We consider the community detection problem in sparse random hypergraphs under the non-uniform hypergraph stochastic block model (HSBM), a general model of random networks with community structure and higher-order interactions. When the random hypergraph has bounded expected degrees, we provide a spectral algorithm that outputs a partition with at least a $γ$ fraction of the vertices classified correctly, where $γ\in (0.5,1)$ depends on the signal-to-noise ratio (SNR) of the model. When the SNR grows slowly as the number of vertices goes to infinity, our algorithm achieves weak consistency, which improves the previous results in Ghoshdastidar and Dukkipati (2017) for non-uniform HSBMs. Our spectral algorithm consists of three major steps: (1) Hyperedge selection: select hyperedges of certain sizes to provide the maximal signal-to-noise ratio for the induced sub-hypergraph; (2) Spectral partition: construct a regularized adjacency matrix and obtain an approximate partition based on singular vectors; (3) Correction and merging: incorporate the hyperedge information from adjacency tensors to upgrade the error rate guarantee. The theoretical analysis of our algorithm relies on the concentration and regularization of the adjacency matrix for sparse non-uniform random hypergraphs, which can be of independent interest.

翻译：我们考虑稀疏随机超图在非一致超图随机块模型（HSBM）下的社区检测问题，该模型是一种具有社区结构和高阶交互作用的随机网络通用模型。当随机超图具有有界期望度时，我们提出了一种谱算法，该算法输出的划分中至少有 $γ$ 比例的顶点被正确分类，其中 $γ\in (0.5,1)$ 取决于模型的信噪比（SNR）。当SNR随顶点数增加而缓慢增长时，我们的算法实现了弱一致性，这改进了Ghoshdastidar和Dukkipati（2017）关于非一致HSBM的先前结果。我们的谱算法包含三个主要步骤：（1）超边选择：选择特定大小的超边，以为诱导子超图提供最大信噪比；（2）谱划分：构造正则化邻接矩阵，并基于奇异向量获得近似划分；（3）修正与合并：结合来自邻接张量的超边信息，以提升错误率保证。算法理论分析依赖于稀疏非一致随机超图邻接矩阵的集中性与正则化，这一点可能具有独立的研究价值。