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 $\gamma$ fraction of the vertices classified correctly, where $\gamma\in (0.5,1)$ 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:在非统一的超高测深块模型(HSBM)下,我们考虑在随机随机高光谱中的社区探测问题,这种高光谱模型是带有社区结构和更高级互动的随机网络的一般模型。当随机高光谱模型已经将预期温度折叠起来时,我们提供一种光谱算法,以至少等于负负负1美元的比例来输出一个至少为正分类的脊椎部分的分区,其中$\gamma=in(0.5,1美元)取决于该模型的信号-噪声比率。当SNR随着顶部数量变得不精确而缓慢增长时,我们的算法就会变得不连贯,从而改善了Ghoshdastidar和Dukkipati(2017年)的先前结果。我们的光谱算法包括三个主要步骤:(1) 超直径选择:选择某些大小的超高屏障,为诱导子波波波波波波波波波比比比比率(SNRRR) ;(2) 频谱分割:构建一个固定的对齐矩阵矩阵矩阵,并获得基于单层矢控矢量矢量矢量矢中度的近的不全局利率分析;(3) 校正校正和制,将一个高的机压率率分析纳入:从我们的高分辨率校正压率率率率率率率率率分析,将一个对准率率率率率率率率率率率率率,将一个高。