基于正则化张量幂迭代的超图网络社区检测
Community Detection for Hypergraph Networks via Regularized Tensor Power Iteration
摘要：
到目前为止，社会网络分析主要集中在成对的相互作用。通过超图网络对高阶相互作用的研究带来了新的见解。我们研究超图网络中的社区发现。一种流行的方法是将超图投影到图上，然后应用图网络的社区检测方法，但我们表明这种方法可能会导致不必要的信息丢失。我们提出了一种新的方法，直接对超图的社区检测。我们方法的核心是正则化高阶正交迭代（reg-HOOI）算法，该算法计算网络邻接张量的近似低秩分解。与现有的张量分解方法，如HOSVD和香草HOOI相比，reg-HOOI产生更好的性能，特别是当超图是稀疏的。给定张量分解的输出，我们然后将社区检测方法SCORE（Jin，2015）从图网络推广到超图网络。我们将新方法称为Tensor-SCORE。在理论上，我们引入了超图的度校正块模型（hDCBM），并证明了Tensor-SCORE对广泛的网络稀疏性和度异质性产生一致的社区检测。作为一个副产品，我们得到的reg-HOOI估计主子空间的收敛速度，不同的初始化，包括我们提出的两个新的初始化方法，对角删除HOSVD和随机图投影。我们将我们的方法应用到几个真正的超图网络，产生令人鼓舞的结果。这表明，探索高阶交互提供了图形表示中看不到的额外信息。
英文摘要：
To date, social network analysis has been largely focused on pairwise interactions. The study of higher-order interactions, via a hypergraph network, brings in new insights. We study community detection in a hypergraph network. A popular approach is to project the hypergraph to a graph and then apply community detection methods for graph networks, but we show that this approach may cause unwanted information loss. We propose a new method for community detection that operates directly on the hypergraph. At the heart of our method is a regularized higher-order orthogonal iteration (reg-HOOI) algorithm that computes an approximate low-rank decomposition of the network adjacency tensor. Compared with existing tensor decomposition methods such as HOSVD and vanilla HOOI, reg-HOOI yields better performance, especially when the hypergraph is sparse. Given the output of tensor decomposition, we then generalize the community detection method SCORE (Jin, 2015) from graph networks to hypergraph networks. We call our new method Tensor-SCORE. In theory, we introduce a degree-corrected block model for hypergraphs (hDCBM), and show that Tensor-SCORE yields consistent community detection for a wide range of network sparsity and degree heterogeneity. As a byproduct, we derive the rates of convergence on estimating the principal subspace by reg-HOOI, with different initializations, including the two new initialization methods we propose, a diagonal-removed HOSVD and a randomized graph projection. We apply our method to several real hypergraph networks which yields encouraging results. It suggests that exploring higher-order interactions provides additional information not seen in graph representations.
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