Note: The hitting and cover times of Metropolis walks

  • Authors:

  • Yoshiaki Nonaka;Hirotaka Ono;Kunihiko Sadakane;Masafumi Yamashita

  • Affiliations:

  • Department of Informatics, Kyushu University, Fukuoka 812-8581, Japan;Department of Informatics, Kyushu University, Fukuoka 812-8581, Japan and Institute of Systems, Information Technologies and Nanotechnologies (ISIT), Fukuoka 814-0001, Japan;Principles of Informatics Research Division, National Institute of Informatics, Tokyo 101-8430, Japan;Department of Informatics, Kyushu University, Fukuoka 812-8581, Japan and Institute of Systems, Information Technologies and Nanotechnologies (ISIT), Fukuoka 814-0001, Japan

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2010

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Abstract

Given a finite graph G=(V,E) and a probability distribution @p=(@p"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={1d"umin{d"u@p"vd"v@p"u,1}if v@?N(u),1-@?wup"u"wif u=v,0otherwise , and are guaranteed to have @p 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(fn^2) and O(fn^2logn), respectively, for any graph G of order n and any probability distribution @p such that f=max"u","v"@?"V@p"u/@p"v