发表机构
Chennai Mathematical Institute（钦奈数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究电阻拉普拉斯矩阵的谱性质，建立其结构理论，推导规范分解等性质，构建基于电阻的图划分目标，实验验证其在图划分、聚类与异常检测中的有效性。
AI 中文摘要
电阻拉普拉斯矩阵是与有效电阻度量相关联的图矩阵，是经典图拉普拉斯矩阵的全局对应物。尽管它继承了普通拉普拉斯矩阵的若干基本性质，包括与Fiedler定理类似的连通图划分定理，但其内在谱结构在很大程度上仍未被探究。本文中，我们建立了电阻拉普拉斯矩阵的结构理论，推导了一种规范分解，将其内在分量、平均分量和偏差分量分离开来，从而揭示了由有效电阻诱导的全局几何与普通拉普拉斯矩阵编码的局部几何的差异。基于该分解，我们确定了相关偏差算子的若干结构与谱性质，得到了最大特征值及其对应特征空间的变分刻画，并将电阻拉普拉斯矩阵表示为拉普拉斯坐标形式，阐明了两种算子特征空间之间的关系。最后，我们构建了基于电阻的图划分目标，其谱松弛形式可得到电阻拉普拉斯矩阵的主导特征向量，为连通划分定理提供了变分解释。在合成数据集与真实数据集上的实验结果表明，所提框架在图划分、数据聚类和探索性异常检测中具有有效性。
英文摘要
The resistance Laplacian is a graph matrix associated with the effective resistance metric and provides a global counterpart of the classical graph Laplacian. Although it inherits several fundamental properties of the ordinary Laplacian, including a connected graph partitioning theorem analogous to that of Fiedler, its intrinsic spectral structure has remained largely unexplored. In this paper, we develop a structural theory of the resistance Laplacian. We derive a canonical decomposition that separates its intrinsic, average, and deviation components, thereby revealing how the global geometry induced by effective resistance differs from the local geometry encoded by the ordinary Laplacian. Building upon this decomposition, we establish several structural and spectral properties of the associated deviation operator, obtain variational characterizations of the largest eigenvalue and its corresponding eigenspace, and express the resistance Laplacian in Laplacian coordinates, thereby elucidating the relationship between the eigenspaces of the two operators. Finally, we formulate resistance-based graph partitioning objectives whose spectral relaxations recover the dominant eigenvector of the resistance Laplacian, providing a variational interpretation of the connected partition theorem. Experimental results on synthetic and real-life datasets demonstrate the effectiveness of the proposed framework for graph partitioning, data clustering, and exploratory anomaly detection.
