A Local Switch Markov Chain on Given Degree Graphs with Application in Connectivity of Peer-to-Peer Networks

Authors:
Tomas Feder;Adam Guetz;Milena Mihail;Amin Saberi
Affiliations:
Stanford University, USA;Stanford University, USA;Georgia Tech, USA;Stanford University, USA
Venue:
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Year:
2006

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Abstract

We study a switch Markov chain on regular graphs, where switches are allowed only between links that are at distance 2; we call this the Flip. The motivation for studying the Flip Markov chain arises in the context of unstructured peer-to-peer networks, which constantly perform such flips in an effort to randomize. We show that the Flip Markov chain on regular graphs is rapidly mixing, thus justifying this widely used peer-topeer networking practice. Our mixing argument uses the Markov chain comparison technique. In particular, we extend this technique to embedding arguments where the compared Markov chains are defined on different state spaces. We give several conditions which generalize our results beyond regular graphs.