7,003 research outputs found
Spatial Mixing and Non-local Markov chains
We consider spin systems with nearest-neighbor interactions on an -vertex
-dimensional cube of the integer lattice graph . We study the
effects that exponential decay with distance of spin correlations, specifically
the strong spatial mixing condition (SSM), has on the rate of convergence to
equilibrium distribution of non-local Markov chains. We prove that SSM implies
mixing of a block dynamics whose steps can be implemented
efficiently. We then develop a methodology, consisting of several new
comparison inequalities concerning various block dynamics, that allow us to
extend this result to other non-local dynamics. As a first application of our
method we prove that, if SSM holds, then the relaxation time (i.e., the inverse
spectral gap) of general block dynamics is , where is the number of
blocks. A second application of our technology concerns the Swendsen-Wang
dynamics for the ferromagnetic Ising and Potts models. We show that SSM implies
an bound for the relaxation time. As a by-product of this implication we
observe that the relaxation time of the Swendsen-Wang dynamics in square boxes
of is throughout the subcritical regime of the -state
Potts model, for all . We also prove that for monotone spin systems
SSM implies that the mixing time of systematic scan dynamics is . Systematic scan dynamics are widely employed in practice but have
proved hard to analyze. Our proofs use a variety of techniques for the analysis
of Markov chains including coupling, functional analysis and linear algebra
Mixing Properties of CSMA Networks on Partite Graphs
We consider a stylized stochastic model for a wireless CSMA network.
Experimental results in prior studies indicate that the model provides
remarkably accurate throughput estimates for IEEE 802.11 systems. In
particular, the model offers an explanation for the severe spatial unfairness
in throughputs observed in such networks with asymmetric interference
conditions. Even in symmetric scenarios, however, it may take a long time for
the activity process to move between dominant states, giving rise to potential
starvation issues. In order to gain insight in the transient throughput
characteristics and associated starvation effects, we examine in the present
paper the behavior of the transition time between dominant activity states. We
focus on partite interference graphs, and establish how the magnitude of the
transition time scales with the activation rate and the sizes of the various
network components. We also prove that in several cases the scaled transition
time has an asymptotically exponential distribution as the activation rate
grows large, and point out interesting connections with related exponentiality
results for rare events and meta-stability phenomena in statistical physics. In
addition, we investigate the convergence rate to equilibrium of the activity
process in terms of mixing times.Comment: Valuetools, 6th International Conference on Performance Evaluation
Methodologies and Tools, October 9-12, 2012, Carg\`ese, Franc
Marathon: An open source software library for the analysis of Markov-Chain Monte Carlo algorithms
In this paper, we consider the Markov-Chain Monte Carlo (MCMC) approach for
random sampling of combinatorial objects. The running time of such an algorithm
depends on the total mixing time of the underlying Markov chain and is unknown
in general. For some Markov chains, upper bounds on this total mixing time
exist but are too large to be applicable in practice. We try to answer the
question, whether the total mixing time is close to its upper bounds, or if
there is a significant gap between them. In doing so, we present the software
library marathon which is designed to support the analysis of MCMC based
sampling algorithms. The main application of this library is to compute
properties of so-called state graphs which represent the structure of Markov
chains. We use marathon to investigate the quality of several bounding methods
on four well-known Markov chains for sampling perfect matchings and bipartite
graph realizations. In a set of experiments, we compute the total mixing time
and several of its bounds for a large number of input instances. We find that
the upper bound gained by the famous canonical path method is several
magnitudes larger than the total mixing time and deteriorates with growing
input size. In contrast, the spectral bound is found to be a precise
approximation of the total mixing time
Path Coupling Using Stopping Times and Counting Independent Sets and Colourings in Hypergraphs
We give a new method for analysing the mixing time of a Markov chain using
path coupling with stopping times. We apply this approach to two hypergraph
problems. We show that the Glauber dynamics for independent sets in a
hypergraph mixes rapidly as long as the maximum degree Delta of a vertex and
the minimum size m of an edge satisfy m>= 2Delta+1. We also show that the
Glauber dynamics for proper q-colourings of a hypergraph mixes rapidly if m>= 4
and q > Delta, and if m=3 and q>=1.65Delta. We give related results on the
hardness of exact and approximate counting for both problems.Comment: Simpler proof of main theorem. Improved bound on mixing time. 19
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Delay performance in random-access grid networks
We examine the impact of torpid mixing and meta-stability issues on the delay
performance in wireless random-access networks. Focusing on regular meshes as
prototypical scenarios, we show that the mean delays in an toric
grid with normalized load are of the order . This
superlinear delay scaling is to be contrasted with the usual linear growth of
the order in conventional queueing networks. The intuitive
explanation for the poor delay characteristics is that (i) high load requires a
high activity factor, (ii) a high activity factor implies extremely slow
transitions between dominant activity states, and (iii) slow transitions cause
starvation and hence excessively long queues and delays. Our proof method
combines both renewal and conductance arguments. A critical ingredient in
quantifying the long transition times is the derivation of the communication
height of the uniformized Markov chain associated with the activity process. We
also discuss connections with Glauber dynamics, conductance and mixing times.
Our proof framework can be applied to other topologies as well, and is also
relevant for the hard-core model in statistical physics and the sampling from
independent sets using single-site update Markov chains
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