Strong Consistency of Spectral Clustering for the Sparse Degree-Corrected Hypergraph Stochastic Block Model

TL;DR

This work establishes strong consistency for spectral clustering under the degree-corrected hypergraph stochastic block model (DCHSBM) in the sparse regime where the maximum expected hyperdegree can be as small as $Ω(\log n)$. It introduces a novel leave-one-out based entrywise perturbation bound for eigenvectors of the non-uniform hypergraph adjacency, enabling exact recovery by simple spectral procedures without preprocessing or refinement. The main result provides explicit conditions linking sparsity, eigen-gap $|\lambda_K|$, and degree heterogeneity $\gamma$ under which both a $k$-means on row-normalized eigenvectors and a thresholding rule recover the true communities with high probability; corollaries extend to $m$-uniform and non-uniform DCHPPMs. Overall, the paper extends sharp eigenvector perturbation theory to non-uniform hypergraphs and demonstrates that basic spectral clustering can achieve exact recovery in a broad, sparse hypergraph setting with heterogeneous hyperdegrees.

Abstract

We prove strong consistency of spectral clustering under the degree-corrected hypergraph stochastic block model in the sparse regime where the maximum expected hyperdegree is as small as $Ω(\log n)$ with $n$ denoting the number of nodes. We show that the basic spectral clustering without preprocessing or postprocessing is strongly consistent in an even wider range of the model parameters, in contrast to previous studies that either trim high-degree nodes or perform local refinement. At the heart of our analysis is the entry-wise eigenvector perturbation bound derived by the leave-one-out technique. To the best of our knowledge, this is the first entry-wise error bound for degree-corrected hypergraph models, resulting in the strong consistency for clustering non-uniform hypergraphs with heterogeneous hyperdegrees.

Figures (1)

• Figure 1: Clustering result for Algorithms 1 and 2 in the DCHPPM. Each value is the average over 10 independent trials. The first and second rows show the results corresponding to $\theta=\theta^{(1)}$ and $\theta^{(2)}$, respectively. The first and second columns are the results corresponding to $K=2$ and 3, respectively.

Theorems & Definitions (22)

• Definition 1: Degree-corrected hypergraph stochastic block model
• Definition 2: Degree-corrected hypergraph planted partition model
• Lemma 1: The eigen-structure of $P$
• Theorem 1
• Corollary 1
• Theorem 2
• Lemma 2
• Remark 1
• Theorem 3
• Corollary 2