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A Hitting Time Formula for the Discrete Green's Function

Part of: Graph theory

Published online by Cambridge University Press:  29 June 2015

ANDREW BEVERIDGE
ANDREW BEVERIDGE*
Affiliation:
Department of Mathematics, Statistics and Computer Science, Macalester College, 1600 Grand Avenue, St Paul, MN 55105, USA (e-mail: abeverid@macalester.edu)
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Abstract

The discrete Green's function (without boundary) $\mathbb{G}$ is a pseudo-inverse of the combinatorial Laplace operator of a graph G = (V, E). We reveal the intimate connection between Green's function and the theory of exact stopping rules for random walks on graphs. We give an elementary formula for Green's function in terms of state-to-state hitting times of the underlying graph. Namely, $\mathbb{G}(i,j) = \pi_j \bigl( H(\pi,j) - H(i,j) \bigr),$ where πi is the stationary distribution at vertex i, H(i, j) is the expected hitting time for a random walk starting from vertex i to first reach vertex j, and H(π, j) = ∑k∈V πk H(k, j). This formula also holds for the digraph Laplace operator.

The most important characteristics of a stopping rule are its exit frequencies, which are the expected number of exits of a given vertex before the rule halts the walk. We show that Green's function is, in fact, a matrix of exit frequencies plus a rank one matrix. In the undirected case, we derive spectral formulas for Green's function and for some mixing measures arising from stopping rules. Finally, we further explore the exit frequency matrix point of view, and discuss a natural generalization of Green's function for any distribution τ defined on the vertex set of the graph.

MSC classification

Primary: 05C81: Random walks on graphs

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 3 , May 2016 , pp. 362 - 379
Copyright
Copyright © Cambridge University Press 2015 

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