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A very popular class of models for networks posits that each node isrepresented by a point in a continuous latent space, and that the probabilityof an edge between nodes is a decreasing function of the distance between themin this latent space. We study the embedding problem for these models, ofrecovering the latent positions from the observed graph. Assuming certainnatural symmetry and smoothness properties, we establish the uniformconvergence of the log-likelihood of latent positions as the number of nodesgrows. A consequence is that the maximum likelihood embedding converges on thetrue positions in a certain information-theoretic sense. Extensions of theseresults, to recovering distributions in the latent space, and so distributionsover arbitrarily large graphs, will be treated in the sequel.

Consistency of Maximum Likelihood for Continuous-Space Network Models I