  We study a generalisation of the random recursive tree (RRT) model and its
multigraph counterpart, the uniform directed acyclic graph (DAG). Here,
vertices are equipped with a random vertex-weight representing initial
inhomogeneities in the network, so that a new vertex connects to one of the old
vertices with a probability that is proportional to their vertex-weight. We
first identify the asymptotic degree distribution of a uniformly chosen vertex
for a general vertex-weight distribution. For the maximal degree, we
distinguish several classes that lead to different behaviour: For bounded
vertex-weights we obtain results for the maximal degree that are similar to
those observed for RRTs and DAGs. If the vertex-weights have unbounded support,
then the maximal degree has to satisfy the right balance between having a high
vertex-weight and being born early. For vertex-weights in the Frechet maximum
domain of attraction the first order behaviour of the maximal degree is random,
while for those in the Gumbel maximum domain of attraction the leading order is
deterministic. Surprisingly, in the latter case, the second order is random
when considering vertices in a compact window in the optimal region, while it
becomes deterministic when considering all vertices.
