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Rapid mixing of the switch Markov chain for strongly stable degree sequences
Author(s) -
Amanatidis Georgios,
Kleer Pieter
Publication year - 2020
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20949
Subject(s) - markov chain , degree (music) , mixing (physics) , markov chain mixing time , markov chain monte carlo , mathematics , embedding , chain (unit) , sequence (biology) , discrete mathematics , stability (learning theory) , combinatorics , statistical physics , markov process , variable order markov model , markov model , computer science , monte carlo method , statistics , physics , artificial intelligence , genetics , machine learning , quantum mechanics , astronomy , biology , acoustics
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 show that the switch chain for sampling simple undirected graphs with a given degree sequence is rapidly mixing when the degree sequence is so‐called strongly stable. Strong stability is satisfied by all degree sequences for which the switch chain was known to be rapidly mixing based on Sinclair's multicommodity flow method up until a recent manuscript of Erdős and coworkers in 2019. Our approach relies on an embedding argument, involving a Markov chain defined by Jerrum and Sinclair in 1990. This results in a much shorter proof that unifies (almost) all the rapid mixing results for the switch chain in the literature, and extends them up to sharp characterizations of P‐stable degree sequences. In particular, our work resolves an open problem posed by Greenhill and Sfragara in 2017.