[论文解读] 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 applications参考文献 48被引用 55

一句话总结

该论文提出 Tensor-SCORE，一种使用正则化 Tucker 分解(reg-HOOI)在邻接张量上进行的超图社区检测方法，随后通过 SCORE 归一化处理度异质性，在度纠正的超图 SBM（hDCBM）下具有理论保障。

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.

研究动机与目标

• 在更高阶（超图）网络中引入社区检测的动机，超越成对交互。
• 开发直接的超图方法，避免投影图方法导致的信息损失。
• 引入 reg-HOOI，以可靠估计稀疏超图邻接张量的 Tucker 分解。
• 将 SCORE 归一化推广到超图，以消除度异质性并实现准确聚类。
• 在 hDCBM 下提供一致性理论保障并分析所提算法的收敛性。

提出的方法

• 用邻接张量表示超图并进行 Tucker 分解。
• 引入正则化 HOOI（reg-HOOI），以控制行范数并提高稀疏张量的收敛性。
• 从分解中获得一个因子矩阵，作为超图的“特征向量”。
• 应用 SCORE 型的行归一化以消除度异质性影响。
• 对标准化的分数向量进行 k-means 聚类以恢复社区。

实验结果

研究问题

• RQ1 在超图邻接张量上进行张量基分解是否能在不投影到图的情况下得到准确的社区结构？
• RQ2 相较于原生 HOOI 或 HOSVD，reg-HOOI 是否在稀疏超图上提供更可靠的收敛性和更好的误差率？
• RQ3 是否可以有效将 SCORE 归一化扩展到超图以处理度异质性并实现精确社区恢复？
• RQ4 在所提出的 hDCBM 模型下，Tensor-SCORE 的理论保障有哪些，包括稀疏性和度异质性情形？

主要发现

• Tensor-SCORE 在度纠正超图块模型（hDCBM）下，在不同稀疏性和度异质性范围内实现一致的社区检测。
• 正则化 HOOI 提高了在稀疏超图邻接张量上 Tucker 因子矩阵的收敛性和估计精度。
• SCORE 归一化在超图设定中消除了度异质性的影响，使分解矩阵的行能够有效聚类。
• reg-HOOI 与 SCORE 的组合在稀疏情形下的表现优于图投影和基于 HOSVD 的方法。
• 初始化策略（对角线移除的 HOSVD 和随机图投影）有助于方法的实际表现和理论保障。