Random walks and the effective resistance of networks
- 1.4k Downloads
In this article we present an interpretation ofeffective resistance in electrical networks in terms of random walks on underlying graphs. Using this characterization we provide simple and elegant proofs for some known results in random walks and electrical networks. We also interpret the Reciprocity theorem of electrical networks in terms of traversals in random walks. The byproducts are (a) precise version of thetriangle inequality for effective resistances, and (b) an exact formula for the expectedone-way transit time between vertices.
Key WordsElectrical networks electrical impedance random walks Markov chains
Unable to display preview. Download preview PDF.
- 1.Chandra, A. K., Raghavan, P., Ruzzo, W. L., Smolensky, R., and Tiwari, P. (1989). The electrical resistance of a graph captures its commute and cover times,Proceedings of the 21st Annual ACM Symposium on Theory of Computing, May. ACM Press, Seattle.Google Scholar
- 2.Doyle, P. G., and Snell, J. L. (1984).Random Walks and Electrical Networks, The Mathematical Association of America, Washington D.C.Google Scholar
- 3.Foster, R. M. (1949). The Average Impedance of an Electrical Network,Contributions to Applied Mechanics (Reissner Anniversary Volume), Ann Arbor, pp. 333–340, Edwards Brothers, Inc.Google Scholar
- 4.Göbel, F., and Jagers, A. A. (1974). Random walks on graphs,Stoch. Processes Appl. 2, 311–336.Google Scholar
- 5.Hayt, W. H., and Kemmerly, J. E. (1978).Engineering Circuit Analysis, 3rd ed., McGraw-Hill, New York.Google Scholar
- 6.Kemeny, J. G., Snell, J. J., and Knapp, A. W. (1966).Denumerable Markov Chains, Van Nostrand, New York.Google Scholar