Simple Markov‐chain algorithms for generating bipartite graphs and tournaments

We consider two problems: randomly generating labeled bipartite graphs with a given degree sequence and randomly generating labeled tournaments with a given score sequence. We analyze simple Markov chains for both problems. For the first problem, we cannot prove that our chain is rapidly mixing in general, but in the near‐regular case, i.e., when all the degrees are almost equal, we give a proof of rapid mixing. Our methods also apply to the corresponding problem for general (nonbipartite) regular graphs, which was studied earlier by several researchers. One significant difference in our approach is that our chain has one state for every graph (or bipartite graph) with the given degree sequence; in particular, there are no auxiliary states as in the chain used by Jerrum and Sinclair. For the problem of generating tournaments, we are able to prove that our Markov chain on tournaments is rapidly mixing, if the score sequence is near‐regular. The proof techniques we use for the two problems are similar. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 14: 293–308, 1999     

A bounding chain for Swendsen‐Wang

The greatest drawback of Monte Carlo Markov chain methods is lack of knowledge of the mixing time of the chain. The use of bounding chains solves this difficulty for some chains by giving theoretical and experimental upper bounds on the mixing time. Moreover, when used with methodologies such as coupling from the past, bounding chains allow the user to take samples drawn exactly from the stationary distribution without knowledge of the mixing time. Here we present a bounding chain for the Swendsen‐Wang process. The Swendsen‐Wang bounding chain allow us to efficiently obtain exact samples from the ferromagnetic Q‐state Potts model for certain classes of graphs. Also, by analyzing this bounding chain, we will show that Swendsen‐Wang is rapidly mixing over a slightly larger range of parameters than was known previously. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 22: 43–59, 2002     

Sampling Eulerian orientations of triangular lattice graphs

We consider the problem of sampling from the uniform distribution on the set of Eulerian orientations of subgraphs of the triangular lattice. Although Mihail and Winkler (1989) showed that this can be achieved in polynomial time for any graph, the algorithm studied here is more natural in the context of planar Eulerian graphs. We analyse the mixing time of a Markov chain on the Eulerian orientations of a planar graph which moves between orientations by reversing the edges of directed faces. Using path coupling and the comparison method we obtain a polynomial upper bound on the mixing time of this chain for any solid subgraph of the triangular lattice. By considering the conductance of the chain we show that there exist non-solid subgraphs (subgraphs with holes) for which the chain will always take an exponential amount of time to converge. Finally, we show that the problem of counting Eulerian orientations remains #P-complete when restricted to planar graphs (Mihail and Winkler had already established this for general graphs).     

On the hardness of sampling independent sets beyond the tree threshold

We consider local Markov chain Monte–Carlo algorithms for sampling from the weighted distribution of independent sets with activity λ, where the weight of an independent set I is λ^|I|. A recent result has established that Gibbs sampling is rapidly mixing in sampling the distribution for graphs of maximum degree d and λ < λ _c (d), where λ _c (d) is the critical activity for uniqueness of the Gibbs measure (i.e., for decay of correlations with distance in the weighted distribution over independent sets) on the d-regular infinite tree. We show that for d ≥ 3, λ just above λ _c (d) with high probability over d-regular bipartite graphs, any local Markov chain Monte–Carlo algorithm takes exponential time before getting close to the stationary distribution. Our results provide a rigorous justification for “replica” method heuristics. These heuristics were invented in theoretical physics and are used in order to derive predictions on Gibbs measures on random graphs in terms of Gibbs measures on trees. A major theoretical challenge in recent years is to provide rigorous proofs for the correctness of such predictions. Our results establish such rigorous proofs for the case of hard-core model on bipartite graphs. We conjecture that λ _c is in fact the exact threshold for this computational problem, i.e., that for λ > λ _c it is NP-hard to approximate the above weighted sum over independent sets to within a factor polynomial in the size of the graph.     

Local uniformity properties for glauber dynamics on graph colorings

We investigate some local properties which hold with high probability for randomly selected colorings of a fixed graph with no short cycles. In a number of related works, establishing these particular properties has been a crucial step towards proving rapid convergence for the single‐site Glauber dynamics, a Markov chain for sampling colorings uniformly at random. For a large class of graphs, this approach yields the most efficient known algorithms for sampling random colorings. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Genus polynomials and crosscap‐number polynomials for ring‐like graphs

An H‐linear graph is obtained by transforming a collection of copies of a fixed graph H into a chain. An H‐ring‐like graph is formed by binding the two end‐copies of H in such a chain to each other. Genus polynomials have been calculated for bindings of several kinds. In this paper, we substantially generalize the rules for constructing sequences of H‐ring‐like graphs from sequences of H‐linear graphs, and we give a general method for obtaining a recursion for the genus polynomials of the graphs in a sequence of ring‐like graphs. We use Chebyshev polynomials to obtain explicit formulas for the genus polynomials of several such sequences. We also give methods for obtaining recursions for partial genus polynomials and for crosscap‐number polynomials of a bar‐ring of a sequence of disjoint graphs.     

Torpid mixing of the Wang–Swendsen–Kotecký algorithm for sampling colorings

We study the problem of sampling uniformly at random from the set of k-colorings of a graph with maximum degree Δ. We focus attention on the Markov chain Monte Carlo method, particularly on a popular Markov chain for this problem, the Wang–Swendsen–Kotecký (WSK) algorithm. The second author recently proved that the WSK algorithm quickly converges to the desired distribution when k11Δ/6. We study how far these positive results can be extended in general. In this note we prove the first non-trivial results on when the WSK algorithm takes exponentially long to reach the stationary distribution and is thus called torpidly mixing. In particular, we show that the WSK algorithm is torpidly mixing on a family of bipartite graphs when 3k<Δ/(20logΔ), and on a family of planar graphs for any number of colors. We also give a family of graphs for which, despite their small chromatic number, the WSK algorithm is not ergodic when kΔ/2, provided k is larger than some absolute constant k₀.     

Mixing of the Glauber dynamics for the ferromagnetic Potts model

We present several results on the mixing time of the Glauber dynamics for sampling from the Gibbs distribution in the ferromagnetic Potts model. At a fixed temperature and interaction strength, we study the interplay between the maximum degree (Δ) of the underlying graph and the number of colours or spins (q) in determining whether the dynamics mixes rapidly or not. We find a lower bound L on the number of colours such that Glauber dynamics is rapidly mixing if at least L colours are used. We give a closely‐matching upper bound U on the number of colours such that with probability that tends to 1, the Glauber dynamics mixes slowly on random Δ‐regular graphs when at most U colours are used. We show that our bounds can be improved if we restrict attention to certain types of graphs of maximum degree Δ, e.g. toroidal grids for Δ = 4. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 48, 21–52, 2016     

Left and right convergence of graphs with bounded degree

The theory of convergent graph sequences has been worked out in two extreme cases, dense graphs and bounded degree graphs. One can define convergence in terms of counting homomorphisms from fixed graphs into members of the sequence (left‐convergence), or counting homomorphisms into fixed graphs (right‐convergence). Under appropriate conditions, these two ways of defining convergence was proved to be equivalent in the dense case by Borgs, Chayes, Lovász, Sós and Vesztergombi. In this paper a similar equivalence is established in the bounded degree case, if the set of graphs in the definition of right‐convergence is appropriately restricted. In terms of statistical physics, the implication that left convergence implies right convergence means that for a left‐convergent sequence, partition functions of a large class of statistical physics models converge. The proof relies on techniques from statistical physics, like cluster expansion and Dobrushin Uniqueness. © 2012 Wiley Periodicals, Inc. Random Struct. 2012     

Cover time and mixing time of random walks on dynamic graphs

The application of simple random walks on graphs is a powerful tool that is useful in many algorithmic settings such as network exploration, sampling, information spreading, and distributed computing. This is due to the reliance of a simple random walk on only local data, its negligible memory requirements, and its distributed nature. It is well known that for static graphs the cover time, that is, the expected time to visit every node of the graph, and the mixing time, that is, the time to sample a node according to the stationary distribution, are at most polynomial relative to the size of the graph. Motivated by real world networks, such as peer‐to‐peer and wireless networks, the conference version of this paper was the first to study random walks on arbitrary dynamic networks. We study the most general model in which an oblivious adversary is permitted to change the graph after every step of the random walk. In contrast to static graphs, and somewhat counter‐intuitively, we show that there are adversary strategies that force the expected cover time and the mixing time of the simple random walk on dynamic graphs to be exponentially long, even when at each time step the network is well connected and rapidly mixing. To resolve this, we propose a simple strategy, the lazy random walk, which guarantees, under minor conditions, polynomial cover time and polynomial mixing time regardless of the changes made by the adversary.     

Mover‐stayer model with covariate effects on stayer's probability and mover's transitions

A discrete time Markov chain assumes that the population is homogeneous, each individual in the population evolves according to the same transition matrix. In contrast, a discrete mover‐stayer (MS) model postulates a simple form of population heterogeneity; in each initial state, there is a proportion of individuals who never leave this state (stayers) and the complementary proportion of individuals who evolve according to a Markov chain (movers). The MS model was extended by specifying the stayer's probability to be a logistic function of an individual's covariates but leaving the same transition matrix for all movers. We further extend the MS model by allowing each mover to have her/his covariates dependent transition matrix. The model for a mover's transition matrix is related to the extant Markov chains mixture model with mixing on the speed of movement of Markov chains. The proposed model is estimated using the expectation‐maximization algorithm and illustrated with a large data set on car loans and the simulation.     

Comparison of Swendsen‐Wang and heat‐bath dynamics

We prove that the spectral gap of the Swendsen‐Wang process for the Potts model on graphs with bounded degree is bounded from below by some constant times the spectral gap of any single‐spin dynamics. This implies rapid mixing for the two‐dimensional Potts model at all temperatures above the critical one, as well as rapid mixing at the critical temperature for the Ising model. After this we introduce a modified version of the Swendsen‐Wang algorithm for planar graphs and prove rapid mixing for the two‐dimensional Potts models at all non‐critical temperatures. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 42, 520–535, 2013     

On 2-switches and isomorphism classes

A 2-switch is an edge addition/deletion operation that changes adjacencies in the graph while preserving the degree of each vertex. A well-known result states that graphs with the same degree sequence may be changed into each other via sequences of 2-switches. We show that if a 2-switch changes the isomorphism class of a graph, then it must take place in one of four configurations. We also present a sufficient condition for a 2-switch to change the isomorphism class of a graph. As consequences, we give a new characterization of matrogenic graphs and determine the largest hereditary graph family whose members are all the unique realizations (up to isomorphism) of their respective degree sequences.     

Very rapidly mixing Markov chains for 2Δ‐colorings and for independent sets in a graph with maximum degree 4

We introduce a new technique for analyzing the mixing rate of Markov chains. We use it to prove that the Glauber dynamics on 2Δ‐colorings of a graph with maximum degree Δ mixes in O(n log n) time. We prove the same mixing rate for the Insert/Delete/Drag chain of Dyer and Greenhill (Random Structures Algorithms 13 , 285–317 (1998)) on independent sets of graphs with maximum degree 4. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 101–115, 2001     

Markov chain mixing time on cycles

Mixing time quantifies the convergence speed of a Markov chain to the stationary distribution. It is an important quantity related to the performance of MCMC sampling. It is known that the mixing time of a reversible chain can be significantly improved by lifting, resulting in an irreversible chain, while changing the topology of the chain. We supplement this result by showing that if the connectivity graph of a Markov chain is a cycle, then there is an Ω(n²) lower bound for the mixing time. This is the same order of magnitude that is known for reversible chains on the cycle.     

Marginal AMP chain graphs

We present a new family of models that is based on graphs that may have undirected, directed and bidirected edges. We name these new models marginal AMP (MAMP) chain graphs because each of them is Markov equivalent to some AMP chain graph under marginalization of some of its nodes. However, MAMP chain graphs do not only subsume AMP chain graphs but also multivariate regression chain graphs. We describe global and pairwise Markov properties for MAMP chain graphs and prove their equivalence for compositional graphoids. We also characterize when two MAMP chain graphs are Markov equivalent.For Gaussian probability distributions, we also show that every MAMP chain graph is Markov equivalent to some directed and acyclic graph with deterministic nodes under marginalization and conditioning on some of its nodes. This is important because it implies that the independence model represented by a MAMP chain graph can be accounted for by some data generating process that is partially observed and has selection bias. Finally, we modify MAMP chain graphs so that they are closed under marginalization for Gaussian probability distributions. This is a desirable feature because it guarantees parsimonious models under marginalization.     

The principal Erdős–Gallai differences of a degree sequence

The Erd?s–Gallai criteria for recognizing degree sequences of simple graphs involve a system of inequalities. Given a fixed degree sequence, we consider the list of differences of the two sides of these inequalities. These differences have appeared in varying contexts, including characterizations of the split and threshold graphs, and we survey their uses here. Then, enlarging upon properties of these graph families, we show that both the last term and the maximum term of the principal Erd?s–Gallai differences of a degree sequence are preserved under graph complementation and are monotonic under the majorization order and Rao's order on degree sequences.     

On Rauzy graph sequences of infinite words

Under study are the sequences of Rauzy graphs (i.e., the graphs of subwords overlapping) of infinite words. The k-stretching of a graph is the graph we obtain by replacing each edge with a chain of length k. Considering a sequence of strongly connected directed graphs of maximal in and out vertex degrees equal to s, we prove that it is, up to stretchings, a subsequence of a Rauzy graphs sequence of some uniformly recurrent infinite word on s-letter alphabet. The language of a word of this kind and stretching for a given sequence of graphs are constructed explicitly.     

Cutoff for General Spin Systems with Arbitrary Boundary Conditions

The cutoff phenomenon describes a sharp transition in the convergence of a Markov chain to equilibrium. In recent work, the authors established cutoff and its location for the stochastic Ising model on the d‐dimensional torus (?/n?)^d for any d ≥ ;1. The proof used the symmetric structure of the torus and monotonicity in an essential way. Here we enhance the framework and extend it to general geometries, boundary conditions, and external fields to derive a cutoff criterion that involves the growth rate of balls and the log‐Sobolev constant of the Glauber dynamics. In particular, we show there is cutoff for the stochastic Ising model on any sequence of bounded‐degree graphs with subexponential growth under arbitrary external fields provided the inverse log‐Sobolev constant is bounded. For lattices with homogenous boundary, such as all‐plus, we identify the cutoff location explicitly in terms of spectral gaps of infinite‐volume dynamics on half‐plane intersections. Analogous results establishing cutoff are obtained for nonmonotone spin systems at high temperatures, including the gas hard‐core model, the Potts model, the antiferromagnetic Potts model, and the coloring model. © 2014 Wiley Periodicals, Inc.     

Note on the knapsack Markov chain

We show that for sufficiently large knapsacks the associated Markov chain on the state space of the admissible packings of the knapsack is rapidly mixing. Our condition basically states that at least half of all items should fit into the knapsack. This is much weaker than the condition assumed by Saloff-Coste (1997).