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A Hilbert Space Approach to Effective Resistance Metric
Journal article   Peer reviewed

A Hilbert Space Approach to Effective Resistance Metric

Palle Jorgensen and Erin Pearse
Complex Analysis and Operator Theory, Vol.4(4), pp.975-1013
11/2010
DOI: 10.1007/s11785-009-0041-1

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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 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” 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 ≠  R W in terms of Neumann versus 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, and is related to R F by a probabilistic argument.
 Harmonic Analysis Mathematics Markov process Resistance forms 05C75 Graph energy Primary 05C50 31C20 47B32 Unbounded linear operators Graph Laplacian Operator Theory Electrical resistance network 47B25 Dirichlet form Mathematics, general Hilbert space Boundary theory Trees 60J10 Random walk Effective resistance Secondary 31C35 Transience 82C41 Orthogonality Weighted graph Spectral graph theory Boundary representation 47B39 46E22 Analysis Discrete potential theory Reproducing kernels Martin boundary

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