Optimal Bayesian estimation for random dot product graphs

Summary We propose and prove the optimality of a Bayesian approach for estimating the latent positions in random dot product graphs, which we call posterior spectral embedding. Unlike classical spectral-based adjacency, or Laplacian spectral embedding, posterior spectral embedding is a fully likelihood-based graph estimation method that takes advantage of the Bernoulli likelihood information of the observed adjacency matrix. We develop a minimax lower bound for estimating the latent positions, and show that posterior spectral embedding achieves this lower bound in the following two senses: it both results in a minimax-optimal posterior contraction rate and yields a point estimator achieving the minimax risk asymptotically. The convergence results are subsequently applied to clustering in stochastic block models with positive semidefinite block probability matrices, strengthening an existing result concerning the number of misclustered vertices. We also study a spectral-based Gaussian spectral embedding as a natural Bayesian analogue of adjacency spectral embedding, but the resulting posterior contraction rate is suboptimal by an extra logarithmic factor. The practical performance of the proposed methodology is illustrated through extensive synthetic examples and the analysis of Wikipedia graph data.