The mixing time of switch Markov chains: A unified approach

Approximate Sampling of Graphs with Near-$P$-stable Degree Intervals

“…In this relatively short, highly technical paper we prove a substantial extension of a recent result of Amanatidis and Kleer [1,Theorem 1.3]. Our proof is based on the unified approach that was developed in [4] for P -stable degree sequences. For sake of brevity in this section we concisely describe the problem itself, but we will not give a detailed description of the background.…”

“…For sake of brevity in this section we concisely describe the problem itself, but we will not give a detailed description of the background. For further details, the diligent reader is referred to [1,4].…”

“…One basic way to study them is the switch Markov chain method, made popular by Kannan, Tetali, and Vempala [9]. The currently best result via this method is [4] where it is proved that the switch Markov chain is rapidly mixing on P -stable degree sequences. The notion of P -stability was introduced by Jerrum and Sinclair [8] and studied for its own sake at first by Jerrum, McKay, and Sinclair [6].…”

“…To our great surprise, it turned out that this result can be derived from the proof of the main theorem of [4]. For that end we had to analyse in detail the auxiliary structures of the proof and to extend to cover this setup.…”

“…Theorem 2.17 ( [4]). The switch Markov chain is rapidly mixing on the realizations of any degree sequence in a set of P -stable degree sequences (the rate of convergence depends on the set).…”