基于Kron缩减和有效电阻的广义电阻几何
Generalized Resistance Geometry from Kron Reduction and Effective Resistance
AI总结：
本文基于Kron缩减和有效电阻理论，研究有向图的广义电阻几何，推导Fiedler-Bapat恒等式并定义电阻曲率与半径，提出广义电阻度量并证明其与严格负类型度量一致，统一了模型降阶、图方差和电阻嵌入理论。
摘要：
我们基于Kron缩减和有效电阻理论，为有向图发展了广义电阻几何，类比经典无向图理论。对于强连通有向图，我们证明了一个Fiedler--Bapat恒等式，将电阻矩阵与拉普拉斯矩阵通过对称伪逆联系起来。该恒等式为强连通有向设置提供了电阻曲率和电阻半径的规范定义。在强连通权衡平衡情况下，它还表明通过伪逆将无向拉普拉斯矩阵与有向拉普拉斯矩阵关联的操作与Kron缩减可交换。我们进一步引入了一类符号无向拉普拉斯矩阵，其中有效电阻定义了节点间的距离。我们称此距离为广义电阻度量，并证明其与严格负类型度量一致。在此框架内，我们研究了电阻曲率和电阻半径的分析与几何性质，刻画了最大图方差问题，并推广了电阻嵌入。这些结果使符号无向电阻几何与经典无符号无向理论并列，并提供了模型降阶、图方差和基于电阻的嵌入理论的统一视角。
英文摘要：
We develop a generalized resistance geometry for unsigned directed graphs and signed undirected graphs based on Kron reduction and effective resistance. For unsigned strongly connected weight-balanced directed graphs, we establish a generalized Fiedler--Bapat identity involving an associated signed undirected Laplacian constructed from the symmetrized pseudoinverse. We identify this Laplacian as the sum of the symmetrized directed Laplacian and a positive-semidefinite correction induced by graph asymmetry. This decomposition yields an effective-resistance comparison that extends a previous normality-based result, and we prove that the construction commutes with Kron reduction. For general unsigned strongly connected directed graphs satisfying a positivity condition, we define resistance curvature and resistance radius through a weight-balanced representation. Motivated by the associated signed undirected graphs, we introduce a class whose effective resistances form a metric and show that the resulting resistance matrices are precisely the strict negative type metric matrices. Within this framework, we characterize the unique solution of the maximum graph-variance problem and develop generalized resistive embeddings into Euclidean space.
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