  In this paper we study weighted distances in scale-free spatial network
models: hyperbolic random graphs (HRG), geometric inhomogeneous random graphs
(GIRG) and scale-free percolation (SFP). In HRGs, $n=\Theta(\mathrm{e}^{R/2})$
vertices are sampled independently from the hyperbolic disk with radius $R$ and
two vertices are connected either when they are within hyperbolic distance $R$,
or independently with a probability depending on the hyperbolic distance. In
GIRGs and SFP, each vertex is given an independent weight and location from an
underlying measured metric space and $\mathbb{Z}^d$, respectively, and two
vertices are connected independently with a probability that is a function of
their distance and weights. We assign i.i.d. weights to the edges of the random
graphs and study the weighted distance between two uniformly chosen vertices.
In SFP, we study the weighted distance from the origin of vertex-sequences with
norm tending to infinity. In particular, we study the case when the parameters
are so that the degree distribution in the graph follows a power law with
exponent $\tau\in(2,3)$ (infinite variance), and the edge-weight distribution
is such that it produces an explosive age-dependent branching process with
power-law offspring distribution. We show that in all three models, typical
distances within the giant/infinite component converge in distribution, solving
an open question in [Explosion and distances in scale-free percolation (2017)].
The main tools of our proof are to couple the models to infinite versions, to
follow the shortest paths to infinity and to connect these paths using
weight-dependent percolation on the graphs: delete edges attached to vertices
with higher weight with higher probability. We realise this using the
edge-weights: only short edges connected to high weight vertices will stay,
yielding arbitrarily short upper bounds for the connections.
