Catherine Greenhill
Published 2011-05-02, updated 2011-10-14Version 4
The switch chain is a well-known Markov chain for sampling directed graphs with a given degree sequence. While not ergodic in general, we show that it is ergodic for regular degree sequences. We then prove that the switch chain is rapidly mixing for regular directed graphs of degree d, where d is any positive integer-valued function of the number of vertices. We bound the mixing time by bounding the eigenvalues of the chain. A new result is presented and applied to bound the smallest (most negative) eigenvalue. This result is a modification of a lemma by Diaconis and Stroock, and by using it we avoid working with a lazy chain. A multicommodity flow argument is used to bound the second-largest eigenvalue of the chain. This argument is based on the analysis of a related Markov chain for undirected regular graphs by Cooper, Dyer and Greenhill, but with significant extension required.
Comments: 48 pages, 18 figures. Version 4: new bound on smallest eigenvalue of chain given instead (Lemma 1.5), transition procedure adjusted, definition of useful arcs generalised to correct the proof of Lemma 2.3, clarified the worked example in final section, some other minor changes
The mixing time of the swap (switch) Markov chains: a unified approach
The simple random walk and max-degree walk on a directed graph
Removal Lemmas with Polynomial Bounds
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