1,159 research outputs found
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
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
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
Towards random uniform sampling of bipartite graphs with given degree sequence
In this paper we consider a simple Markov chain for bipartite graphs with
given degree sequence on vertices. We show that the mixing time of this
Markov chain is bounded above by a polynomial in in case of {\em
semi-regular} degree sequence. The novelty of our approach lays in the
construction of the canonical paths in Sinclair's method.Comment: 47 pages, submitted for publication. In this version we explain
explicitly our main contribution and corrected a serious flaw in the cycle
decompositio
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