2,433 research outputs found
The mixing time of the switch Markov chains: a unified approach
Since 1997 a considerable effort has been spent to study the mixing time of
switch Markov chains on the realizations of graphic degree sequences of simple
graphs. Several results were proved on rapidly mixing Markov chains on
unconstrained, bipartite, and directed sequences, using different mechanisms.
The aim of this paper is to unify these approaches. We will illustrate the
strength of the unified method by showing that on any -stable family of
unconstrained/bipartite/directed degree sequences the switch Markov chain is
rapidly mixing. This is a common generalization of every known result that
shows the rapid mixing nature of the switch Markov chain on a region of degree
sequences. Two applications of this general result will be presented. One is an
almost uniform sampler for power-law degree sequences with exponent
. The other one shows that the switch Markov chain on the
degree sequence of an Erd\H{o}s-R\'enyi random graph is asymptotically
almost surely rapidly mixing if is bounded away from 0 and 1 by at least
.Comment: Clarification
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
Rapid Mixing of the Switch Markov Chain for Strongly Stable Degree Sequences and 2-Class Joint Degree Matrices
The switch Markov chain has been extensively studied as the most natural
Markov Chain Monte Carlo approach for sampling graphs with prescribed degree
sequences. We use comparison arguments with other, less natural but simpler to
analyze, Markov chains, to show that the switch chain mixes rapidly in two
different settings. We first study the classic problem of uniformly sampling
simple undirected, as well as bipartite, graphs with a given degree sequence.
We apply an embedding argument, involving a Markov chain defined by Jerrum and
Sinclair (TCS, 1990) for sampling graphs that almost have a given degree
sequence, to show rapid mixing for degree sequences satisfying strong
stability, a notion closely related to -stability. This results in a much
shorter proof that unifies the currently known rapid mixing results of the
switch chain and extends them up to sharp characterizations of -stability.
In particular, our work resolves an open problem posed by Greenhill (SODA,
2015).
Secondly, in order to illustrate the power of our approach, we study the
problem of uniformly sampling graphs for which, in addition to the degree
sequence, a joint degree distribution is given. Although the problem was
formalized over a decade ago, and despite its practical significance in
generating synthetic network topologies, small progress has been made on the
random sampling of such graphs. The case of a single degree class reduces to
sampling of regular graphs, but beyond this almost nothing is known. We fully
resolve the case of two degree classes, by showing that the switch Markov chain
is always rapidly mixing. Again, we first analyze an auxiliary chain for
strongly stable instances on an augmented state space and then use an embedding
argument.Comment: Accepted to SODA 201
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