[Paper Review] Partial recovery and weak consistency in the non-uniform hypergraph Stochastic Block Model

Ioana Dumitriu, Hai‐Xiao Wang+1 more|arXiv (Cornell University)|Dec 22, 2021

Complex Network Analysis Techniques4 citations

TL;DR

This paper proposes a spectral algorithm for partial recovery and weak consistency in the non-uniform hypergraph stochastic block model (HSBM) with bounded expected degrees. By selecting high-signal hyperedges, regularizing the adjacency tensor, and iteratively correcting partitions using hyperedge information, the method achieves partial recovery with a fraction $\gamma \in (0.5,1)$ of vertices correctly classified, and weak consistency when the signal-to-noise ratio grows slowly with $n$. The theoretical analysis relies on concentration and regularization of sparse non-uniform hypergraph adjacency matrices.

ABSTRACT

We consider the community detection problem in sparse random hypergraphs under the non-uniform 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 $γ\in (0.5,1)$ 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 Ghoshdastidar and Dukkipati (2017) 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.

Motivation & Objective

• To address partial recovery and weak consistency in sparse non-uniform hypergraph stochastic block models (HSBM), where higher-order interactions are modeled through varying hyperedge types.
• To extend existing results in uniform HSBM to the more realistic non-uniform setting, where hyperedges of different sizes have distinct connection probabilities.
• To develop a spectral algorithm that maintains high accuracy even when expected degrees are bounded, improving on prior work in weak consistency regimes.
• To establish theoretical guarantees via concentration and regularization of adjacency tensors in sparse, non-uniform hypergraphs.

Proposed method

• Hyperedge selection: Identify and retain hyperedges of specific sizes that maximize the signal-to-noise ratio (SNR) in the induced sub-hypergraph.
• Spectral partitioning: Construct a regularized adjacency matrix from selected hyperedges and compute leading singular vectors to obtain an initial approximate partition.
• Correction and merging: Use the full adjacency tensor to refine the initial partition, reducing misclassification error through iterative correction.
• Regularization: Apply matrix regularization techniques to stabilize spectral decomposition under sparsity and non-uniformity.
• Concentration analysis: Prove that the adjacency tensor concentrates around its mean under bounded expected degrees, enabling theoretical guarantees.
• Use of Wedin’s $\sin\Theta$ theorem to bound perturbation in singular subspaces, ensuring robustness of the partition to noise.

Experimental results

Research questions

• RQ1 Can partial recovery be achieved in the non-uniform HSBM when expected degrees are bounded, even as the signal-to-noise ratio grows slowly?
• RQ2 Does the proposed spectral algorithm achieve weak consistency in the non-uniform HSBM under slowly growing SNR?
• RQ3 How can hyperedge selection be optimized to maximize SNR and improve clustering accuracy in higher-order networks?
• RQ4 What theoretical tools are needed to analyze concentration and regularization in sparse, non-uniform hypergraph adjacency tensors?
• RQ5 Can spectral methods be adapted to handle the complexity of non-uniform hypergraphs while maintaining strong error guarantees?

Key findings

• The proposed spectral algorithm achieves partial recovery with a fraction $\gamma \in (0.5,1)$ of vertices correctly classified, where $\gamma$ increases with the signal-to-noise ratio.
• When the signal-to-noise ratio grows slowly with $n$, the algorithm achieves weak consistency, meaning at most $o(n)$ vertices are misclassified with high probability.
• The algorithm’s performance is guaranteed via a novel concentration and regularization analysis of the adjacency tensor in sparse, non-uniform hypergraphs.
• Hyperedge selection based on SNR maximization significantly improves the signal strength and enables accurate spectral partitioning.
• The use of Wedin’s $\sin\Theta$ theorem ensures that perturbations in the singular subspace are bounded, enabling robust recovery under noise.
• The theoretical framework provides a foundation for analyzing spectral methods in non-uniform hypergraphs, which may be of independent interest beyond community detection.