An Extension of Foster's Network Theorem

Abstract

Consider an electrical network onnnodes with resistorsrijbetween nodesiandj. LetRijdenote theeffective resistancebetween the nodes. Then Foster's Theorem [5] asserts thatwherei∼jdenotesiandjare connected by a finiterij. 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 aresistive inverseidentity. 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 thereciprocity theoremin single-source resistive networks, thus allowing us to establish the equivalence ofreversibilityin Markov chains andreciprocityin electrical networks.