A Hilbert Space Approach to Effective Resistance Metric

By Palle E. T. Jorgensen, Erin P. J. Pearse. 2009

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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 $${\mathcal E}$$ 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 {vx} constructed in a previous work 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 versus Dirichlet boundary conditions. We show that the metric space (G, RF) embeds isometrically into $${{\mathcal H}_{\mathcal E}}$$, and the metric space (G, RW) embeds isometrically into the closure of the space of finitely supported functions; a subspace of $${{\mathcal H}_{\mathcal E}}$$. 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 $${\mathcal E}$$ to finite subnetworks, and is related to RF by a probabilistic argument.

Subfields: Geometry and Topology, Materials Chemistry, Statistics and Probability

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