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An Extension of Foster’s Network Theorem

. Combinatorics, Probability and Computing 3 (3): 421-–427 (September 1994)

Abstract

Consider an electrical network on n nodes with resistors rij between nodes i and j. Let Rij denote the effective resistance between the nodes. Then Foster’s Theorem 5 asserts thatwhere i ∼ j denotes i and j are connected by a finite rij. In 10 this theorem is proved by making use of random walks. The classical connection between electrical networks and reversible random walks implies a corresponding statement for reversible Markov chains. In this paper we prove an elementary identity for ergodic Markov chains, and show that this yields Foster's theorem when the chain is time-reversible.We also prove a generalization of a resistive inverse identity. This identity was known for resistive networks, but we prove a more general identity for ergodic Markov chains. We show that time-reversibility, once again, yields the known identity. Among other results, this identity also yields an alternative characterization of reversibility of Markov chains (see Remarks 1 and 2 below). This characterization, when interpreted in terms of electrical currents, implies the reciprocity theorem in single-source resistive networks, thus allowing us to establish the equivalence of reversibility in Markov chains and reciprocity in electrical networks.

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