Markov chain algorithms for planar lattice structures

Luby, Michael; Randall, Dana; Sinclair, Alistair

doi:10.1109/sfcs.1995.492472

Cited by 100 publications

(187 citation statements)

References 18 publications

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“…The case r = 1 is more complicated as Lemma 5 shows that the expectation is constant. However, this allows us to use standard martingale techniques and the proof of the following is partly adapted from the proof of Lemma 3.4 in [22]. …”

Section: Bounding the Absorption Timementioning

confidence: 99%

Approximating Fixation Probabilities in the Generalized Moran Process

et al. 2012

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Citation for published item:h¡ % zD tF nd qold ergD vFeF nd wertziosD qFfF nd i her yD hF nd ern D wF nd pir kisD FqF @PHIRA 9epproxim ting (x tion pro ilities in the gener lized wor n pro essF9D elgorithmi FD TW @IAF ppF UVEWIFFurther information on publisher's website:The nal publication is available at Springer via http://dx.doi.org/10.1007/s00453-012-9722-7.Additional information:A preliminary version of this work appeared in Proceedings of the ACM SIAM Symposium on Discrete Algorithms (SODA), pp. 954 960, 2012. Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO• the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractWe consider the Moran process, as generalized by Lieberman, Hauert and Nowak (Nature, 433:312-316, 2005). A population resides on the vertices of a finite, connected, undirected graph and, at each time step, an individual is chosen at random with probability proportional to its assigned "fitness" value. It reproduces, placing a copy of itself on a neighbouring vertex chosen uniformly at random, replacing the individual that was there. The initial population consists of a single mutant of fitness r > 0 placed uniformly at random, with every other vertex occupied by an individual of fitness 1. The main quantities of interest are the probabilities that the descendants of the initial mutant come to occupy the whole graph (fixation) and that they die out (extinction); almost surely, these are the only possibilities. In general, exact computation of these quantities by standard Markov chain techniques requires solving a system of linear equations of size exponential in the order of the graph so is not feasible. We show that, with high probability, the number of steps needed to reach fixation or extinction is bounded by a polynomial in the number of vertices in the graph. This bound allows us to construct fully polynomial randomized approximation schemes (FPRAS) for the probability of fixation (when r 1) and of extinction (for all r > 0). * A preliminary version of this work appeared in

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Section: Bounding the Absorption Timementioning

confidence: 99%

Approximating Fixation Probabilities in the Generalized Moran Process

et al. 2012

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“…For example, to sample perfect matchings using M(B), one could restrict to subgraphs in which every vertex has odd degree and reject moves that increase degrees above one. This idea generalizes an approach that has been proven to be fast on the 2-dimensional lattice [14].…”

Section: With Probabilitymentioning

confidence: 94%

“…More generally, M(B) can sample from any state space that is comprised of all subgraphs with any fixed set of odd-degree vertices, since moves of M(B) do not change the parity of the degree of any vertex. Subgraphs with a fixed set of odd-degree vertices are important in a variety of applications in statistical physics and computing, such as the Ising model with an external field (not considered here), as well as perfect matchings, Eulerian orientations, and 3-colorings [14]. For example, to sample perfect matchings using M(B), one could restrict to subgraphs in which every vertex has odd degree and reject moves that increase degrees above one.…”

Section: With Probabilitymentioning

confidence: 99%

Cycle Basis Markov Chains for the Ising Model

Streib

2017

2017 Proceedings of the Fourteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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A very challenging problem from statistical physics is computing the partition function of the ferromagnetic Ising model, even in the relatively simple case of no applied field. In this case, the partition function can be written as a function of the subgraphs of the underlying graph in which all vertices have even degree. In their seminal work, Jerrum and Sinclair showed that this quantity can be approximated by a rapidly converging Markov chain on all subgraphs. However, their chain frequently leaves the space of even subgraphs. Our aim is to devise and analyze a new class of Markov chains that do not leave this space, in the hopes of finding a faster sampling algorithm.We define Markov chains by viewing the even subgraphs as a vector space (often called the cycle space) whose transitions are defined by the addition of basis elements. The rate of convergence depends on the basis chosen, and our analysis proceeds by dividing bases into two types, short and long. The classical singlesite update Markov chain known as Glauber dynamics is a special case of our short cycle basis Markov chains. We show that for any graph and any long basis, there is a temperature for which the corresponding Markov chain requires an exponential time to mix. Moreover, we show that for d-dimensional grids with d ≥ 2-those of the most physical importance-all fundamental bases (a natural class of bases derived from spanning trees) are long. For the 2-dimensional grid on the torus, we show that there is a temperature for which the Markov chain requires exponential time for any chosen basis. * Official contribution of the National Institute of Standards and Technology; not subject to copyright in the United States.† ampasco@super.org ‡ nsstrei@super.org

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“…When γ = 0, the only allowable configurations are Eulerian orientations (known as the 6-vertex model) where every internal vertex has in-degree = outdegree = 2, and the local Markov chain is known to be efficient [6,10]. When γ is close to 1, we can use simple coupling arguments to show that the chain is again rapidly mixing.…”

Section: Non-bipartite Independent Sets and Weighted Even Orientationsmentioning

confidence: 99%

Slow Mixing of Markov Chains Using Fault Lines and Fat Contours

Greenberg

Randall

2008

Algorithmica

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We show that local dynamics require exponential time for two sampling problems motivated by statistical physics: independent sets on the triangular lattice (the hard-core lattice gas model) and weighted even orientations of the two-dimensional Cartesian lattice (the 8-vertex model). For each problem, there is a parameter λ known as the fugacity, such that local Markov chains are expected to be fast when λ is small and slow when λ is large. Unfortunately, establishing slow mixing for these models has been a challenge, as standard contour arguments typically used to show that a chain has small conductance do not seem to apply. We modify this approach by introducing the notion of fat contours that can have nontrivial area, and use these to establish slow mixing of local chains defined for these models.

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Markov chain algorithms for planar lattice structures

Cited by 100 publications

References 18 publications

Approximating Fixation Probabilities in the Generalized Moran Process

Approximating Fixation Probabilities in the Generalized Moran Process

Cycle Basis Markov Chains for the Ising Model

Slow Mixing of Markov Chains Using Fault Lines and Fat Contours

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