The hitting and cover times of Metropolis walks

Yoshiaki Nonaka, Hirotaka Ono, Kunihiko Sadakane, Masafumi Yamashita

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Abstract

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.

Original languageEnglish
Pages (from-to)1889-1894
Number of pages6
JournalTheoretical Computer Science
Volume411
Issue number16-18
DOIs
Publication statusPublished - Mar 28 2010

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All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Nonaka, Y., Ono, H., Sadakane, K., & Yamashita, M. (2010). The hitting and cover times of Metropolis walks. Theoretical Computer Science, 411(16-18), 1889-1894. https://doi.org/10.1016/j.tcs.2010.01.032