We establish asymptotic normality results for estimation of the block probability matrix B in stochastic blockmodel graphs using spectral embedding when the average degrees grows at the rate of in n, the number of vertices. As a corollary, we show that when B is of full-rank, estimates of B obtained from spectral embedding are asymptotically efficient. When B is singular the estimates obtained from spectral embedding can have smaller mean square error than those obtained from maximizing the log-likelihood under no rank assumption, and furthermore, can be almost as efficient as the true MLE that assumes the rank of B is known. Our results indicate, in the context of stochastic blockmodel graphs, that spectral embedding is not just computationally tractable, but that the resulting estimates are also admissible, even when compared to the purportedly optimal but computationally intractable maximum likelihood estimation under no rank assumption.
The authors were supported in part by Johns Hopkins University Human Language Technology Center of Excellence and the XDATA and D3M programs of the Defense Advanced Research Projects Agency as administered through contract FA8750-12-2-0303 and contract FA8750-17-2-0112.
The authors would like to thank the anonymous referees, an Associate Editor and the Editor for their constructive comments that considerably improved the quality of this paper.
"Asymptotically efficient estimators for stochastic blockmodels: The naive MLE, the rank-constrained MLE, and the spectral estimator." Bernoulli 28 (2) 1049 - 1073, May 2022. https://doi.org/10.3150/21-BEJ1376