  A very popular class of models for networks posits that each node is
represented by a point in a continuous latent space, and that the probability
of an edge between nodes is a decreasing function of the distance between them
in this latent space. We study the embedding problem for these models, of
recovering the latent positions from the observed graph. Assuming certain
natural symmetry and smoothness properties, we establish the uniform
convergence of the log-likelihood of latent positions as the number of nodes
grows. A consequence is that the maximum likelihood embedding converges on the
true positions in a certain information-theoretic sense. Extensions of these
results, to recovering distributions in the latent space, and so distributions
over arbitrarily large graphs, will be treated in the sequel.
