We make the first steps towards generalizing the theory of stochastic block
models, in the sparse regime, towards a model where the discrete community
structure is replaced by an underlying geometry. We consider a geometric random
graph over a homogeneous metric space where the probability of two vertices to
be connected is an arbitrary function of the distance. We give sufficient
conditions under which the locations can be recovered (up to an isomorphism of
the space) in the sparse regime. Moreover, we define a geometric counterpart of
the model of flow of information on trees, due to Mossel and Peres, in which
one considers a branching random walk on a sphere and the goal is to recover
the location of the root based on the locations of leaves. We give some
sufficient conditions for percolation and for non-percolation of information in
this model.Comment: 21 page