  We use the framework of multivariate regular variation to analyse the
extremal behaviour of preferential attachment models. To this end, we follow a
directed linear preferential attachment model for a random, heavy-tailed number
of steps in time and treat the incoming edge count of all existing nodes as a
random vector of random length. By combining martingale properties, moment
bounds and a Breiman type theorem we show that the resulting quantity is
multivariate regularly varying, both as a vector of fixed length formed by the
edge counts of a finite number of oldest nodes, and also as a vector of random
length viewed in sequence space. A P\'{o}lya urn representation allows us to
explicitly describe the extremal dependence between the degrees with the help
of Dirichlet distributions. As a by-product of our analysis we establish new
results for almost sure convergence of the edge counts in sequence space as the
number of nodes goes to infinity.
