Recovering Unbalanced Communities in the Stochastic Block Model With Application to Clustering with a Faulty Oracle

The stochastic block model (SBM) is a fundamental model for studying graph clustering or community detection in networks. It has received great attention in the last decade and the balanced case, i.e., assuming all clusters have large size, has been well studied. However, our understanding of SBM with unbalanced communities (arguably, more relevant in practice) is still very limited. In this paper, we provide a simple SVD-based algorithm for recovering the communities in the SBM with communities of varying sizes. Under the KS-threshold conjecture, the tradeoff between the parameters in our algorithm is nearly optimal up to polylogarithmic factors for a wide range of regimes. As a byproduct, we obtain a time-efficient algorithm with improved query complexity for a clustering problem with a faulty oracle, which improves upon a number of previous work (Mazumdarand Saha [NIPS 2017], Larsen, Mitzenmacher and Tsourakakis [WWW 2020], Peng and Zhang[COLT 2021]). Under the KS-threshold conjecture, the query complexity of our algorithm is nearly optimal up to polylogarithmic factors.