  We study a class of branching processes in which a population consists of
immortal individuals equipped with a fitness value. Individuals produce
offspring with a rate given by their fitness, and offspring may either belong
to the same family, sharing the fitness of their parent, or be founders of new
families, with a fitness sampled from a fitness distribution. Examples that can
be embedded in this class are stochastic house-of-cards models, urn models with
reinforcement, and the preferential attachment tree of Bianconi and Barabasi.
Our focus is on the case when the fitness distribution has bounded support and
regularly varying tail at the essential supremum. In this case there exists a
condensation phase, in which asymptotically a proportion of mass in the
empirical fitness distribution of the overall population condenses in the
maximal fitness value. Our main results describe the asymptotic behaviour of
the size and fitness of the largest family at a given time. In particular, we
show that as time goes to infinity the size of the largest family is always
negligible compared to the overall population size. This implies that
condensation, when it arises, is non-extensive and emerges as a collective
effort of several families none of which can create a condensate on its own.
Our result disproves claims made in the physics literature in the context of
preferential attachment trees.
