A Hilbert space approach to effective resistance metric

TL;DR

This paper employs a Hilbert space framework to analyze effective resistance metrics in electrical networks, clarifying the differences between free and wired resistances through reproducing kernels and boundary conditions.

Contribution

It introduces a Hilbert space approach to characterize and compare free and wired effective resistance metrics using reproducing kernels and boundary condition interpretations.

Findings

R^F and R^W are expressed as norms of operators.

R^F embeds isometrically into the energy space.

R^{tr} coincides with R^F in the limit.

Abstract

A resistance network is a connected graph $(G,c)$. The conductance function $c_{xy}$ weights the edges, which are then interpreted as conductors of possibly varying strengths. The Dirichlet energy form $\mathcal E$ produces a Hilbert space structure (which we call the energy space ${\mathcal H}_{\mathcal E}$) on the space of functions of finite energy. We use the reproducing kernel $\{v_x\}$ constructed in \cite{DGG} to analyze the effective resistance $R$, which is a natural metric for such a network. It is known that when $(G,c)$ supports nonconstant harmonic functions of finite energy, the effective resistance metric is not unique. The two most natural choices for $R(x,y)$ are the ``free resistance'' $R^F$, and the ``wired resistance'' $R^W$. We define $R^F$ and $R^W$ in terms of the functions $v_x$ (and certain projections of them). This provides a way to express $R^F$ and $R^W$ as norms of certain operators, and explain $R^F \neq R^W$ in terms of Neumann vs. Dirichlet boundary conditions. We show that the metric space $(G,R^F)$ embeds isometrically into ${\mathcal H}_{\mathcal E}$, and the metric space $(G,R^W)$ embeds isometrically into the closure of the space of finitely supported functions; a subspace of ${\mathcal H}_{\mathcal E}$. Typically, $R^F$ and $R^W$ are computed as limits of restrictions to finite subnetworks. A third formulation $R^{tr}$ is given in terms of the trace of the Dirichlet form $\mathcal E$ to finite subnetworks. A probabilistic approach shows that in the limit, $R^{tr}$ coincides with $R^F$. This suggests a comparison between the probabilistic interpretations of $R^F$ vs. $R^W$.