  Many real-world networks of interest are embedded in physical space. We
present a new random graph model aiming to reflect the interplay between the
geometries of the graph and of the underlying space. The model favors
configurations with small average graph distance between vertices, but adding
an edge comes at a cost measured according to the geometry of the ambient
physical space. In most cases, we identify the order of magnitude of the
average graph distance as a function of the parameters of the model. As the
proofs reveal, hierarchical structures naturally emerge from our simple
modeling assumptions. Moreover, a critical regime exhibits an infinite number
of discontinuous phase transitions.
