[Paper Review] Community Detection for Hypergraph Networks via Regularized Tensor Power Iteration

Zheng Tracy Ke, Feng Shi+1 more|arXiv (Cornell University)|Sep 14, 2019

Tensor decomposition and applications48 references55 citations

TL;DR

The paper proposes Tensor-SCORE, a hypergraph community detection method using regularized Tucker decomposition (reg-HOOI) on the adjacency tensor, followed by SCORE normalization to handle degree heterogeneity, with theoretical guarantees under a degree-corrected hypergraph SBM (hDCBM).

ABSTRACT

To date, social network analysis has been largely focused on pairwise interactions. The study of higher-order interactions, via a hypergraph network, brings in new insights. We study community detection in a hypergraph network. A popular approach is to project the hypergraph to a graph and then apply community detection methods for graph networks, but we show that this approach may cause unwanted information loss. We propose a new method for community detection that operates directly on the hypergraph. At the heart of our method is a regularized higher-order orthogonal iteration (reg-HOOI) algorithm that computes an approximate low-rank decomposition of the network adjacency tensor. Compared with existing tensor decomposition methods such as HOSVD and vanilla HOOI, reg-HOOI yields better performance, especially when the hypergraph is sparse. Given the output of tensor decomposition, we then generalize the community detection method SCORE (Jin, 2015) from graph networks to hypergraph networks. We call our new method Tensor-SCORE. In theory, we introduce a degree-corrected block model for hypergraphs (hDCBM), and show that Tensor-SCORE yields consistent community detection for a wide range of network sparsity and degree heterogeneity. As a byproduct, we derive the rates of convergence on estimating the principal subspace by reg-HOOI, with different initializations, including the two new initialization methods we propose, a diagonal-removed HOSVD and a randomized graph projection. We apply our method to several real hypergraph networks which yields encouraging results. It suggests that exploring higher-order interactions provides additional information not seen in graph representations.

Motivation & Objective

• Motivate community detection in higher-order (hypergraph) networks beyond pairwise interactions.
• Develop a direct hypergraph method that avoids information loss from projected-graph approaches.
• Introduce reg-HOOI to reliably estimate the Tucker decomposition of sparse hypergraph adjacency tensors.
• Generalize SCORE normalization to hypergraphs to remove degree heterogeneity and enable accurate clustering.
• Provide theoretical guarantees for consistency under hDCBM and analyze convergence of the proposed algorithm.

Proposed method

• Represent the hypergraph with an adjacency tensor and perform a Tucker decomposition.
• Introduce regularized HOOI (reg-HOOI) to control row-wise norms and improve convergence on sparse tensors.
• Obtain a factor matrix from the decomposition that serves as hypergraph 'eigenvectors'.
• Apply SCORE-type row-normalization to remove degree heterogeneity effects.
• Cluster nodes with k-means on the normalized score vectors to recover communities.

Experimental results

Research questions

• RQ1 Can tensor-based decomposition on hypergraph adjacency tensors yield accurate community structure without projecting to graphs?
• RQ2 Does reg-HOOI provide reliable convergence and better error rates for sparse hypergraphs compared to vanilla HOOI or HOSVD?
• RQ3 Can SCORE normalization be effectively extended to hypergraphs to handle degree heterogeneity and enable exact community recovery?
• RQ4 What are the theoretical guarantees for Tensor-SCORE under the proposed hDCBM model, including sparsity and degree-heterogeneity regimes?

Key findings

• Tensor-SCORE yields consistent community detection under a degree-corrected hypergraph block model (hDCBM) across a range of sparsity and degree heterogeneity.
• Regularized HOOI improves convergence and estimation accuracy of the Tucker factor matrix on sparse hypergraph adjacency tensors.
• SCORE normalization removes degree heterogeneity effects in the hypergraph setting, enabling effective clustering of rows of the factor matrix.
• The combination of reg-HOOI and SCORE achieves better performance than graph-projection and HOSVD-based methods in sparse regimes.
• Initialization strategies (diagonal-removed HOSVD and randomized graph projection) facilitate the method’s practical performance and theoretical guarantees.