Consistency of Maximum Likelihood for Continuous-Space Network Models I

TL;DR

This paper proves that maximum likelihood estimation for continuous-space network models reliably recovers true latent positions as the network size increases, under certain symmetry and smoothness assumptions.

Contribution

It establishes the uniform convergence of the log-likelihood and the consistency of maximum likelihood embeddings for continuous latent space network models.

Findings

Maximum likelihood embeddings converge to true positions

Uniform convergence of the log-likelihood is proven

Results hold under symmetry and smoothness assumptions

Abstract

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.