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To date, social network analysis has been largely focused on pairwiseinteractions. The study of higher-order interactions, via a hypergraph network,brings in new insights. We study community detection in a hypergraph network. Apopular approach is to project the hypergraph to a graph and then applycommunity detection methods for graph networks, but we show that this approachmay cause unwanted information loss. We propose a new method for communitydetection that operates directly on the hypergraph. At the heart of our methodis a regularized higher-order orthogonal iteration (reg-HOOI) algorithm thatcomputes an approximate low-rank decomposition of the network adjacency tensor.Compared with existing tensor decomposition methods such as HOSVD and vanillaHOOI, reg-HOOI yields better performance, especially when the hypergraph issparse. Given the output of tensor decomposition, we then generalize thecommunity detection method SCORE (Jin, 2015) from graph networks to hypergraphnetworks. 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 awide range of network sparsity and degree heterogeneity. As a byproduct, wederive the rates of convergence on estimating the principal subspace byreg-HOOI, with different initializations, including the two new initializationmethods we propose, a diagonal-removed HOSVD and a randomized graph projection.  We apply our method to several real hypergraph networks which yieldsencouraging results. It suggests that exploring higher-order interactionsprovides additional information not seen in graph representations.

Community Detection for Hypergraph Networks via Regularized Tensor Power Iteration