Mixing time and eigenvalues of the abelian sandpile Markov chain

We consider the problem of sampling ‘unlabelled structures’, i.e., sampling combinatorial structures modulo a group of symmetries. The main tool which has been used for this sampling problem is Burnside’s lemma. In situations where a significant proportion of the structures have no nontrivial symmetries, it is already fairly well understood how to apply this tool. More generally, it is possible to obtain nearly uniform samples by simulating a Markov chain that we call the Burnside process: this is a random walk on a bipartite graph which essentially implements Burnside’s lemma. For this approach to be feasible, the Markov chain ought to be ‘rapidly mixing’, i.e., converge rapidly to equilibrium. The Burnside process was known to be rapidly mixing for some special groups, and it has even been implemented in some computational group theory algorithms. In this paper, we show that the Burnside process is not rapidly mixing in general. In particular, we construct an infinite family of permutation groups for which we show that the mixing time is exponential in the degree of the group.

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Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings

Combinatorics Probability Computing ◽

10.1017/s0963548318000470 ◽

2018 ◽

Vol 28 (3) ◽

pp. 365-387

Author(s):

S. CANNON ◽

D. A. LEVIN ◽

A. STAUFFER

Keyword(s):

Markov Chain ◽

Relaxation Time ◽

Lower Bound ◽

Upper Bound ◽

Open Problem ◽

Mixing Time ◽

Perpendicular Bisector ◽

Unit Square

We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall and Spencer in 2002 [14]. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2−s, (a + 1)2−s] × [b2−t, (b + 1)2−t] for a, b, s, t ∈ ℤ⩾ 0. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n4.09), which implies that the mixing time is at most O(n5.09). We complement this by showing that the relaxation time is at least Ω(n1.38), improving upon the previously best lower bound of Ω(n log n) coming from the diameter of the chain.

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On the mixing time of the SIS Markov chain model for epidemic spread

53rd IEEE Conference on Decision and Control ◽

10.1109/cdc.2014.7040364 ◽

2014 ◽

Cited By ~ 7

Author(s):

Hyoung Jun Ahn ◽

Babak Hassibi

Keyword(s):

Markov Chain ◽

Markov Chain Model ◽

Mixing Time ◽

Chain Model ◽

Epidemic Spread

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In search of lost mixing time: adaptive Markov chain Monte Carlo schemes for Bayesian variable selection with very large p

Biometrika ◽

10.1093/biomet/asaa055 ◽

2020 ◽

Author(s):

J E Griffin ◽

K G Łatuszyński ◽

M F J Steel

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Variable Selection ◽

Adaptive Design ◽

Mixing Time ◽

Bayesian Variable Selection ◽

Monte Carlo Algorithms ◽

Large Numbers ◽

Large P Small N

Summary The availability of datasets with large numbers of variables is rapidly increasing. The effective application of Bayesian variable selection methods for regression with these datasets has proved difficult since available Markov chain Monte Carlo methods do not perform well in typical problem sizes of interest. We propose new adaptive Markov chain Monte Carlo algorithms to address this shortcoming. The adaptive design of these algorithms exploits the observation that in large-$p$, small-$n$ settings, the majority of the $p$ variables will be approximately uncorrelated a posteriori. The algorithms adaptively build suitable nonlocal proposals that result in moves with squared jumping distance significantly larger than standard methods. Their performance is studied empirically in high-dimensional problems and speed-ups of up to four orders of magnitude are observed.

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Assessing significance in a Markov chain without mixing

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.1617540114 ◽

2017 ◽

Vol 114 (11) ◽

pp. 2860-2864 ◽

Cited By ~ 30

Author(s):

Maria Chikina ◽

Alan Frieze ◽

Wesley Pegden

Keyword(s):

Markov Chain ◽

Random Walk ◽

Stationary Distribution ◽

Potential Application ◽

Null Hypothesis ◽

Value Function ◽

Mixing Time ◽

Statistical Test ◽

Reversible Markov Chain ◽

A Value

We present a statistical test to detect that a presented state of a reversible Markov chain was not chosen from a stationary distribution. In particular, given a value function for the states of the Markov chain, we would like to show rigorously that the presented state is an outlier with respect to the values, by establishing a p value under the null hypothesis that it was chosen from a stationary distribution of the chain. A simple heuristic used in practice is to sample ranks of states from long random trajectories on the Markov chain and compare these with the rank of the presented state; if the presented state is a 0.1% outlier compared with the sampled ranks (its rank is in the bottom 0.1% of sampled ranks), then this observation should correspond to a p value of 0.001. This significance is not rigorous, however, without good bounds on the mixing time of the Markov chain. Our test is the following: Given the presented state in the Markov chain, take a random walk from the presented state for any number of steps. We prove that observing that the presented state is an ε-outlier on the walk is significant at p=2ε under the null hypothesis that the state was chosen from a stationary distribution. We assume nothing about the Markov chain beyond reversibility and show that significance at p≈ε is best possible in general. We illustrate the use of our test with a potential application to the rigorous detection of gerrymandering in Congressional districting.

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Rapid mixing of path integral Monte Carlo for 1D stoquastic Hamiltonians

Quantum ◽

10.22331/q-2021-02-11-395 ◽

2021 ◽

Vol 5 ◽

pp. 395

Author(s):

Elizabeth Crosson ◽

Aram W. Harrow

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Path Integral ◽

Equilibrium Distribution ◽

Mixing Time ◽

Gibbs Distribution ◽

Convergence Time ◽

Spin Models ◽

Classical Spin ◽

Classical Spin Models

Path integral quantum Monte Carlo (PIMC) is a method for estimating thermal equilibrium properties of stoquastic quantum spin systems by sampling from a classical Gibbs distribution using Markov chain Monte Carlo. The PIMC method has been widely used to study the physics of materials and for simulated quantum annealing, but these successful applications are rarely accompanied by formal proofs that the Markov chains underlying PIMC rapidly converge to the desired equilibrium distribution. In this work we analyze the mixing time of PIMC for 1D stoquastic Hamiltonians, including disordered transverse Ising models (TIM) with long-range algebraically decaying interactions as well as disordered XY spin chains with nearest-neighbor interactions. By bounding the convergence time to the equilibrium distribution we rigorously justify the use of PIMC to approximate partition functions and expectations of observables for these models at inverse temperatures that scale at most logarithmically with the number of qubits. The mixing time analysis is based on the canonical paths method applied to the single-site Metropolis Markov chain for the Gibbs distribution of 2D classical spin models with couplings related to the interactions in the quantum Hamiltonian. Since the system has strongly nonisotropic couplings that grow with system size, it does not fall into the known cases where 2D classical spin models are known to mix rapidly.

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Towards Random Uniform Sampling of Bipartite Graphs with given Degree Sequence

The Electronic Journal of Combinatorics ◽

10.37236/3028 ◽

2013 ◽

Vol 20 (1) ◽

Cited By ~ 4

Author(s):

István Miklós ◽

Péter L Erdős ◽

Lajos Soukup

Keyword(s):

Markov Chain ◽

Mixing Time ◽

Degree Sequence ◽

Bipartite Graphs ◽

Multicommodity Flow ◽

Uniform Sampling

In this paper we consider a simple Markov chain for bipartite graphs with given degree sequence on $n$ vertices. We show that the mixing time of this Markov chain is bounded above by a polynomial in $n$ in case of half-regular degree sequence. The novelty of our approach lies in the construction of the multicommodity flow in Sinclair's method.

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On the convergence rates of some adaptive Markov chain Monte Carlo algorithms

Journal of Applied Probability ◽

10.1239/jap/1445543848 ◽

2015 ◽

Vol 52 (3) ◽

pp. 811-825

Author(s):

Yves Atchadé ◽

Yizao Wang

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Convergence Rates ◽

Mixing Time ◽

Regularity Conditions ◽

Monte Carlo Algorithms ◽

Mcmc Algorithms ◽

Variation Distance ◽

Importance Resampling

In this paper we study the mixing time of certain adaptive Markov chain Monte Carlo (MCMC) algorithms. Under some regularity conditions, we show that the convergence rate of importance resampling MCMC algorithms, measured in terms of the total variation distance, is O(n-1). By means of an example, we establish that, in general, this algorithm does not converge at a faster rate. We also study the interacting tempering algorithm, a simplified version of the equi-energy sampler, and establish that its mixing time is of order O(n-1/2).

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A note on perfect simulation for Exponential Random Graph Models

ESAIM Probability and Statistics ◽

10.1051/ps/2019024 ◽

2020 ◽

Vol 24 ◽

pp. 138-147 ◽

Cited By ~ 1

Author(s):

Andressa Cerqueira ◽

Aurélien Garivier ◽

Florencia Leonardi

Keyword(s):

Markov Chain ◽

Random Graph ◽

Mixing Time ◽

Graph Model ◽

Glauber Dynamics ◽

Perfect Simulation ◽

Exponential Random Graph Models ◽

Graph Models ◽

Random Graph Model ◽

Exponential Random Graph

In this paper, we propose a perfect simulation algorithm for the Exponential Random Graph Model, based on the Coupling from the past method of Propp and Wilson (1996). We use a Glauber dynamics to construct the Markov Chain and we prove the monotonicity of the ERGM for a subset of the parametric space. We also obtain an upper bound on the running time of the algorithm that depends on the mixing time of the Markov chain.

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Mixing Time for a Markov Chain on Cladograms

Combinatorics Probability Computing ◽

10.1017/s096354830000417x ◽

2000 ◽

Vol 9 (3) ◽

pp. 191-204 ◽

Cited By ~ 13

Author(s):

DAVID J. ALDOUS

Keyword(s):

Markov Chain ◽

Stationary Distribution ◽

Mixing Time ◽

Incident Edge

A cladogram is a tree with labelled leaves and unlabelled degree-3 branchpoints. A certain Markov chain on the set of n-leaf cladograms consists of removing a random leaf (and its incident edge) and re-attaching it to a random edge. We show that the mixing time (time to approach the uniform stationary distribution) for this chain is at least O(n2) and at most O(n3).

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