[Paper Review] Community Detection in Hypergraphs, Spiked Tensor Models, and Sum-of-Squares

Chiheon Kim, Afonso S. Bandeira+1 more|arXiv (Cornell University)|May 8, 2017

Complex Network Analysis Techniques6 citations

TL;DR

This paper studies community detection in hypergraphs under a stochastic block model, linking it to a spiked tensor model distinct from standard tensor PCA. Using the Sum-of-Squares (SoS) hierarchy, it reveals a significant computational-statistical gap: while exact recovery is information-theoretically possible under mild conditions, SoS algorithms fail well before this threshold, highlighting a fundamental gap in computational efficiency for hypergraph models compared to classical matrix models.

ABSTRACT

We study the problem of community detection in hypergraphs under a stochastic block model. Similarly to how the stochastic block model in graphs suggests studying spiked random matrices, our model motivates investigating statistical and computational limits of exact recovery in a certain spiked tensor model. In contrast with the matrix case, the spiked model naturally arising from community detection in hypergraphs is different from the one arising in the so-called tensor Principal Component Analysis model. We investigate the effectiveness of algorithms in the Sum-of-Squares hierarchy on these models. Interestingly, our results suggest that these two apparently similar models exhibit significantly different computational to statistical gaps.

Motivation & Objective

• To understand the statistical and computational limits of exact community recovery in hypergraphs under a stochastic block model.
• To investigate how the Sum-of-Squares (SoS) hierarchy performs on a hypergraph-inspired spiked tensor model, distinct from standard tensor PCA.
• To compare the computational thresholds of SoS algorithms with information-theoretic limits in hypergraph community detection.
• To identify and quantify the computational-statistical gap in hypergraph models, contrasting them with classical matrix-based models.

Proposed method

• Formulates a hypergraph stochastic block model (HSBM) with 4-uniform hyperedges and community-dependent edge probabilities.
• Derives a Gaussian analogue of the HSBM, leading to a spiked tensor model where the signal is a rank-1 4-tensor formed from the community vector.
• Applies the Sum-of-Squares (SoS) hierarchy to analyze the feasibility of exact recovery via pseudo-expectations of degree 4.
• Uses block-diagonalization of the moment matrix and concentration bounds to analyze the spectral norm of the noise component.
• Constructs a valid pseudo-expectation functional that satisfies constraints of degree 4, enabling analysis of the SoS relaxation.
• Employs tail bounds on Gaussian chaos and spectral norms to show that the SoS matrix remains positive semidefinite under certain noise levels.

Experimental results

Research questions

• RQ1 What is the computational threshold for exact community recovery in hypergraphs under the stochastic block model?
• RQ2 How does the performance of the Sum-of-Squares hierarchy compare to information-theoretic limits in hypergraph community detection?
• RQ3 Why does the spiked tensor model arising from hypergraphs differ fundamentally from the standard tensor PCA model in terms of computational complexity?
• RQ4 Can the SoS hierarchy detect communities in hypergraphs when the signal-to-noise ratio is below the information-theoretic threshold?
• RQ5 What is the precise relationship between the noise level and the failure of SoS algorithms in this hypergraph model?

Key findings

• The SoS hierarchy fails to recover communities when the noise level is below $ O(1/ninom{n}{4}^{-1/2}) $, indicating a computational barrier.
• The information-theoretic threshold for exact recovery is lower than the SoS threshold, revealing a significant computational-statistical gap in hypergraph models.
• The spiked tensor model derived from hypergraph SBM is structurally different from standard tensor PCA, leading to distinct algorithmic behavior.
• With high probability, the moment matrix $ X_{ ho} $ remains positive semidefinite under noise levels $ ho = o_n(1/ninom{n}{4}^{-1/2}) $, supporting the existence of a valid pseudo-expectation.
• The spectral norm of the noise component in the SoS matrix is bounded by $ O(ninom{n}{4}^{1/2}) $, which is critical for establishing the failure threshold.
• The SoS relaxation of degree 4 fails to detect the community structure even when the signal is strong enough for information-theoretic recovery, demonstrating a gap in computational power.