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We study a generalisation of the random recursive tree (RRT) model and itsmultigraph counterpart, the uniform directed acyclic graph (DAG). Here,vertices are equipped with a random vertex-weight representing initialinhomogeneities in the network, so that a new vertex connects to one of the oldvertices with a probability that is proportional to their vertex-weight. Wefirst identify the asymptotic degree distribution of a uniformly chosen vertexfor a general vertex-weight distribution. For the maximal degree, wedistinguish several classes that lead to different behaviour: For boundedvertex-weights we obtain results for the maximal degree that are similar tothose 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 highvertex-weight and being born early. For vertex-weights in the Frechet maximumdomain 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 isdeterministic. Surprisingly, in the latter case, the second order is randomwhen considering vertices in a compact window in the optimal region, while itbecomes deterministic when considering all vertices.

The maximal degree in random recursive graphs with random weights