Abstract
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.
Similar content being viewed by others
References
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.
Doyle, P. G., and Snell, J. L. (1984).Random Walks and Electrical Networks, The Mathematical Association of America, Washington D.C.
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.
Göbel, F., and Jagers, A. A. (1974). Random walks on graphs,Stoch. Processes Appl. 2, 311–336.
Hayt, W. H., and Kemmerly, J. E. (1978).Engineering Circuit Analysis, 3rd ed., McGraw-Hill, New York.
Kemeny, J. G., Snell, J. J., and Knapp, A. W. (1966).Denumerable Markov Chains, Van Nostrand, New York.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Tetali, P. Random walks and the effective resistance of networks. J Theor Probab 4, 101–109 (1991). https://doi.org/10.1007/BF01046996
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01046996