  Continuous-time branching processes describe the evolution of a population
whose individuals generate a random number of children according to a birth
process. Such branching processes can be used to understand preferential
attachment models in which the birth rates are linear functions. We are
motivated by citation networks, where power-law citation counts are observed as
well as aging in the citation patterns. To model this, we introduce fitness and
age-dependence in these birth processes. The multiplicative fitness moderates
the rate at which children are born, while the aging is integrable, so that
individuals receives a finite number of children in their lifetime. We show the
existence of a limiting degree distribution for such processes. In the
preferential attachment case, where fitness and aging are absent, this limiting
degree distribution is known to have power-law tails. We show that the limiting
degree distribution has exponential tails for bounded fitnesses in the presence
of integrable aging, while the power-law tail is restored when integrable aging
is combined with fitness with unbounded support with at most exponential tails.
In the absence of integrable aging, such processes are explosive.
