Random Graphs.pdf

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CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 73 EDITORIAL BOARD B. BOLLOBAS, W. FULTON, A. KATOK, F. KIRWAN, P. SARNAK RANDOM GRAPHS

J.L. Alperin Local representation theory P. Koosis The logarithmic integral II J.E. Humphreys Reflection groups and Coxeter groups P. Wojtaszczyk Banach spaces.1or analysts J.E. Gilbert & M.A.M. Murray Clifford algebras and Dirac operators in harmonic analysis J.-P. Kahane Some random series of functions, 2nd edition H. Cohn Introduction to the construction ofclassfields J. Lambek & P.J. Scott Introduction to higher-order categorical logic H. Matsumura Commutative ring theory C.B. Thomas Characteristic classes and the cohomology of/finite groups Already published I W.M.L. Holcombe Algebraic automata theory 2 K. Petersen Ergodic theory 3 P.T. Johnstone Stone spaces 4 W.H. Schikhof Ultrametric calculus 5 6 7 8 9 10 M. Aschbacher Finite group theory I1 12 P. Koosis The logarithmic integral I 13 A. Pietsch Eigenvalues and s-numbers 14 S.J. Patterson An introduction to the theory of the Riemann zeta-function 15 H.J. Baues Algebraic homotopy 16 V.S. Varadarajan Introduction to harmonic analysis on semisimple Lie groups 17 W. Dicks & M. Dunwoody Groups acting on graphs 18 L.J. Corwin & F.P. Greenleaf Representations of nilpotent Lie groups and their applications 19 R. Fritsch & R. Piccinini Cellular structures in topology 20 H. Klingen Introductory lectures on Siegel modular forms 21 22 M.J. Collins Representations and characters of finite groups 24 H. Kunita Stochastic flows and stochastic differential equations 25 26 27 A. Frohlich & M.J. Taylor Algebraic number theory 28 K. Goebel & W.A. Kirk Topics in metric fixed point theory 29 30 D.J. Benson Representations and cohomology I 31 D.J. Benson Representations and cohomology II 32 C. Allday & V. Puppe Cohomological methods in transformation groups 33 C. Soul& et al Lectures on Arakelov geometry 34 A. Ambrosetti & G. Prodi A primer of nonlinear analysis 35 37 Y. Meyer Wavelets and operators I 38 C. Weibel An introduction to homological algebra 39 W. Bruns & J. Herzog Cohen-Macaulay rings 40 V. Snaith Explicit Brauer induction 41 G. Laumon Cohomology of Drinfield modular varieties I 42 E.B. Davies Spectral theory and differential operators 43 44 45 R. Pinsky Positive harmonicfunctions and diffusion 46 G. Tenenbaum Introduction to analytic and probabilistic number theory 47 C. Peskine An algebraic introduction to complex projectile geometry I 48 Y. Meyer & R. Coifman Wavelets and operators II 49 R. Stanley Enumerative combinatories 50 51 M. Audin Spinning tops 52 V. Jurdjevic Geometric control theory 53 H. Voelklein Groups as Galois groups J. Le Potier Lectures on vector bundles 54 55 D. Bump Automorphic.1orms 56 G. Laumon Cohomology of Drinfeld modular varieties 11 57 D.M. Clarke & B.A. Davey Natural dualities ibr the working algebraist 59 P. Taylor Practical foundations of mathematics 60 M. Brodmann & R. Sharp Local cohomology 61 62 R. Stanley Enumerative combinatorics II 64 68 Ken-iti Sato LUvy processes and infinitely divisible distributions 71 R. Blei Anal'ysis in integer and tiactional dimensions J.D. Dixon, M.P.F. Du Sautoy, A. Mann & D. Segal Analytic pro-p groups, 2nd edition J. Jost & X. Li-Jost Calculus of variations J. Palis & F. Takes Hyperbolicity, stability and chaos at homoclinic bifiarcations J. Diestel, H. Jarchow & A. Tonge Absolutely summing operators P. Mattila Geometry of sets and measures in euclidean spaces 1. Porteous Clifford algebras and the classical groups

Random Graphs Second Edition BELA BOLLOBAS University of Memphis and Trinity College, Cambridge CAMBRIDGE UNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, VIC 3166, Australia Ruiz de Alarc6n 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org © Cambridge University Press 2001 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published 1985 by Academic Press Second edition published 2001 by Cambridge University Press Printed in the United Kingdom at the University Press, Cambridge Typeface Times 1/14pt System LATEX2e [UPH] A catalogue record Jor this book is available from the British Library Library of Congress Cataloguing in Publication data Bollob~s, Bela Random graphs / BMla Bollobis. - 2nd ed. p. cm. - (Cambridge mathematical library) Includes bibliographical references and index. ISBN 0 521 79722 5 (pbk.) 1. Random graphs. 1. Title II. Series QA166.17.B66 2001 511.'5-dc2l 00-068952 IBSN 0 521 80920 7 hardback ISBN 0 521 79722 5 paperback

To Gabriella and Mark

'I learn so as to be contented.' After the inscription on 'Tsukubai', the stone wash-basin in Ryoanji temple.

Contents Probability Theoretic Preliminaries Notation and Basic Facts Some Basic Distributions Preface Notation 1 1.1 1.2 1.3 Normal Approximation 1.4 1.5 2 2.1 2.2 2.3 2.4 Inequalities Convergence in Distribution Models of Random Graphs The Basic Models Properties of Almost All Graphs Large Subsets of Vertices Random Regular Graphs 3 3.1 3.2 3.3 3.4 3.5 The Degree Sequence The Distribution of an Element of the Degree Sequence Almost Determined Degrees The Shape of the Degree Sequence Jumps and Repeated Values Fast Algorithms for the Graph Isomorphism Problem Small Subgraphs Strictly Balanced Graphs 4 4.1 4.2 Arbitrary Subgraphs 4.3 Poisson Approximation 5 5.1 5.2 The Evolution of Random Graphs-Sparse Components Trees of Given Sizes As Components The Number of Vertices on Tree Components vii page xi xvii 1 1 5 9 15 25 34 34 43 46 50 60 60 65 69 72 74 78 79 85 91 96 96 102

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