### Abstract

Given a finite graph G = (V, E) and a probability distribution π = (π_{v})_{v ∈ V} on V, Metropolis walks, i.e., random walks on G building on the Metropolis-Hastings algorithm, obey a transition probability matrix P = (p_{u v})_{u, v ∈ V} defined by, for any u, v ∈ V, p_{u v} = {(frac(1, d_{u}) min {frac(d_{u} π_{v}, d_{v} π_{u}), 1}, if v ∈ N (u),; 1 - under(∑, w ≠ u) p_{u 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 d_{u} = | N (u) | is the degree of u. This paper shows that the hitting and the cover times of Metropolis walks are O (f n^{2}) and O (f n^{2} log n), respectively, for any graph G of order n and any probability distribution π such that f = max_{u, v ∈ V} π_{u} / π_{v} < ∞. 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 n^{2}) hitting and Ω (f n^{2} log n) cover times. It follows that the hitting and the cover times of Metropolis walks are Θ (f n^{2}) and Θ (f n^{2} log n), respectively.

Original language | English |
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Pages (from-to) | 1889-1894 |

Number of pages | 6 |

Journal | Theoretical Computer Science |

Volume | 411 |

Issue number | 16-18 |

DOIs | |

Publication status | Published - Mar 28 2010 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Theoretical Computer Science*,

*411*(16-18), 1889-1894. https://doi.org/10.1016/j.tcs.2010.01.032