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Spectral clustering has become one of the most popular algorithms in data clustering and community detection. We study the performance of classical two-step spectral clustering via the graph Laplacian to learn the stochastic block model. Our aim is to answer the following question: when is spectral clustering via the graph Laplacian able to achieve strong consistency, i.e., the exact recovery of the underlying hidden communities? Our work provides an entrywise analysis (an ℓ∞-norm perturbation bound) of the Fiedler eigenvector of both the unnormalized and the normalized Laplacian associated with the adjacency matrix sampled from the stochastic block model. We prove that spectral clustering is able to achieve exact recovery of the planted community structure under conditions that match the information-theoretic limits. 

Leptospirosis is a life-threatening, zoonotic disease with various clinical presentations, including renal injury, hepatic injury, pancreatitis, and pulmonary hemorrhage. With prompt recognition of the disease and treatment, 90% of infected dogs have a positive outcome. Therefore, rapid, early diagnosis of leptospirosis is crucial. Testing for Leptospira-specific serum antibodies using the microscopic agglutination test (MAT) lacks sensitivity early in the disease process, and diagnosis can take >2 wk because of the need to demonstrate a rise in titer. We applied machine-learning algorithms to clinical variables from the first day of hospitalization to create machine-learning prediction models (MLMs). The models incorporated patient signalment, clinicopathologic data (CBC, serum chemistry profile, and urinalysis = blood work [BW] model), with or without a MAT titer obtained at patient intake (=BW + MAT model). The models were trained with data from 91 dogs with confirmed leptospirosis and 322 dogs without leptospirosis. Once trained, the models were tested with a cohort of dogs not included in the model training (9 leptospirosis-positive and 44 leptospirosis-negative dogs), and performance was assessed. Both models predicted leptospirosis in the test set with 100% sensitivity (95% CI: 70.1–100%). Specificity was 90.9% (95% CI: 78.8–96.4%) and 93.2% (95% CI: 81.8–97.7%) for the BW and BW + MAT models, respectively. Our MLMs outperformed traditional acute serologic screening and can provide accurate early screening for the probable diagnosis of leptospirosis in dogs.