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.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

Maths lecturers in denial about their own maths practice? A case of teaching matrix operations to undergraduate students

This case study provides evidence of an apparent disparity in the way that certain mathematics topics are taught compared to the way that they are used in professional practice. In particular, we focus on the topic of matrices by comparing sources from published research articles against typical undergraduate textbooks and lecture notes. Our results show that the most important operation when using matrices in research is that of matrix multiplication, with 33 of the 40 publications which we surveyed utilising this as the most prominent operation and the remainder of the publications instead opting not to use matrix multiplication at all rather than offering weighting to alternative operations. This is in contrast to the way in which matrices are taught, with very few of these teaching sources highlighting that matrix multiplication is the most important operation for mathematicians. We discuss the implications of this discrepancy and offer an insight as to why it can be beneficial to consider the professional uses of such topics when teaching mathematics to undergraduate students

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

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

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 $P$-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 $\gamma>1+\sqrt{3}$. The other one shows that the switch Markov chain on the degree sequence of an Erd\H{o}s-R\'enyi random graph $G(n,p)$ is asymptotically almost surely rapidly mixing if $p$ is bounded away from 0 and 1 by at least $\frac{5\log n}{n-1}$.Comment: Clarification

Sharp Poincar\'e and log-Sobolev inequalities for the switch chain on regular bipartite graphs

Consider the switch chain on the set of $d$-regular bipartite graphs on $n$ vertices with $3\leq d\leq n^{c}$, for a small universal constant $c>0$. We prove that the chain satisfies a Poincar\'e inequality with a constant of order $O(nd)$; moreover, when $d$ is fixed, we establish a log-Sobolev inequality for the chain with a constant of order $O_d(n\log n)$. We show that both results are optimal. The Poincar\'e inequality implies that in the regime $3\leq d\leq n^c$ the mixing time of the switch chain is at most $O\big((nd)^2 \log(nd)\big)$, improving on the previously known bound $O\big((nd)^{13} \log(nd)\big)$ due to Kannan, Tetali and Vempala and $O\big(n^7d^{18} \log(nd)\big)$ obtained by Dyer et al. The log-Sobolev inequality that we establish for constant $d$ implies a bound $O(n\log^2 n)$ on the mixing time of the chain which, up to the $\log n$ factor, captures a conjectured optimal bound. Our proof strategy relies on building, for any fixed function on the set of $d$-regular bipartite simple graphs, an appropriate extension to a function on the set of multigraphs given by the configuration model. We then establish a comparison procedure with the well studied random transposition model in order to obtain the corresponding functional inequalities. While our method falls into a rich class of comparison techniques for Markov chains on different state spaces, the crucial feature of the method - dealing with chains with a large distortion between their stationary measures - is a novel addition to the theory.Comment: references and abstract update