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Unformatted text preview: Sampling regular graphs and a peer-to-peer network (Extended Abstract) Colin Cooper * , Martin Dyer and Catherine Greenhill Abstract We consider a simple Markov chain for d-regular graphs on n vertices, and show that the mixing time of this Markov chain is bounded above by a polynomial in n and d . A related Markov chain for d-regular graphs on a varying number of vertices is introduced, for even degree d . We use this to model a certain peer-to-peer network structure. We prove that the related chain has mixing time which is bounded by a polynomial in N , the expected number of vertices, under reasonable assumptions about the arrival and departure process. 1 Introduction In this paper we prove rapid mixing of a simple and natural Markov chain for generating random regular graphs. We use this to model a peer-to-peer network implementation of Bourassa and Holt , which is based on random walks. Bourassa and Holt conjectured that their network quickly becomes a random regular graph, and consequently has good properties with high probability. Their claim was supported by computer simulation. We extend our Markov chain analysis to give theoretical justification of their conjecture. 1.1 Sampling regular graphs The problem of sam- pling graphs with a given degree sequence (on a fixed number of vertices) has been well studied, especially the case of sampling d-regular graphs on n vertices, where d = d ( n ) may grow with n . Expected polynomial- time algorithms for uniform generation were described by Bollob as , Frieze , McKay and Wormald  and Steger and Wormald  (for degrees O ( log n ), o ( n 1 / 5 ), o ( n 1 / 3 ) and o ( n 1 / 11 ), respectively). Simpler al- gorithms can be used if we are content to sample approx- imately uniformly. Tinhofer  describes one such but does not bound how far away the resulting probability distribution is from uniform. Jerrum and Sinclair  * firstname.lastname@example.org , Department of Computer Science, Kings College, London WC2R 2LS, UK. email@example.com , School of Computing, University of Leeds, Leeds LS2 9JT, UK. firstname.lastname@example.org , School of Mathematics, UNSW, Sydney 2052, Australia. give a polynomial time approximately uniform gener- ator with running time polynomial in n and log( - 1 ), for any regular degree sequence d = d ( n ) n/ 2 (and by complementation, higher degrees can be handled). Their algorithm uses a Markov chain which samples per- fect and near-perfect matchings of a related graph, and can be extended to certain non-regular degree sequences (see ). Moreover, the analysis of Jerrum, Sinclair and Vigoda  shows that the approach of  gives a polynomial time approximately uniform generator for bipartite graphs with any given degree sequence (see [10, Section 6])....
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- Spring '07