On the Commute Time of Random Walks on Graphs
Palacios, Jose Luis
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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]
This article has been published in the November 1995 issue of Brazilian Journal of Probability and Statistics.