Generalized Resistance Geometry from Kron Reduction and Effective Resistance

Abstract

We develop a generalized resistance geometry based on Kron reduction and effective resistance for directed graphs, paralleling classical undirected graph theory. For strongly connected directed graphs, we prove a Fiedler--Bapat identity that links the resistance matrix and the Laplacian through the symmetrized pseudoinverse. This identity provides a canonical definition of the resistance curvature and resistance radius in the strongly connected directed setting. In the strongly connected weight-balanced case, it also implies that the operation of associating an undirected Laplacian with a directed Laplacian via the pseudoinverse of the symmetrized pseudoinverse commutes with Kron reduction. We further introduce a class of signed undirected Laplacians for which effective resistance defines a distance between nodes. We call this distance the generalized resistance metric and prove that it coincides with the class of strict negative type metrics. Within this framework, we investigate analytical and geometric properties of resistance curvature and resistance radius, characterize the maximum graph-variance problem, and generalize resistive embeddings. These results place signed undirected resistance geometry on a footing parallel to the classical unsigned undirected theory and provide a unified perspective on model reduction, graph variance, and resistance-based embedding.

Figures (6)

• Figure 1: The hierarchical relations among three classes of metric spaces: Negative type metric $\supsetneq$ Strict negative type metric $\supsetneq$ Resistance metric space. For a detailed explanation, see Devriendt2022graphFiedler2011_2.
• Figure 1: 1. Commutativity of two operations on an SCWB-directed Laplacian: taking the pseudoinverse of symmetrized pseudoinverse: $\mathcal{L} \to (\mathcal{L}^{\dagger}_{s} )^{\dagger}$ (specified by blue arrows) and Kron reduction: $\mathcal{L} \to \mathcal{L}/ \mathcal{L}[\alpha^{c}, \alpha^{c}]$ (specified by red arrows). 2. We can properly define the resistance curvature $\bm{p}$ and the resistance radius $\sigma^2$ in SC-directed setting. 3. We can observe the changes in $\bm{p}$ and $\sigma^2$ before and after Kron reduction and making the graph weight-balanced. The resistance radius for SCWB-directed graphs is non-increasing w.r.t. Kron reduction when it is viewed as a function on the node set.
• Figure 1: Examples of signed undirected graphs excluded from the Laplacian class \ref{['eq:consider_Laplacian_class']}. The effective resistance between nodes can be zero or negative, so in these cases the effective resistance is not a metric between nodes.
• Figure 2: An example of a signed undirected Laplacian $\mathcal{Q}$ in \ref{['eq:consider_Laplacian_class']} where some nodes with negative resistance curvature are included in the support of the maximum variance distribution.
• Figure 3: The positions of the center of gravity and the two vertices $\varphi(\bm{1}_i),\ \varphi(\bm{1}_j)$ and their relationship to the two angles, $\theta_{ij}$ ane $\phi_{ij}$.

Theorems & Definitions (24)

• Example 2.1
• Definition 2.2: reachability, Def.I I I.2 of Sugiyama2023graph
• Theorem 3.1: FBI for strongly connected weight-balanced directed graphs
• Proof 1
• Corollary 3.2
• Theorem 3.3
• Proof 2
• Example 3.4
• Definition 3.5
• Lemma 3.6