A Hilbert Space Approach to Effective Resistance Metric

作者：Jorgensen Palle E T; Pearse Erin Peter James^*

来源：Complex Analysis and Operator Theory, 2010, 4(4): 975-1013.

DOI：10.1007/s11785-009-0041-1

• Dirichlet form
• Graph energy
• Discrete potential theory
• Graph Laplacian
• Weighted graph
• Trees
• Spectral graph theory
• Electrical resistance network
• Effective resistance
• Resistance forms
• Markov process
• Random walk
• Transience
• Martin boundary
• Boundary theory
• Boundary representation
• Harmonic analysis
• Hilbert space
• Orthogonality
• Unbounded linear operators
• Reproducing kernels

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 E produces a Hilbert space structure (which we call the energy space H(epsilon)) on the space of functions of finite energy. We use the reproducing kernel {nu(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 nu(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) not equal R(W) in terms of Neumann versus Dirichlet boundary conditions. We show that the metric space (G, R(F)) embeds isometrically into H(epsilon), and the metric space (G, R(W)) embeds isometrically into the closure of the space of finitely supported functions; a subspace of H(epsilon). 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 E to finite subnetworks, and is related to R(F) by a probabilistic argument.

• 出版日期2010