph
model. New vertices are introduced consecutively and a new vertex is attached
to an old vertex with probability proportional to the degree of the old one
multiplied by a random fitness. We concentrate on the typical behaviour of the
graph by calculating the fitness distribution of a vertex chosen proportional
to its degree. For a particular variant of the model, this analysis was first
carried out by Borgs, Chayes, Daskalakis and Roch. However, we present a new
method, which is robust in the sense that it does not depend on the exact
specification of the attachment law. In particular, we show that a peculiar
phenomenon, referred to as Bose-Einstein condensation, can be observed in a
wide variety of models. Finally, we also compute the joint degree and fitness
distribution of a uniformly chosen vertex.
