TY - JOUR

T1 - The hitting and cover times of Metropolis walks

AU - Nonaka, Yoshiaki

AU - Ono, Hirotaka

AU - Sadakane, Kunihiko

AU - Yamashita, Masafumi

N1 - Funding Information:
This work is supported in part by the Asahi Glass Foundation and a Grant-in-Aid of the Ministry of Education, Science, Sports and Culture of Japan. Corresponding author at: Department of Informatics, Kyushu University, Fukuoka 812-8581, Japan. Tel.: +81 92 802 3637. E-mail addresses: nonaka@tcslab.csce.kyushu-u.ac.jp (Y. Nonaka), ono@inf.kyushu-u.ac.jp (H. Ono), sada@nii.ac.jp (K. Sadakane), mak@inf.kyushu-u.ac.jp (M. Yamashita).
Copyright:
Copyright 2010 Elsevier B.V., All rights reserved.

PY - 2010/3/28

Y1 - 2010/3/28

N2 - Given a finite graph G = (V, E) and a probability distribution π = (πv)v ∈ V on V, Metropolis walks, i.e., random walks on G building on the Metropolis-Hastings algorithm, obey a transition probability matrix P = (pu v)u, v ∈ V defined by, for any u, v ∈ V, pu v = {(frac(1, du) min {frac(du πv, dv πu), 1}, if v ∈ N (u),; 1 - under(∑, w ≠ u) pu w, if u = v,; 0, otherwise,) and are guaranteed to have π as the stationary distribution, where N (u) is the set of adjacent vertices of u ∈ V and du = | N (u) | is the degree of u. This paper shows that the hitting and the cover times of Metropolis walks are O (f n2) and O (f n2 log n), respectively, for any graph G of order n and any probability distribution π such that f = maxu, v ∈ V πu / πv < ∞. We also show that there are a graph G and a stationary distribution π such that any random walk on G realizing π attains Ω (f n2) hitting and Ω (f n2 log n) cover times. It follows that the hitting and the cover times of Metropolis walks are Θ (f n2) and Θ (f n2 log n), respectively.

AB - Given a finite graph G = (V, E) and a probability distribution π = (πv)v ∈ V on V, Metropolis walks, i.e., random walks on G building on the Metropolis-Hastings algorithm, obey a transition probability matrix P = (pu v)u, v ∈ V defined by, for any u, v ∈ V, pu v = {(frac(1, du) min {frac(du πv, dv πu), 1}, if v ∈ N (u),; 1 - under(∑, w ≠ u) pu w, if u = v,; 0, otherwise,) and are guaranteed to have π as the stationary distribution, where N (u) is the set of adjacent vertices of u ∈ V and du = | N (u) | is the degree of u. This paper shows that the hitting and the cover times of Metropolis walks are O (f n2) and O (f n2 log n), respectively, for any graph G of order n and any probability distribution π such that f = maxu, v ∈ V πu / πv < ∞. We also show that there are a graph G and a stationary distribution π such that any random walk on G realizing π attains Ω (f n2) hitting and Ω (f n2 log n) cover times. It follows that the hitting and the cover times of Metropolis walks are Θ (f n2) and Θ (f n2 log n), respectively.

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U2 - 10.1016/j.tcs.2010.01.032

DO - 10.1016/j.tcs.2010.01.032

M3 - Article

AN - SCOPUS:77949273917

VL - 411

SP - 1889

EP - 1894

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 16-18

ER -