Sparse random hypergraphs: Non-backtracking spectra and community detection

Abstract

We consider the community detection problem in a sparse q-uniform hypergraph G, assuming that G is generated according to the so-called Hypergraph Stochastic Block Model (HSBM). We prove that a spectral method based on the non-backtracking operator for hypergraphs works with high probability down to the generalized Kesten-Stigum detection threshold conjectured by Angelini et al. in [12]. We characterize the spectrum of the non-backtracking operator for the sparse HSBM, and provide an efficient dimension reduction procedure using the Ihara-Bass formula for hypergraphs. As a result, community detection for the sparse HSBM on n vertices can be reduced to an eigenvector problem of a 2n × 2n non-normal matrix constructed from the adjacency matrix and the degree matrix of the hypergraph. To the best of our knowledge, this is the first provable and efficient spectral algorithm that achieves the conjectured threshold for HSBMs with k blocks generated according to a general symmetric probability tensor.

Nonbacktracking spectral clustering of nonuniform hypergraphs

“…We now extend this theorem to nonuniform hypergraphs. Our result generalizes a computation by Angelini et al [4] and theorem by Stephan and Zhu [61].…”

“…6: end for 7: z = Cluster( X) 8: return z related formulations exist in the literature [29,4,18,23,61,19]. We prove in-expectation results for eigenpairs of the matrices B k , and pose conjectures generalizing recent proofs by Stephan and Zhu [61] of eigenpair concentration results in the uniform case. These conjectures will also inform our development of a spectral method based on belief-propagation in Section 5.…”

“…The authors offered a conjecture on the detectability threshold in a hypergraph planted partition model. Several of the conjectures for uniform hypergraphs were recently formalized and proved by Stephan and Zhu [61].…”

“…In the case of uniform hypergraphs, there exist recent probabilistic guarantees ensuring that x is correlated with planted communities under a certain choice of data generating process [61]. Extending these guarantees to the nonuniform cases is an avenue of future work.…”

“…In this paper, we study spectral methods based on the hypergraph nonbacktracking operator [62]. Several spectral clustering algorithms exist for uniform hypergraphs-hypergraphs in which all edges contain the same number of nodes-including one based on the nonbacktracking operator [4,61]. Many interesting hypergraph data sets, however, are nonuniform.…”