随机点积图中度-α拉普拉斯算子的谱嵌入

中文摘要 AI 辅助

网络数据的谱聚类方法通常基于邻接矩阵、对称拉普拉斯等少数矩阵表示。本文研究一类连续的度归一化谱嵌入，涵盖上述常用选择作为特例。在随机点积图模型下，本文为该嵌入族建立了行级中心极限定理，明确描述了度归一化如何影响总体几何与嵌入节点的局部不确定性。利用该极限分布，本文通过投影高斯贝叶斯误差诊断，在双群落随机块模型中比较不同归一化方式。结果表明，不存在统一最优的归一化方式，最优选择取决于网络密度、群落不平衡度及块概率结构：通常在低密度或更不平衡场景中，更强的归一化更受青睐。这些结果为理解何时及为何替代归一化可改进谱聚类提供了统一的分布视角。

英文摘要

Spectral clustering methods for network data are commonly based on a few matrix representations, such as the adjacency matrix and the symmetric Laplacian. We study a continuum of degree-normalized spectral embeddings that includes these commonly used choices as special cases. Under a random dot product graph model, we establish a row-wise central limit theorem for this family of embeddings. The result provides an explicit description of how degree normalization affects both population geometry and the local uncertainty of embedded nodes. We use the limiting distributions to compare different normalizations in two-community stochastic block models through a projected-Gaussian Bayes-error diagnostic. These comparisons show that no single normalization is uniformly preferred. Instead, the favored normalization depends on network density, community imbalance, and block-probability structure. Typically, stronger normalization is favored in lower-density or more imbalanced settings. These results provide a unified distributional understanding of when and why alternative normalizations may improve spectral clustering.

发表机构

• University of Hong Kong（香港大学）
• University of Arizona（亚利桑那大学）

机构由 AI 辅助整理，请以论文原文为准。