Consistency of Maximum Likelihood for Continuous-Space Network Models I
Cosma Rohilla Shalizi, Dena Marie Asta
TL;DR
This work proves that non-parametric maximum likelihood embeddings for continuous latent space network models are statistically consistent in an information-theoretic sense, by establishing uniform convergence of the normalized log-likelihood over latent positions as the network grows. Under mild regularity (infinite-homogeneous latent spaces, smooth injective link functions, and bounded logit), the maximum-likelihood embedding converges to the true latent configuration up to isometry, in terms of KL divergence to the true graph distribution. The key technical contributions include a finite pseudo-dimension bound for the log-likelihood class, pointwise and uniform concentration results, and a decomposition of the expected log-likelihood into entropy and KL terms. The results pave the way for coordinate-wise consistency and density-estimation extensions in subsequent work, and hold under mis-specification with appropriate regularity assumptions, providing a general framework for consistent geometric network inference.
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.