%0 Journal Article %1 tetali94 %A Tetali, Prasad %C Cambridge, UK %D 1994 %I Cambridge University Press %J Combinatorics, Probability and Computing %K circuit effective.resistance foster.theorem graph.theory physics %N 3 %P 421-–427 %R 10.1017/S0963548300001309 %T An Extension of Foster’s Network Theorem %V 3 %X 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.

@article{tetali94, 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.}, added-at = {2017-01-24T08:27:47.000+0100}, address = {Cambridge, UK}, author = {Tetali, Prasad}, biburl = {https://www.bibsonomy.org/bibtex/2031be1afefa878faed2063492c01b5a2/ytyoun}, doi = {10.1017/S0963548300001309}, interhash = {4d66b6493d41ea8e14251e0291b24dc6}, intrahash = {031be1afefa878faed2063492c01b5a2}, journal = {Combinatorics, Probability and Computing}, keywords = {circuit effective.resistance foster.theorem graph.theory physics}, month = sep, number = 3, pages = {421-–427}, publisher = {Cambridge University Press}, timestamp = {2017-02-01T04:27:21.000+0100}, title = {An Extension of {Foster’s} Network Theorem}, volume = 3, year = 1994 }