  We study the asymptotic growth rate of the label size of high-degree vertices
in weighted recursive graphs (WRG) when the weights are i.i.d. almost surely
bounded random variables, and as a result confirm a conjecture by Lodewijks and
Ortgiese. WRGs are a generalisation of the random recursive tree (RRT) and
directed acyclic graph model (DAG), in which vertices are assigned
vertex-weights and where new vertices attach to $m\in\mathbb N$ predecessors,
each selected independently with a probability proportional to the
vertex-weight of the predecessor. Prior work established the asymptotic growth
rate of the maximum degree of the WRG model and here we show that there exists
a critical exponent $\mu_m$, such that the typical label size of the maximum
degree vertex equals $n^{\mu_m(1+o(1))}$ almost surely as $n$, the size of the
graph, tends to infinity. These results extend and improve on the asymptotic
behaviour of the location of the maximum degree, formerly only known for the
RRT model, to the more general weighted multigraph case of the WRG model.
Moreover, for the Weighted Recursive Tree (WRT) model, that is, the WRG model
with $m=1$, we prove the joint convergence of the rescaled degree and label of
high-degree vertices under additional assumptions on the vertex-weight
distribution, and also extend results on the growth rate of the maximum degree
obtained by Eslava, Lodewijks, and Ortgiese.
