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  • Cited by 3
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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    MONTENEGRO, RAVI 2014. Intersection Conductance and Canonical Alternating Paths: Methods for General Finite Markov Chains. Combinatorics, Probability and Computing, Vol. 23, Issue. 04, p. 585.

    Montenegro, Ravi 2009. The simple random walk and max-degree walk on a directed graph. Random Structures and Algorithms, Vol. 34, Issue. 3, p. 395.

    Wei, Fang Qian, Weining Wang, Chen and Zhou, Aoying 2009. Detecting Overlapping Community Structures in Networks. World Wide Web, Vol. 12, Issue. 2, p. 235.

  • Combinatorics, Probability and Computing, Volume 15, Issue 4
  • July 2006, pp. 541-570

Blocking Conductance and Mixing in Random Walks

  • R. KANNAN (a1), L. LOVÁSZ (a2) and R. MONTENEGRO (a3)
  • DOI:
  • Published online: 07 June 2006

The notion of conductance introduced by Jerrum and Sinclair [8] has been widely used to prove rapid mixing of Markov chains. Here we introduce a bound that extends this in two directions. First, instead of measuring the conductance of the worst subset of states, we bound the mixing time by a formula that can be thought of as a weighted average of the Jerrum–Sinclair bound (where the average is taken over subsets of states with different sizes). Furthermore, instead of just the conductance, which in graph theory terms measures edge expansion, we also take into account node expansion. Our bound is related to the logarithmic Sobolev inequalities, but it appears to be more flexible and easier to compute.

In the case of random walks in convex bodies, we show that this new bound is better than the known bounds for the worst case. This saves a factor of $O(n)$ in the mixing time bound, which is incurred in all proofs as a ‘penalty’ for a ‘bad start’. We show that in a convex body in $\mathbb{R}^n$, with diameter $D$, random walk with steps in a ball with radius $\delta$ mixes in $O^*(nD^2/\delta^2)$ time (if idle steps at the boundary are not counted). This gives an $O^*(n^3)$ sampling algorithm after appropriate preprocessing, improving the previous bound of $O^*(n^4)$.

The application of the general conductance bound in the geometric setting depends on an improved isoperimetric inequality for convex bodies.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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