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The electrical resistance of a graph captures its commute and cover times
Abstract
View an n-vertex, m-edge undirected graph as an electrical network with unit resistors as edges. We extend known relations between random walks and electrical networks by showing that resistance in this network is intimately connected with the lengths of random walks on the graph. For example, the commute time between two vertices s and t (the expected length of a random walk from s to t and back) is precisely characterized by the effective resistance Rst between s and t: commute time = 2mRst. Additionally, the cover time (the expected length of a random walk visiting all vertices) is characterized by the maximum resistance R in the graph to within a factor of log n: mR ≤ cover time ≤ O (mR log n). For many graphs, the bounds on cover time obtained in this manner are better than those obtained from previous techniques such as the eigenvalues of the adjacency matrix. In particular, using this approach, we improve known bounds on cover times for various classes of graphs, including high-degree graphs, expanders, and multi-dimensional meshes. Moreover, resistance seems to provide an intuitively appealing and tractable approach to these problems.
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Cited by
Fortunato, S. (2010) · Physics Reports · doi:10.1016/j.physrep.2009.11.002 · Cited by 11,293
Fouss, F., Pirotte, A., Renders, J.-M., & Saerens, M. (2007) · IEEE Transactions on Knowledge and Data Engineering · doi:10.1109/tkde.2007.46 · Cited by 1,265
The electrical resistance of a graph captures its commute and cover times
Chandra, A. K., Raghavan, P., Ruzzo, W. L., Smolensky, R., & Tiwari, P. (1996) · Computational Complexity · doi:10.1007/bf01270385 · Cited by 328
Self-citationRandom walks and the effective resistance of networks
Tetali, P. (1991) · Journal of Theoretical Probability · doi:10.1007/bf01046996 · Cited by 310
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Cushman, S. A., McRae, B., Adriaensen, F., Beier, P., Shirley, M., & Zeller, K. (2013) · Key Topics in Conservation Biology 2 · doi:10.1002/9781118520178.ch21 · Cited by 179
Metadata
- Authors
- Chandra, A. K., Raghavan, P., Ruzzo, W. L., & Smolensky, R.
- Venue
- Proceedings of the twenty-first annual ACM symposium on Theory of computing - STOC '89
- Publisher
- ACM Press
- Published
- 1989
- Pages
- 574-586
- OpenAlex
- W1972636593
- Cited by
- 360
- References
- 0
- FWCI
- 6.35
- Type
- conference paper
- Language
- en
- Open access
- closed
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- Chandra, A. K., Raghavan, P., Ruzzo, W. L., & Smolensky, R. (1989). The electrical resistance of a graph captures its commute and cover times. Proceedings of the twenty-first annual ACM symposium on Theory of computing - STOC '89. https://doi.org/10.1145/73007.73062
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