  We identify the upper large deviation probability for the number of edges in
scale-free geometric random graph models as the space volume goes to infinity.
Our result covers the models of scale-free percolation, the Boolean model with
heavy-tailed radius distribution, and the age-dependent random connection
model. In all these cases the mechanism behind the large deviation is based on
a condensation effect. Loosely speaking, the mechanism randomly selects a
finite number of vertices and increases their power, so that they connect to a
macroscopic number of vertices in the graph, while the other vertices retain a
degree close to their expectation and thus make no more than the expected
contribution to the large deviation event. We verify this intuition by means of
limit theorems for the empirical distributions of degrees and edge-lengths
under the conditioning. We observe that at large finite volumes, the
edge-length distribution splits into a bulk and travelling wave part of
asymptotically positive proportions.
