Eigenvalue Bounds on Convergence to Stationarity for Nonreversible Markov Chains, with an Application to the Exclusion Process

Fill, James Allen

doi:10.1214/aoap/1177005981

Cited by 272 publications

(205 citation statements)

References 15 publications

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“…Many applications of such comparison techniques are given in Diaconis and Stroock [7] and numerous other papers [8,10] in particular for card shuffling games. These comparison theorems are sometimes called "Poincaré" inequalities [7].…”

Section: Comparison Theoremsmentioning

confidence: 99%

“…Motivated by non-reversible Markov chains, Fill [10] derived bounds for the rate of convergence by using eigenvalues of certain Hermitian matrices associated with a directed graph, such as the sum and product of the transition probability matrix and its transpose. The Cheeger inequality for directed graphs provides methods for further bounding the rate of convergence.…”

Section: Introductionmentioning

confidence: 99%

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Laplacians and the Cheeger Inequality for Directed Graphs

2005

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We consider Laplacians for directed graphs and examine their eigenvalues. We introduce a notion of a circulation in a directed graph and its connection with the Rayleigh quotient. We then define a Cheeger constant and establish the Cheeger inequality for directed graphs. These relations can be used to deal with various problems that often arise in the study of non-reversible Markov chains including bounding the rate of convergence and deriving comparison theorems.

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Section: Comparison Theoremsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Laplacians and the Cheeger Inequality for Directed Graphs

2005

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“…When G is undirected, this definition is equivalent to the second-largest eigenvalue of P being at most λ in absolute value -see, e.g., [Mih89,Fil91].…”

Section: -Biased Sets and Generatorsmentioning

confidence: 99%

Randomness-Efficient Sampling Within NC 1

Healy

2006

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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We construct a randomness-efficient averaging sampler that is computable by uniform constantdepth circuits with parity gates (i.e., in uniform AC 0 [⊕]). Our sampler matches the parameters achieved by random walks on constant-degree expander graphs, allowing us to apply a variety expander-based techniques within NC 1 . For example, we obtain the following results:• Randomness-efficient error-reduction for uniform probabilistic NC 1 , TC 0 , AC 0 [⊕] and AC 0

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“…Many key properties of digraphs can then be bounded by the eigenvalues ofL and the degree of asymmetry. For instance, by accounting for the asymmetry of digraphs, we are able to obtain a tighter bound (than that of Fill's in [9] and Chung's in [6]) on (non-reversible) Markov chain mixing rate.…”

Section: Introductionmentioning

confidence: 99%

Random Walks on Digraphs, the Generalized Digraph Laplacian and the Degree of Asymmetry

Zhang

2010

Algorithms and Models for the Web-Graph

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Abstract. In this paper we extend and generalize the standard random walk theory (or spectral graph theory) on undirected graphs to digraphs. In particular, we introduce and define a (normalized) digraph Laplacian matrix, and prove that 1) its Moore-Penrose pseudo-inverse is the (discrete) Green's function of the digraph Laplacian matrix (as an operator on digraphs), and 2) it is the normalized fundamental matrix of the Markov chain governing random walks on digraphs. Using these results, we derive new formula for computing hitting and commute times in terms of the Moore-Penrose pseudo-inverse of the digraph Laplacian, or equivalently, the singular values and vectors of the digraph Laplacian. Furthermore, we show that the Cheeger constant defined in [6] is intrinsically a quantity associated with undirected graphs. This motivates us to introduce a metric -the largest singular value of ∆ := (L −L T )/2 -to quantify and measure the degree of asymmetry in a digraph. Using this measure, we establish several new results, such as a tighter bound (than that of Fill's in [9] and Chung's in [6]) on the Markov chain mixing rate, and a bound on the second smallest singular value ofL.

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Eigenvalue Bounds on Convergence to Stationarity for Nonreversible Markov Chains, with an Application to the Exclusion Process

Cited by 272 publications

References 15 publications

Laplacians and the Cheeger Inequality for Directed Graphs

Laplacians and the Cheeger Inequality for Directed Graphs

Randomness-Efficient Sampling Within NC 1

Random Walks on Digraphs, the Generalized Digraph Laplacian and the Degree of Asymmetry

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