scholarly journals Towards Random Uniform Sampling of Bipartite Graphs with given Degree Sequence

10.37236/3028 ◽  
2013 ◽  
Vol 20 (1) ◽  
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.

scholarly journals A Polynomial Bound on the Mixing Time of a Markov Chain for Sampling Regular Directed Graphs

10.37236/721 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Catherine Greenhill
Keyword(s):  
Markov Chain ◽  
Directed Graphs ◽  
Mixing Time ◽  
Degree Sequence ◽  
Negative Eigenvalue ◽  
Multicommodity Flow ◽  
Degree Sequences ◽  
Regular Graphs ◽  
Significant Extension ◽  
Second Largest Eigenvalue

The switch chain is a well-known Markov chain for sampling directed graphs with a given degree sequence. While not ergodic in general, we show that it is ergodic for regular degree sequences. We then prove that the switch chain is rapidly mixing for regular directed graphs of degree $d$, where $d$ is any positive integer-valued function of the number of vertices. We bound the mixing time by bounding the eigenvalues of the chain. A new result is presented and applied to bound the smallest (most negative) eigenvalue. This result is a modification of a lemma by Diaconis and Stroock [Annals of Applied Probability 1991], and by using it we avoid working with a lazy chain. A multicommodity flow argument is used to bound the second-largest eigenvalue of the chain. This argument is based on the analysis of a related Markov chain for undirected regular graphs by Cooper, Dyer and Greenhill [Combinatorics, Probability and Computing 2007], but with significant extension required.

scholarly journals On Sampling Colorings of Bipartite Graphs

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
R. Balasubramanian ◽  
C.R. Subramanian
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Mixing Time ◽  
Bipartite Graphs ◽  
Exponential Time ◽  
Negative Results ◽  
Bipartite Subgraph ◽  
Single Transition ◽  
International Audience ◽  
Complete Bipartite Subgraph

International audience We study the problem of efficiently sampling k-colorings of bipartite graphs. We show that a class of markov chains cannot be used as efficient samplers. Precisely, we show that, for any k, 6 ≤ k ≤ n^\1/3-ε \, ε > 0 fixed, \emphalmost every bipartite graph on n+n vertices is such that the mixing time of any markov chain asymptotically uniform on its k-colorings is exponential in n/k^2 (if it is allowed to only change the colors of O(n/k) vertices in a single transition step). This kind of exponential time mixing is called \emphtorpid mixing. As a corollary, we show that there are (for every n) bipartite graphs on 2n vertices with Δ (G) = Ω (\ln n) such that for every k, 6 ≤ k ≤ Δ /(6 \ln Δ ), each member of a large class of chains mixes torpidly. While, for fixed k, such negative results are implied by the work of CDF, our results are more general in that they allow k to grow with n. We also show that these negative results hold true for H-colorings of bipartite graphs provided H contains a spanning complete bipartite subgraph. We also present explicit examples of colorings (k-colorings or H-colorings) which admit 1-cautious chains that are ergodic and are shown to have exponential mixing time. While, for fixed k or fixed H, such negative results are implied by the work of CDF, our results are more general in that they allow k or H to vary with n.

scholarly journals On the Swap-Distances of Different Realizations of a Graphical Degree Sequence

2013 ◽  
Vol 22 (3) ◽  
pp. 366-383 ◽  
Author(s):  
PÉTER L. ERDŐS ◽  
ZOLTÁN KIRÁLY ◽  
ISTVÁN MIKLÓS
Keyword(s):  
Social Networks ◽  
Markov Chain ◽  
Degree Sequence ◽  
Upper Bounds ◽  
Simple Graph ◽  
Related Question ◽  
Degree Sequences ◽  
Uniform Sampling ◽  
Integer Sequence ◽  
The Greedy Algorithm

One of the first graph-theoretical problems to be given serious attention (in the 1950s) was the decision whether a given integer sequence is equal to the degree sequence of a simple graph (orgraphical, for short). One method to solve this problem is the greedy algorithm of Havel and Hakimi, which is based on theswapoperation. Another, closely related question is to find a sequence of swap operations to transform one graphical realization into another of the same degree sequence. This latter problem has received particular attention in the context of rapidly mixing Markov chain approaches to uniform sampling of all possible realizations of a given degree sequence. (This becomes a matter of interest in the context of the study of large social networks, for example.) Previously there were only crude upper bounds on the shortest possible length of such swap sequences between two realizations. In this paper we develop formulae (Gallai-type identities) for theswap-distances of any two realizations of simple undirected or directed degree sequences. These identities considerably improve the known upper bounds on the swap-distances.

scholarly journals The Mixing of Markov Chains on Linear Extensions in Practice

2017 ◽  
Author(s):  
Topi Talvitie ◽  
Teppo Niinimäki ◽  
Mikko Koivisto
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Mixing Time ◽  
Mixing Times ◽  
Perfect Simulation ◽  
Linear Extensions ◽  
Worst Case ◽  
Uniform Sampling ◽  
Markov Chain Approach ◽  
Time Bounds

We investigate almost uniform sampling from the set of linear extensions of a given partial order. The most efficient schemes stem from Markov chains whose mixing time bounds are polynomial, yet impractically large. We show that, on instances one encounters in practice, the actual mixing times can be much smaller than the worst-case bounds, and particularly so for a novel Markov chain we put forward. We circumvent the inherent hardness of estimating standard mixing times by introducing a refined notion, which admits estimation for moderate-size partial orders. Our empirical results suggest that the Markov chain approach to sample linear extensions can be made to scale well in practice, provided that the actual mixing times can be realized by instance-sensitive upper bounds or termination rules. Examples of the latter include existing perfect simulation algorithms, whose running times in our experiments follow the actual mixing times of certain chains, albeit with significant overhead.

scholarly journals The ‘Burnside Process’ Converges Slowly

2002 ◽  
Vol 11 (1) ◽  
pp. 21-34 ◽  
Author(s):  
LESLIE ANN GOLDBERG ◽  
MARK JERRUM
Keyword(s):  
Markov Chain ◽  
Mixing Time ◽  
Significant Proportion ◽  
Main Tool ◽  
Infinite Family ◽  
Combinatorial Structures ◽  
Special Groups ◽  
Sampling Problem ◽  
Burnside's Lemma ◽  
Burnside’S Lemma

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.

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

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.

scholarly journals Mixing time and eigenvalues of the abelian sandpile Markov chain

2019 ◽  
Vol 372 (12) ◽  
pp. 8307-8345
Author(s):  
Daniel C. Jerison ◽  
Lionel Levine ◽  
John Pike
Keyword(s):  
Markov Chain ◽  
Mixing Time ◽  
Abelian Sandpile

Methods of simulating random patterns of non-spherical objects and their application

1996 ◽  
Vol 28 (02) ◽  
pp. 342-343
Author(s):  
Masaharu Tanemura
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Spatial Patterns ◽  
Hard Core ◽  
Spherical Particles ◽  
Finite Region ◽  
Uniform Sampling ◽  
Regular Manner ◽  
Random Patterns

We consider two mechanisms for simulating spatial patterns of hard-core non-spherical particles, namely the random sequential packing (RSP) and the Markov chain Monte Carlo (MCMC) procedures. The former is described as follows: we put a particle one-by-one into a finite region by sampling its location x and direction θ uniformly at random; if it does not overlap with other particles put before, it is put successfully, otherwise, we discard it and try another uniform sampling of (x, θ); by repeating the above, we can obtain a set of non-overlapping particles. The MCMC procedure is the following: we first give a certain non-overlapping pattern of non-spherical particles prepared in a random or a regular manner; then we select a particle and sample its new trial location x and direction θ at random; if the new sample (x, θ) is accepted, i.e. it does not overlap with other particles, the selected particle is moved to the new ‘position’, otherwise the particle is retained at the old position; by repeating the above, a series of a set of non-overlapping particles is generated.

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

Biometrika ◽  
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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