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Sampling Regular Graphs and a Peer-to-Peer Network

Published online by Cambridge University Press:  01 July 2007

COLIN COOPER ,
MARTIN DYER  and
CATHERINE GREENHILL
COLIN COOPER
Affiliation:
Department of Computer Science, Kings College, London WC2R 2LS, UK (e-mail: [email protected])
MARTIN DYER
Affiliation:
School of Computing, University of Leeds, Leeds LS2 9JT, UK (e-mail: [email protected])
CATHERINE GREENHILL
Affiliation:
School of Mathematics, The University of New South Wales, Sydney NSW 2052, Australia (e-mail: [email protected])
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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.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 16 , Issue 4 , July 2007 , pp. 557 - 593
Copyright
Copyright © Cambridge University Press 2006

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