Rapid Mixing of the Switch Markov Chain for Strongly Stable Degree Sequences and 2-Class Joint Degree Matrices

Abstract

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.

Switch-Based Markov Chains for Sampling Hamiltonian Cycles in Dense Graphs

“…Remark 2 (Bipartite case). Theorem 1 remains true if we restrict ourselves to bipartite graphs G = (A ∪ B, E), where |A| = |B| = n, and δ(G) 1 2 n + 7. The proofs are almost identical, so we make remarks in footnotes where the proofs differ.…”

“…The mixing time of switch-based Markov chains have been studied extensively for sampling subgraphs of K n with a given degree sequence, see, e.g., [8,2,11,1]. It is well known, see e.g.…”

“…The proof of Theorem 8 is outlined in Appendix A. It is based on the embedding argument introduced in [1] for the switch Markov chain that samples graphs with a given degree sequence. It is perhaps interesting to note that it seems much harder to prove Theorem 8 by using other approaches for that problem, such as [2,11].…”

“…It is perhaps interesting to note that it seems much harder to prove Theorem 8 by using other approaches for that problem, such as [2,11]. These approaches do have the advantage that they get better mixing time bounds than those in [1].…”

“…That is, for a fixed edge b = (F 0 , F 0 ) ∈ A switch , the total number of edges a = (F, F ) ∈ A infeasible that use b in their detour is at most poly(n), as (roughly speaking) F 0 is at most 12/k transitions away from F by construction (and k is constant). This yields the desired flow g. For a precise and detailed outline of this idea, we refer the reader to [1].…”