A Hilbert Space Approach to Effective Resistance Metric

A resistance network is a connected graph (G, c). The conductance function cxy weights the edges, which are then interpreted as conductors of possibly varying strengths. The Dirichlet energy form E produces a Hilbert space structure (which we call the energy space HE) on the space of functions of finite energy. We use the reproducing kernel {vx} constructed in [JP09b] 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” RF , and the “wired resistance” RW . We define RF and RW in terms of the functions vx (and certain projections of them). This provides a way to express RF and RW as norms of certain operators, and explain RF , RW in terms of Neumann vs. Dirichlet boundary conditions. We show that the metric space (G,RF) embeds isometrically into HE, and the metric space (G,RW ) embeds isometrically into the closure of the space of finitely supported functions; a subspace of HE. Typically, RF and RW are computed as limits of restrictions to finite subnetworks. A third formulation Rtr is given in terms of the trace of the Dirichlet form E to finite subnetworks. A probabilistic approach shows that in the limit, Rtr coincides with RF . This suggests a comparison between the probabilistic interpretations of RF vs. RW .