OXFORD STUDIES IN PROBABILITYManaging Editor L. C. G. ROGERS Editorial Board P. BAXENDALE P. GREENWOOD F. P. KELLY J.-F. LE GALL E. PARDOUX D. WILLIAMS OXFORD STUDIES IN PROBABILITY 1. F. B. Knight: Foundations of the prediction process 2. A. D. Barbour, L. Holst, and S. Janson: Poisson approximation 3. J. F. C. Kingman: Poisson processes 4. V. V. Petrov: Limit theorems of probability theory 5. M. Penrose: Random geometric graphs Random Geometric Graphs MATHEW PENROSE University of Bath Great Clarendon Street, Oxford OX26DP Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Bangkok Buenos Aires Cape Town Chennai Dar es Salaam Delhi Hong Kong Istanbul Kaarachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi São Paulo Shanghai Taipei Tokyo Toronto Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Mathew Penrose, 2003 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2003 Reprinted 2004 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organisation. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above. You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer. A catalogue record for this title is available from the British Library (Data available) ISBN 0 19 850626 0 1098765432 PREFACE Random geometric graphs are easily described. A set of points is randomly scattered over a region of space according to some probability distribution, and any two points separated by a distance less than a certain specified value are connected by an edge. This book is an attempt to describe the mathematical theory of the resulting graphs and to give a flavour of some of the applications. I started to contemplate writing this book in the summer of 1998, when it occurred to me, firstly, that random geometric graphs are a natural alternative to the classical Erdös-Rényi random graph schemes, and secondly, that an account of them in monograph form could provide a useful collection of techniques in geometric probability. Although the project has taken longer than expected, I hope these assumptions retain their force, and that resulting book can be useful both to mathematicians with an interest in geometrical probability, and to practitioners in various subjects including communications engineering, classification, and computer science, wishing to see how far the mathematical theory has progressed. This monograph is self-contained, and could be used as the basis of a graduatelevel course (or courses). An overview of the topics covered appears in Section 1.4. The reader will find proofs in the text, and will find prior knowledge of the probabilistic concepts briefly reviewed in Section 1.6 to be useful. Other preliminaries are minimal; a small number of results in adjacent subjects such as measure theory, topology, and graph theory are used and are stated without proof in the text. With regard to citations, I have tried to provide the most useful references for the reader, without always giving full historical details. Thus, there may be some results for which the reader is referred to some standard text, rather than to the original work containing those results. Likewise, any claims made regarding the novelty of work in this book are necessarily subject to the limits of my own knowledge, and I apologize in advance to the authors of any relevant works which I have failed to mention through ignorance. References to related work are generally given in the notes the end of each chapter, along with relevant open problems. It is a pleasure to thank the following people and institutions for their assistance. The Fields Institute in Toronto provided hospitality for ten weeks in the spring of 1999. Jordi Petit provided the software used to produce diagrams of random geometric graphs in this book. Pauline Coolen-Schrner, Joseph Yukich, and Andrew Wade read and commented on earlier drafts of some of the chapters; vi PREFACE however, I wish to take full credit myself for any remaining errors, which I intend to monitor on a web page if and when they come to light. Durham, UK M.P. September 2002 CONTENTS Notation xi 1 Introduction 1 1.1 Motivation and history 1 1.2Statistical background 4 1.3 Computer science background 7 1.4 Outline of results 9 1.5 Some basic definitions 11 1.6 Elements of probability 14 1.7 Poissonization 18 1.8 Notes and open problems 21 2 Probabilistic ingredients 22 2.1 Dependency graphs and Poisson approximation 22 2.2 Multivariate Poisson approximation 25 2.3 Normal approximation 27 2.4 Martingale theory 33 2.5 De-Poissonization 37 2.6 Notes 46 3 Subgraph and component counts 47 3.1 Expectations 48 3.2Poisson approximation 52 3.3 Second moments in a Poisson process 55 3.4 Normal approximation for Poisson processes 60 3.5 Normal approximation: de-Poissonization 65 3.6 Strong laws of large numbers 69 3.7 Notes 73 4 Typical vertex degrees 74 4.1 The setup 75 4.2Laws of large numbers 76 4.3 Asymptotic covariances 78 4.4 Moments for de-Poissonization 82 4.5 Finite-dimensional central limit theorems 87 4.6 Convergence in Skorohod space 91 4.7 Notes and open problems 93 5 Geometrical ingredients 95 5.1 Consequences of the Lebesgue density theorem 95 5.2Covering, packing, and slicing 97 viii CONTENTS 5.3 The Brunn–Minkowski inequality 102 5.4 Expanding sets in the orthant 104 6 Maximum degree, cliques, and colourings 109 6.1 Focusing 110 6.2Subconnective laws of large numbers 118 6.3 More laws of large numbers for maximum degree 120 6.4 Laws of large numbers for clique number 126 6.5 The chromatic number 130 6.6 Notes and open problems 134 7 Minimum degree: laws of large numbers 136 7.1 Thresholds in smoothly bounded regions 136 7.2Strong laws for thresholds in the cube 145 7.3 Strong laws for the minimum degree 151 7.4 Notes 154 8 Minimum degree: convergence in distribution 155 8.1 Uniformly distributed points I 156 8.2Uniformly distributed points II 160 8.3 Normally distributed points I 167 8.4 Normally distributed points II 173 8.5 Notes and open problems 176 9 Percolative ingredients 177 9.1 Unicoherence 177 9.2Connectivity and Peierls arguments 177 9.3 Bernoulli percolation 180 9.4 k-Dependent percolation 186 9.5 Ergodic theory 187 9.6 Continuum percolation: fundamentals 188 10 Percolation and the largest component 194 10.1 The subcritical regime 195 10.2Existence of a crossing component 200 10.3 Uniqueness of the giant component 205 10.4 Sub-exponential decay for supercritical percolation 210 10.5 The second-largest component 216 10.6 Large deviations in the supercritical regime 220 10.7 Fluctuations of the giant component 224 10.8 Notes and open problems 230 11 The largest component for a binomial process 231 11.1 The subcritical case 231 11.2The supercritical case on the cube 234 11.3 Fractional consistency of single-linkage clustering 240 11.4 Consistency of the RUNT test for unimodality 247 CONTENTS ix 11.5 Fluctuations of the giant component 252 11.6 Notes and open problems 257 12 Ordering and partitioning problems 259 12.1 Background on layout problems 259 12.2 The subcritical case 262 12.3 The supercritical case 268 12.4 The superconnectivity regime 275 12.5 Notes and open problems 279 13 Connectivity and the number of components 281 13.1 Multiple connectivity 282 13.2Strong laws for points in the cube or torus 283 13.3 SLLN in smoothly bounded regions 289 13.4 Convergence in distribution 295 13.5 Further results on points in the cube 302 13.6 Normally distributed points 306 13.7 The component count in the thermodynamic limit 309 13.8 Notes and open problems 316 References 318 Index 328 This page intentionally left blank NOTATION In this list, section numbers refer to the places where the notation is defined. If only a chapter number is given, the notation is introduced at the start of that chapter. Some items of notation whose use is localized are omitted from this list. xii NOTATION Symbol Usage Section 0 The origin of Rd 1.5 1 Indicator random variable or indicator function of A 1.6 A B(x;r ) Ball of radius r centred at x 1.5 B*(x;r , η, e) Segment of the ball of radius r centred at x 5.2 B (x;r ) Segment of ball of radius r centred at x 8.3 ▵ B(s) The box [-s/2, s/2]d 9.6 B (m) Lattice box of side m 9.2 z B′ (n) Lattice box of side n centred at the origin 10.5 z C Bernoulli process (random subset of Zd) induced by Zp 9.3 p Bi(n, p) Binomial random variable 1.6 C 13.4 The unit cube C(G), C ,C′ Clique numbers of G,G(X ; r ), and G(P ; r ), respectively 6 n n n n n n c.c.