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We establish asymptotic normality results for estimation of the blockprobability matrix $\mathbf{B}$ in stochastic blockmodel graphs using spectralembedding when the average degrees grows at the rate of $\omega(\sqrt{n})$ in$n$, the number of vertices. As a corollary, we show that when $\mathbf{B}$ isof full-rank, estimates of $\mathbf{B}$ obtained from spectral embedding areasymptotically efficient. When $\mathbf{B}$ is singular the estimates obtainedfrom spectral embedding can have smaller mean square error than those obtainedfrom maximizing the log-likelihood under no rank assumption, and furthermore,can be almost as efficient as the true MLE that assume known$\mathrm{rk}(\mathbf{B})$. Our results indicate, in the context of stochasticblockmodel graphs, that spectral embedding is not just computationallytractable, but that the resulting estimates are also admissible, even whencompared to the purportedly optimal but computationally intractable maximumlikelihood estimation under no rank assumption.

Asymptotically efficient estimators for stochastic blockmodels: the naive MLE, the rank-constrained MLE, and the spectral