  Spectral clustering has been widely used for community detection in network
sciences. While its empirical successes are well-documented, a clear
theoretical understanding, particularly for sparse networks where degrees are
much smaller than $\log n$, remains unclear. In this paper, we address this
significant gap by demonstrating that spectral clustering offers exponentially
small error rates when applied to sparse networks under Stochastic Block
Models. Our analysis provides sharp characterizations of its performance,
backed by matching upper and lower bounds possessing an identical exponent with
the same leading constant. The key to our results is a novel truncated $\ell_2$
perturbation analysis for eigenvectors, coupled with a new analysis idea of
eigenvectors truncation.
