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 Markov chain Monte Carlo approach for sampling graphs with prescribed degree sequences. In this work we study the problem of uniformly sampling graphs for which, in addition to the degree sequence, joint degree constraints are given. These constraints specify how many edges there should be between two given degree classes (i.e., subsets of nodes that all have the same degree). 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. In the case of one degree class, the problem reduces to the sampling of regular graphs (i.e., graphs in which all nodes have the same degree), but beyond this very little is known. We fully resolve the case of two degree classes, by showing that the switch Markov chain is always rapidly mixing. We do this by combining a 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.