The Combinatorics of Effective Resistances and Resistive Inverses

Abstract

Let matrix (σ_ij) denote the edge conductances of an electrical network, so that there is a resistor ofr_ij=1/σ_ijohms between nodesiandj. This uniquely determines the matrix (R_ij) ofeffective resistances, defined such that if a potential of 1 V is applied across nodesiandj, a current of 1/R_ijwill flow. We call (σ_ij) theresistive inverseof (R_ij). One source of interest in the resistive inverse, arising in the design of on-line algorithms, is that it produces an efficient random walk if the walk must pay a cost ofr_ijfor traversing edge (i, j). Coppersmithet al.(1993,J. Assoc. Comput. Mach.40(3), 421–453) showed that the random walk that makes transitions according to (σ_ij) is more efficient—more “competitive”—than the random walk that makes transitions according to (R_ij).  Coppersmithet al.gave a simple but obscure four-step algorithm for computing the resistive inverse. We give a complete self-contained combinatorial explanation of this algorithm, including the classical theorems of Kirchhoff and Foster.