  In Chung-Lu random graphs, a classic model for real-world networks, each
vertex is equipped with a weight drawn from a power-law distribution, and two
vertices form an edge independently with probability proportional to the
product of their weights. Chung-Lu graphs have average distance O(log log n)
and thus reproduce the small-world phenomenon, a key property of real-world
networks. Modern, more realistic variants of this model also equip each vertex
with a random position in a specific underlying geometry. The edge probability
of two vertices then depends, say, inversely polynomial on their distance.
  In this paper we study a generic augmented version of Chung-Lu random graphs.
We analyze a model where the edge probability of two vertices can depend
arbitrarily on their positions, as long as the marginal probability of forming
an edge (for two vertices with fixed weights, one fixed position, and one
random position) is as in Chung-Lu random graphs. The resulting class contains
Chung-Lu random graphs, hyperbolic random graphs, and geometric inhomogeneous
random graphs as special cases.
  Our main result is that every random graph model in this general class has
the same average distance as Chung-Lu random graphs, up to a factor 1+o(1).
This shows in particular that specific choices, such as the underlying geometry
being Euclidean or the dependence on the distance being inversely polynomial,
do not significantly influence the average distance. The proof also yields that
our model has a giant component and polylogarithmic diameter with high
probability.
