Fundamental Limits of Spectral Clustering in Stochastic Block Models

We give a precise characterization of the performance of spectral clustering for community detection under Stochastic Block Models by carrying out sharp statistical analysis. We show spectral clustering has an exponentially small error with matching upper and lower bounds that have the same exponent, including the sharp leading constant. The fundamental limits established for the spectral clustering hold for networks with multiple and imbalanced communities and sparse networks with degrees far smaller than $\log n$. The key to our results is a novel truncated $\ell_2$ perturbation analysis for eigenvectors and a new analysis idea of eigenvectors truncation.