The hitting and cover times of Metropolis walks

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 = { 1 d u min { d u π v d v π u , 1 } if v ∈ N ( u ) , 1 − ∑ 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.