Rate-Optimal Spectral Likelihood-Ratio Clustering in Stochastic Block Models

Spectral clustering commonly assigns communities by clustering a low-dimensional representation, whereas likelihood refinement can improve an initial assignment using the observed edges. We ask whether sharp community-recovery accuracy can be retained when the refinement no longer accesses individual edges. We develop spectral likelihood-ratio clustering for stochastic block models using one regularized adjacency eigensystem and the original vertex degrees. The method estimates connection probabilities and updates each vertex's label by a likelihood-ratio classifier on reconstructed rows. The refinement accepts any sufficiently accurate initialization, including one computed from the same graph. Its expected misclassification proportion is bounded by a constant times the error of a raw-edge oracle classifier, plus the probability that initialization misses the required accuracy. Repeated k-means++ on the retained spectral coordinates supplies one complete procedure with a constant-factor oracle guarantee. The oracle knows the model parameters and the other labels. The guarantee holds for a fixed number of communities, comparable positive block probabilities and community sizes, and a nondegenerate normalized block matrix. It allows arbitrarily slowly diverging expected degrees under the stated upper sparsity condition. We explain the likelihood-ratio construction first, then the regularization and exact-degree correction required by the general result. The proof combines sharp local testing, truncated spectral-row estimates and leave-one-out comparisons.