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In this paper we study weighted distances in scale-free spatial networkmodels: 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 $R$ andtwo vertices are connected either when they are within hyperbolic distance $R$,or independently with a probability depending on the hyperbolic distance. InGIRGs and SFP, each vertex is given an independent weight and location from anunderlying measured metric space and $\mathbb{Z}^d$, respectively, and twovertices are connected independently with a probability that is a function oftheir distance and weights. We assign i.i.d. weights to the edges of the randomgraphs and study the weighted distance between two uniformly chosen vertices.In SFP, we study the weighted distance from the origin of vertex-sequences withnorm tending to infinity. In particular, we study the case when the parametersare so that the degree distribution in the graph follows a power law withexponent $\tau\in(2,3)$ (infinite variance), and the edge-weight distributionis such that it produces an explosive age-dependent branching process withpower-law offspring distribution. We show that in all three models, typicaldistances within the giant/infinite component converge in distribution, solvingan 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, tofollow the shortest paths to infinity and to connect these paths usingweight-dependent percolation on the graphs: delete edges attached to verticeswith higher weight with higher probability. We realise this using theedge-weights: only short edges connected to high weight vertices will stay,yielding arbitrarily short upper bounds for the connections.

Explosion in weighted Hyperbolic Random Graphs and Geometric Inhomogeneous Random Graphs