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.

翻译：[translated abstract in Chinese] 谱聚类已被广泛应用于网络科学中的社区检测。尽管其实证成功已有充分记载，但明确的理论理解（尤其是在度数远小于$\log n$的稀疏网络中）仍不清楚。在本文中，我们通过证明谱聚类应用于稀疏网络（在随机块模型下）时能够实现指数级小的错误率，填补了这一重要空白。我们的分析为其性能提供了精确刻画，并得到匹配的上下界支持，这些上下界具有相同的指数和相同的主导常数。我们结果的关键在于一种针对特征向量的新型截断$\ell_2$扰动分析，并结合了特征向量截断的新分析思路。