Community detection in the sparse hypergraph stochastic block model

Pal, Soumik; Zhu, Yizhe

doi:10.1002/rsa.21006

Cited by 16 publications

(9 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This is the model considered in [70] for r = 2, and in [12] for general r ≥ 2. In this case, the eigenvalues of…”

Section: Hypergraph Stochastic Block Modelmentioning

confidence: 99%

“…Such method was first applied to the non-backtracking operator in [23,20]. For hypergraphs, it was first developed in [70] for the self-avoiding matrix, and we generalize it to the non-backtracking matrix. The moment method is based on a bipartite representation of the non-backtracking walk on hypergraphs, and such connection between hypergraphs and bipartite graphs was also used in [42,41].…”

Section: Organization Of the Papermentioning

confidence: 99%

“…where φ i , φ j are eigenvectors of Q, ∂h is defined in (65), and h φi,t is defined in (70). Recall the definition of χ i in (16).…”

Section: Structure Of Near Eigenvectorsmentioning

confidence: 99%

“…Partial recovery in this regime was studied in [7,40], but the conjectured threshold can not be achieved by their methods. It was shown in [70] that a spectral algorithm based on the self-avoiding walks on hypergraphs could achieve the threshold for the 2-block case with equal size and a probability tensor given by two parameters. However, constructing such self-avoiding matrix is not very efficient and is hard to implement in practice.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Sparse random hypergraphs: Non-backtracking spectra and community detection

Stephan¹,

Zhu²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…This is the model considered in [70] for r = 2, and in [12] for general r ≥ 2. In this case, the eigenvalues of…”

Section: Hypergraph Stochastic Block Modelmentioning

confidence: 99%

Section: Organization Of the Papermentioning

confidence: 99%

“…where φ i , φ j are eigenvectors of Q, ∂h is defined in (65), and h φi,t is defined in (70). Recall the definition of χ i in (16).…”

Section: Structure Of Near Eigenvectorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Sparse random hypergraphs: Non-backtracking spectra and community detection

Stephan¹,

Zhu²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Weak consistency for HSBMs was studied in [14,15,24,25,31]. For detection of the HSBM, the authors of [7] proposed a conjecture that the phase transition occurs in the regime of constant expected degrees, and the positive part of the conjecture for the binary case was solved in [49].…”

Section: Introductionmentioning

confidence: 99%

Partial recovery and weak consistency in the non-uniform hypergraph Stochastic Block Model

Dumitriu¹,

Wang²,

Zhu³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We consider the community detection problem in sparse random hypergraphs under the nonuniform 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 γ fraction of the vertices classified correctly, where γ ∈ (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 [24] 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.

show abstract