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Random Graph Dynamics
Thetheoryofrandomgraphsbeganinthelate1950sinseveralpapersby
Erdo¨sandRe´nyi.Inthelatetwentiethcentury,thenotionofsixdegreesof
separation,meaningthatanytwopeopleontheplanetcanbeconnected
by a short chain of people who know each other, inspired Strogatz and
Wattstodefinethesmallworldrandomgraphinwhicheachsiteiscon-
nectedtokcloseneighbors,butalsohaslong-rangeconnections.Atabout
thesametime,itwasobservedinhumansocialandsexualnetworksand
ontheInternetthatthenumberofneighborsofanindividualorcomputer
hasapowerlawdistribution.ThisinspiredBaraba´siandAlberttodefine
thepreferentialattachmentmodel,whichhastheseproperties.Thesetwo
papers have led to an explosion of research. While this literature is ex-
tensive, many of the papers are based on simulations and nonrigorous
arguments.Thepurposeofthisbookistouseawidevarietyofmathe-
matical argument to obtain insights into the properties of these graphs.
Auniquefeatureofthisbookistheinterestinthedynamicsofprocess
takingplaceonthegraphinadditiontotheirgeometricproperties,such
asconnectednessanddiameter.
Rick Durrett is a Professor of Mathematics at Cornell University. He
receivedhisPh.D.inOperationsResearchfromStanfordin1976.After
9 years at UCLA, he moved to Cornell, where his research turned to
applicationsofprobability,firsttoecologyand,morerecently,togenetics.
Hehaswrittenmorethan150papers,sixotherbooks,andhas33academic
descendants.
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CAMBRIDGE SERIES IN STATISTICAL AND PROBABILISTIC MATHEMATICS
EditorialBoard:
R.Gill,DepartmentofMathematics,UtrechtUniversity
B.D.Ripley,DepartmentofStatistics,UniversityofOxford
S.Ross,EpsteinDepartmentofIndustrial&SystemsEngineering,Universityof
SouthernCalifornia
B.W.Silverman,St.Peter’sCollege,Oxford
M.Stein,DepartmentofStatistics,UniversityofChicago
This series of high-quality upper-division textbooks and expository monographs covers
all aspects of stochastic applicable mathematics. The topics range from pure and applied
statistics to probability theory, operations research, optimization, and mathematical pro-
gramming. The books contain clear presentations of new developments in the field and
alsoofthestateoftheartinclassicalmethods.Whileemphasizingrigoroustreatmentof
theoreticalmethods,thebooksalsocontainapplicationsanddiscussionsofnewtechniques
madepossiblebyadvancesincomputationalpractice.
AlreadyPublished
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2. MarkovChains,byJ.Norris
3. AsymptoticStatistics,byA.W.vanderVaart
4. WaveletMethodsforTimeSeriesAnalysis,byDonaldB.PercivalandAndrew
T.Walden
5. BayesianMethods,byThomasLeonardandJohnS.J.Hsu
6. EmpiricalProcessesinM-Estimation,bySaravandeGeer
7. NumericalMethodsofStatistics,byJohnF.Monahan
8. AUser’sGuidetoMeasureTheoreticProbability,byDavidPollard
9. TheEstimationandTrackingofFrequency,byB.G.QuinnandE.J.Hannan
10. DataAnalysisandGraphicsUsingR,byJohnMaindonaldandJohnBraun
11. StatisticalModels,byA.C.Davison
12. SemiparametricRegression,byD.Ruppert,M.P.Wand,andR.J.Carroll
13. ExercisesinProbability,byLoicChaumontandMarcYor
14. StatisticalAnalysisofStochasticProcessesinTime,byJ.K.Lindsey
15. MeasureTheoryandFiltering,byLakhdarAggounandRobertElliott
16. EssentialsofStatisticalInference,byG.A.YoungandR.L.Smith
17. ElementsofDistributionTheory,byThomasA.Severini
18. StatisticalMechanicsofDisorderedSystems,byAntonBovier
19. TheCoordinate-FreeApproachtoLinearModels,byMichaelJ.Wichura
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Random Graph Dynamics
RICK DURRETT
CornellUniversity
CAMBRIDGEUNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB28RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521866569
© Rick Durrett 2007
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2006
ISBN-13 978-0-511-34983-6 eBook (NetLibrary)
ISBN-10 0-511-34983-1 eBook (NetLibrary)
ISBN-13 978-0-521-86656-9 hardback
ISBN-10 0-521-86656-1 hardback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
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Contents
Preface pagevii
1. Overview 1
1.1 IntroductiontotheIntroduction 1
1.2 Erdo¨s,Re´nyi,Molloy,andReed 3
1.3 SixDegrees,SmallWorlds 7
1.4 PowerLaws,PreferentialAttachment 11
1.5 EpidemicsandPercolation 15
1.6 PottsModelsandtheContactProcess 18
1.7 RandomWalksandVoterModels 20
1.8 CHKNSModel 21
2. Erdo¨s–Re´nyiRandomGraphs 27
2.1 BranchingProcesses 27
2.2 ClusterGrowthasanEpidemic 34
2.3 ClusterGrowthasaRandomWalk 37
2.4 DiameteroftheGiantComponent 43
2.5 CLTfortheGiantComponent 46
2.6 CombinatorialApproach 50
2.7 CriticalRegime 56
2.8 ThresholdforConnectivity 62
3. FixedDegreeDistributions 70
3.1 DefinitionsandHeuristics 70
3.2 ProofofPhaseTransition 75
3.3 SubcriticalEstimates 82
3.4 Distances:FiniteVariance 84
3.5 Epidemics 85
4. PowerLaws 90
4.1 Baraba´si-AlbertModel 90
4.2 RelatedModels 93
4.3 MartingalesandUrns 99
4.4 Scale-FreeTrees 105
4.5 Distances:PowerLaws2<β <3 110
v
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vi Contents
4.6 Diameter:Baraba´si-AlbertModel 116
4.7 Percolation,Resilience 121
4.8 SISEpidemic 125
5. SmallWorlds 132
5.1 WattsandStrogatzModel 132
5.2 PathLengths 134
5.3 Epidemics 140
5.4 IsingandPottsModels 144
5.5 ContactProcess 148
6. RandomWalks 153
6.1 SpectralGap 153
6.2 Conductance 156
6.3 FixedDegreeDistribution 159
6.4 PreferentialAttachmentGraph 164
6.5 ConnectedErdo¨s–Re´nyiGraphs 169
6.6 SmallWorlds 171
6.7 OnlyDegrees2and3 175
6.8 HittingTimes 177
6.9 VoterModels 181
7. CHKNSModel 187
7.1 HeuristicArguments 187
7.2 ProofofthePhaseTransition 190
7.3 SubcriticalEstimates 193
7.4 Kosterlitz-ThoulessTransition 197
7.5 ResultsattheCriticalValue 200
References 203
Index 211
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Preface
Chapter 1 will explain what this book is about. Here I will explain why I chose
towritethebook,howitiswritten,whenandwheretheworkwasdone,andwho
helped.
Why.ItwouldmakeagoodstoryifIwasinspiredtowritethisbookbyanimage
of Paul Erdo¨s magically appearing on a cheese quesadilla, which I later sold for
thousandsofdollarsoneBay.However,thatisnottrue.Thethreemaineventsthat
ledtothisbookwere(i)theuseofrandomgraphsinthesolutionofaproblemthat
waspartofNathanaelBerestycki’sthesis;(ii)atalkthatIheardSteveStrogatzgive
on the CHKNS model, which inspired me to prove some rigorous results about
theirmodel;and(iii)abookreviewIwroteonthebooksbyWattsandBaraba´sifor
theNoticesoftheAmericanMathSociety.
The subject of this book was attractive for me, since many of the papers were
outside the mathematics literature, so the rigorous proofs of the results were, in
somecases,interestingmathematicalproblems.Inaddition,sinceIhadworkedfor
anumberofyearsonthepropertiesofstochasticspatialmodelsonregularlattices,
therewasthenaturalquestionofhowthebehaviorofthesesystemschangedwhen
one introduced long-range connections between individuals or considered power
lawdegreedistributions.Bothofthesemodificationsarereasonableifoneconsiders
thespreadofinfluenzainatownwherechildrenbringthediseasehomefromschool,
or the spread of sexually transmitted diseases through a population of individuals
thathaveawidelyvaryingnumberofcontacts.
How. The aim of this book is to introduce the reader to the subject in the same
way that a walk through Musee´ d’Orsay exposes the visitor to the many styles
of impressionism. We will choose results to highlight the major themes, but we
willnotexamineindetaileveryvariationofpreferentialattachmentthathasbeen
studied. We will concentrate on the ideas, giving the interesting parts of proofs,
and referring the reader to the literature for the missing details. As Tom Liggett
saidafterhehadwrittenhisbookInteractingParticleSystems,thereisnopointin
havingabookthatisjustaunionofpapers.
vii
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viii Preface
Throughout we approach the subject with a probabilistic viewpoint. One prag-
maticreasonisthat,intheabsenceoffuturisticproceduresliketheoneTomCruise’s
characterhadinTheMinorityReport,thesearetheonlyeyesthroughwhichIcan
view the world. For connections to computer algorithms and their analysis, you
will have to ask someone who knows that story. In addition, we will emphasize
topics not found in other mathematical books. I have nothing to add to the treat-
mentofrandomregulargraphsinJanson,Luczak,andRucin´sky(2000),soIwill
not spend much time on this special case of random graphs with a fixed degree
distribution.TheclassicaltheoryofrandomgraphsofErdo¨sandRe´nyiiscovered
nicely by Bolloba´s (2001), so I will keep treatment to the minimum necessary to
prepareformorecomplicatedexamples.
Several reviewers lobbied for an introductory chapter devoted to some of the
tools: branching processes, large deviations, martingales, convergence of Markov
chains, almost exponentiality of waiting times, etc. Personally I do not think it
is necessary (or even desirable) to read the entire Kama Sutra before having sex
the first time, so instead I approach the book as I do my Ph.D. students’ research
projects.Wewillstartlookingatthesubjectandlearnaboutthetoolsastheyarise.
Readers who find these interruptions distracting can note the statement and skip
theproof,advicethatcanbeappliedtomostoftheresultsinthebook.
When and where. The first version of these notes was written for a graduate
seminar, which I gave on the topic at Cornell in the fall of 2004. On the Monday
of the first full week of the semester, as I sat at my desk in Malott Hall, my back
began to hurt, and I thought to myself that I had worked too hard on notes over
the weekend. Two months, new desk chairs, a variety of drugs, physical therapy,
and a lot of pain later, an MRI showed my problem was an infected disk. I still
remembertheradiologist’sexcitingwords:“Wecan’tletyougohomeuntilweget
in touch with your doctor.” Four days in the hospital and four months of sipping
coffeeeverymorningwhileanIV-drippedantibioticsintomyarm,thebugsinme
had been defeated, and I was back to almost normal. Reading papers on random
graphs, figuring out proofs, and organizing the material while lying on my bed
helpedmetogetthroughthatordeal.
Inthesummerof2005,Irevisedthenotesandaddednewmaterialinpreparation
forsixlecturesonthistopic,whichIgaveatthefirstCornellProbabilitySummer
School.Severalmoreiterationsofpolishingfollowed.Whenmybraintoldmethat
themanuscriptwasingreatshape,severalpaidreviewersshowedmethattherewas
stillworktodo.Finally,amonthinParisatE´coleNormaleSupe´rieureinFebruary
2006 provided a pleasant setting for finishing the project. I would like to thank
Jean Francois LeGall for the invitation to visit Paris, radio station 92.1 FM for
providingthebackgroundmusicwhileItypedinmyapartment,andtherestaurants
on and around Rue Mouffetard for giving me something to look forward to at the
endoftheday.
