The switch Markov chain for sampling irregular graphs and digraphs

The problem of efficiently sampling from a set of (undirected, or directed) graphs with a given degree sequence has many applications. One approach to this problem uses a simple Markov chain, which we call the switch chain, to perform the sampling. The switch chain is known to be rapidly mixing for regular degree sequences, both in the undirected and directed setting. We prove that the switch chain for undirected graphs is rapidly mixing for any degree sequence with minimum degree at least 1 and with maximum degree dmax which satisfies 3≤dmax≤[Formula presented]M, where M is the sum of the degrees. The mixing time bound obtained is only a factor n larger than that established in the regular case, where n is the number of vertices. Our result covers a wide range of degree sequences, including power-law density-bounded graphs with parameter γ>5/2 and sufficiently many edges. For directed degree sequences such that the switch chain is irreducible, we prove that the switch chain is rapidly mixing when all in-degrees and out-degrees are positive and bounded above by [Formula presented]m, where m is the number of arcs, and not all in-degrees and out-degrees equal 1. The mixing time bound obtained in the directed case is an order of m2 larger than that established in the regular case

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

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