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Community Detection in the Hypergraph SBM: Optimal Recovery Given the Similarity Matrix

08/23/2022
by   Julia Gaudio, et al.
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Community detection is a fundamental problem in network science. In this paper, we consider community detection in hypergraphs drawn from the hypergraph stochastic block model (HSBM), with a focus on exact community recovery. We study the performance of polynomial-time algorithms for community detection in a case where the full hypergraph is unknown. Instead, we are provided a similarity matrix W, where W_ij reports the number of hyperedges containing both i and j. Under this information model, Kim, Bandeira, and Goemans [KBG18] determined the information-theoretic threshold for exact recovery, and proposed a semidefinite programming relaxation which they conjectured to be optimal. In this paper, we confirm this conjecture. We also show that a simple, highly efficient spectral algorithm is optimal, establishing the spectral algorithm as the method of choice. Our analysis of the spectral algorithm crucially relies on strong entrywise bounds on the eigenvectors of W. Our bounds are inspired by the work of Abbe, Fan, Wang, and Zhong [AFWZ20], who developed entrywise bounds for eigenvectors of symmetric matrices with independent entries. Despite the complex dependency structure in similarity matrices, we prove similar entrywise guarantees.

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