On the Commute Time of Random Walks on Graphs

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Authors
Northshield, Sam
Palacios, Jose Luis
Issue Date
1995
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Article
Language
en_US
Keywords
commute time , cover time , escape probability , lollipop graph
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Abstract
Given a simple random walk on an undirected connected graph, the commute time of the vertices x and y is defined as C(x,y) = ExTy+EyTx. We give a new proof, based on the optional sampling theorem for martingales, of the formula C(x,y) = 1/(Π(y)e(y,x)) in terms of the escape probability e(y,x ) (the probability that once the random walk leaves x, it hits y before it returns to x) and the stationary distribution Π(·). We use this formula for C(x,y) to show that the maximum commute time among all barbell-type graphs in N vertices is attained for the lollipop graph and its value is O[(4N3)/27]
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This article has been published in the November 1995 issue of Brazilian Journal of Probability and Statistics.
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Northshield, S. & Palacios, J.L. (1995). On the Commute Time of Random Walks on Graphs. Brazilian Journal of Probability and Statistics, 9(2).
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Brazilian Journal of Probability and Statistics
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