  We investigate spatial random graphs defined on the points of a Poisson
process in $d$-dimensional space, which combine scale-free degree distributions
and long-range effects. Every Poisson point is assigned an independent weight.
Given the weight and position of the points, we form an edge between any pair
of points independently with a probability depending on the two weights of the
points and their distance. Preference is given to short edges and connections
to vertices with large weights. We characterize the parameter regime where
there is a nontrivial percolation phase transition and show that it depends not
only on the power-law exponent of the degree distribution but also on a
geometric model parameter. We apply this result to characterize robustness of
age-based spatial preferential attachment networks.
