  Preferential attachment models form a popular class of growing networks,
where incoming vertices are preferably connected to vertices with high degree.
We consider a variant of this process, where vertices are equipped with a
random initial fitness representing initial inhomogeneities among vertices and
the fitness influences the attractiveness of a vertex in an additive way. We
consider a heavy-tailed fitness distribution and show that the model exhibits a
phase transition depending on the tail exponent of the fitness distribution. In
the weak disorder regime, one of the old vertices has maximal degree
irrespective of fitness, while for strong disorder the vertex with maximal
degree has to satisfy the right balance between fitness and age. Our methods
use martingale methods to show concentration of degree evolutions as well as
extreme value theory to control the fitness landscape.
