Markov Chains and Electrical Networks

Random Paths and Cuts, Electrical Networks, and Reversible Markov Chains

SIAM Journal on Discrete Mathematics ◽

10.1137/0403026 ◽

1990 ◽

Vol 3 (3) ◽

pp. 311-319 ◽

Cited By ~ 13

Author(s):

Kenneth A. Berman ◽

Mokhtar H. Konsowa

Keyword(s):

Markov Chains ◽

Electrical Networks ◽

Reversible Markov Chains

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Markov Chains and Electrical Networks

Probability Theory - Universitext ◽

10.1007/978-3-030-56402-5_19 ◽

2020 ◽

pp. 461-492

Author(s):

Achim Klenke

Keyword(s):

Markov Chains ◽

Electrical Networks

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An Extension of Foster's Network Theorem

Combinatorics Probability Computing ◽

10.1017/s0963548300001309 ◽

1994 ◽

Vol 3 (3) ◽

pp. 421-427 ◽

Cited By ~ 19

Author(s):

Prasad Tetali

Keyword(s):

Markov Chains ◽

Random Walks ◽

Electrical Network ◽

Electrical Networks ◽

Time Reversibility ◽

Electrical Currents ◽

Resistive Networks ◽

Image Position ◽

Elementary Identity

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.

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