907 research outputs found
A decomposition based proof for fast mixing of a Markov chain over balanced realizations of a joint degree matrix
A joint degree matrix (JDM) specifies the number of connections between nodes
of given degrees in a graph, for all degree pairs and uniquely determines the
degree sequence of the graph. We consider the space of all balanced
realizations of an arbitrary JDM, realizations in which the links between any
two degree groups are placed as uniformly as possible. We prove that a swap
Markov Chain Monte Carlo (MCMC) algorithm in the space of all balanced
realizations of an {\em arbitrary} graphical JDM mixes rapidly, i.e., the
relaxation time of the chain is bounded from above by a polynomial in the
number of nodes . To prove fast mixing, we first prove a general
factorization theorem similar to the Martin-Randall method for disjoint
decompositions (partitions). This theorem can be used to bound from below the
spectral gap with the help of fast mixing subchains within every partition and
a bound on an auxiliary Markov chain between the partitions. Our proof of the
general factorization theorem is direct and uses conductance based methods
(Cheeger inequality).Comment: submitted, 18 pages, 4 figure
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
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
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 P-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 P-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.</p
New Classes of Degree Sequences with Fast Mixing Swap Markov Chain Sampling
In network modelling of complex systems one is often required to sample random realizations of networks that obey a given set of constraints, usually in the form of graph measures. A much studied class of problems targets uniform sampling of simple graphs with given degree sequence or also with given degree correlations expressed in the form of a Joint Degree Matrix. One approach is to use Markov chains based on edge switches (swaps) that preserve the constraints, are irreducible (ergodic) and fast mixing. In 1999, Kannan, Tetali and Vempala (KTV) proposed a simple swap Markov chain for sampling graphs with given degree sequence, and conjectured that it mixes rapidly (in polynomial time) for arbitrary degree sequences. Although the conjecture is still open, it has been proved for special degree sequences, in particular for those of undirected and directed regular simple graphs, half-regular bipartite graphs, and graphs with certain bounded maximum degrees. Here we prove the fast mixing KTV conjecture for novel, exponentially large classes of irregular degree sequences. Our method is based on a canonical decomposition of degree sequences into split graph degree sequences, a structural theorem for the space of graph realizations and on a factorization theorem for Markov chains. After introducing bipartite ‘splitted’ degree sequences, we also generalize the canonical split graph decomposition for bipartite and directed graphs. Copyright © Cambridge University Press 201
Approximate Sampling and Counting of Graphs with Near-Regular Degree Intervals
The approximate uniform sampling of graphs with a given degree sequence is a well-known, extensively studied problem in theoretical computer science and has significant applications, e.g., in the analysis of social networks. In this work we study an extension of the problem, where degree intervals are specified rather than a single degree sequence. We are interested in sampling and counting graphs whose degree sequences satisfy the degree interval constraints. A natural scenario where this problem arises is in hypothesis testing on social networks that are only partially observed. In this work, we provide the first fully polynomial almost uniform sampler (FPAUS) as well as the first fully polynomial randomized approximation scheme (FPRAS) for sampling and counting, respectively, graphs with near-regular degree intervals, in which every node has a degree from an interval not too far away from a given . In order to design our FPAUS, we rely on various state-of-the-art tools from Markov chain theory and combinatorics. In particular, we provide the first non-trivial algorithmic application of a breakthrough result of Liebenau and Wormald (2017) regarding an asymptotic formula for the number of graphs with a given near-regular degree sequence. Furthermore, we also make use of the recent breakthrough of Anari et al. (2019) on sampling a base of a matroid under a strongly log-concave probability distribution. As a more direct approach, we also study a natural Markov chain recently introduced by Rechner, Strowick and M\"uller-Hannemann (2018), based on three simple local operations: Switches, hinge flips, and additions/deletions of a single edge. We obtain the first theoretical results for this Markov chain by showing it is rapidly mixing for the case of near-regular degree intervals of size at most one
Rapid Mixing of the Switch Markov Chain for 2-Class Joint Degree Matrices
The switch Markov chain has been extensively studied as the most natural Markovchain Monte Carlo approach for sampling graphs with prescribed degree sequences. In this work westudy the problem of uniformly sampling graphs for which, in addition to the degree sequence, jointdegree constraints are given. These constraints specify how many edges there should be between twogiven degree classes (i.e., subsets of nodes that all have the same degree). Although the problem wasformalized over a decade ago, and despite its practical significance in generating synthetic networktopologies, small progress has been made on the random sampling of such graphs. In the case of onedegree class, the problem reduces to the sampling of regular graphs (i.e., graphs in which all nodeshave the same degree), but beyond this very little is known. We fully resolve the case of two degreeclasses, by showing that the switch Markov chain is always rapidly mixing. We do this by combininga recent embedding argument developed by the authors in combination with ideas of Bhatnagar et al.[Algorithmica, 50 (2008), pp. 418--445] introduced in the context of sampling bichromatic matchings
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