In this paper, we study the community detection problem in the stochastic block model (SBM) under privacy constraints. We introduce private and highly efficient algorithms for exact community detection within the SBM framework. Our algorithms represent the first differentially private methods capable of achieving exact recovery in a wide range of model parameters with near-linear time and space complexity. This is a significant improvement over previous SBM recovery algorithms, which either required pseudo-polynomial time or a quadratic scaling of resources for a constant privacy budget. Central to our approach is the introduction of a new concept, adaptive disjoint-star algorithms. These algorithms efficiently explore the graph's structure by querying node degrees on edge-disjoint subgraphs. We demonstrate that this general class of algorithms inherently offers strong privacy guarantees, a result that potentially holds value beyond the scope of SBM community detection. Finally, in we perform an empirical analysis of our algorithms showing that they can scale exact recovery on graphs with two orders of magnitude more nodes than prior work.

翻译：在本文中，我们研究了隐私约束下随机块模型（SBM）中的社区检测问题。我们提出了在SBM框架内实现精确社区检测的隐私保护且高效的算法。这些算法是首批能够在广泛模型参数范围内以近线性时间和空间复杂度实现精确恢复的差分隐私方法。相比于以往的SBM恢复算法（这些算法在恒定隐私预算下要么需要伪多项式时间，要么资源需求呈二次方增长），这是一个重大改进。我们方法的核心是引入一个新概念——自适应不交并星形算法。这些算法通过查询边不交子图上的节点度来高效探索图结构。我们证明了这类通用算法本质上具有强隐私保证，这一结果可能具有超越SBM社区检测范围的潜在价值。最后，我们通过实证分析表明，与先前工作相比，这些算法能够在节点数量多两个数量级的图上实现精确恢复的规模化。