Community detection for large networks poses challenges due to the high computational cost as well as heterogeneous community structures. In this paper, we consider widely existing real-world networks with ``grouped communities'' (or ``the group structure''), where nodes within grouped communities are densely connected and nodes across grouped communities are relatively loosely connected. We propose a two-step community detection approach for such networks. Firstly, we leverage modularity optimization methods to partition the network into groups, where between-group connectivity is low. Secondly, we employ the stochastic block model (SBM) or degree-corrected SBM (DCSBM) to further partition the groups into communities, allowing for varying levels of between-community connectivity. By incorporating this two-step structure, we introduce a novel divide-and-conquer algorithm that asymptotically recovers both the group structure and the community structure. Numerical studies confirm that our approach significantly reduces computational costs while achieving competitive performance. This framework provides a comprehensive solution for detecting community structures in networks with grouped communities, offering a valuable tool for various applications.

翻译：大型网络的社区检测面临高计算成本以及异构社区结构的挑战。本文针对广泛存在的具有"分组社区"（或称"组结构"）的现实网络展开研究，其中分组社区内部节点连接紧密，而跨分组社区节点连接相对稀疏。我们为此类网络提出了一种两步式社区检测方法。首先，利用模块度优化方法将网络划分为组间连接度较低的群组。其次，采用随机块模型（SBM）或度修正随机块模型（DCSBM）将群组进一步划分为社区，允许不同层级的社区间连接度。通过引入这种两步结构，我们提出了一种新颖的分治算法，能够渐近地恢复组结构和社区结构。数值研究证实，我们的方法在保持竞争性性能的同时显著降低了计算成本。该框架为具有分组社区的网络中的社区结构检测提供了全面解决方案，为各类应用提供了有价值的工具。