Sampling Regular Graphs and a Peer-to-Peer Network

Authors:
Colin Cooper;Martin Dyer;Catherine Greenhill
Affiliations:
Department of Computer Science, Kings College, London WC2R 2LS, UK (e-mail: ccooper@dcs.kcl.ac.uk);School of Computing, University of Leeds, Leeds LS2 9JT, UK (e-mail: dyer@comp.leeds.ac.uk);School of Mathematics, The University of New South Wales, Sydney NSW 2052, Australia (e-mail: csg@unsw.edu.au)
Venue:
Combinatorics, Probability and Computing
Year:
2007

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Abstract

This paper has two parts. In the first part we consider a simple Markov chain for d-regular graphs on n vertices, where d = d(n) may grow with n. We show that the mixing time of this Markov chain is bounded above by a polynomial in n and d. In the second part of the paper, a related Markov chain for d-regular graphs on a varying number of vertices is introduced, for even constant d. This is a model for a certain peer-to-peer network. We prove that the related chain has mixing time which is bounded above by a polynomial in N, the expected number of vertices, provided certain assumptions are met about the rate of arrival and departure of vertices.