Epidemics and Rumours in Complex Networks

130 Pages·2010·0.596 MB·English

by  Moez Draief, Laurent Massoulié

LONDONMATHEMATICALSOCIETYLECTURENOTESERIES
ManagingEditor:ProfessorM.Reid,MathematicsInstitute,
UniversityofWarwick,CoventryCV47AL,UnitedKingdom
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LondonMathematicalSocietyLectureNoteSeries:369
Epidemics and Rumours in
Complex Networks
MOEZ DRAIEF
ImperialCollege,London
LAURENT MASSOULIE´
ThomsonCorporateResearch,Paris
cambridge university press
Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,Sa˜oPaulo,Delhi
CambridgeUniversityPress
TheEdinburghBuilding,CambridgeCB28RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Informationonthistitle:www.cambridge.org/9780521734431
(cid:1)C M.DraiefandL.Massoulie´2010
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2010
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloguinginPublicationdata
Draief,Moez,1978–
Epidemicsandrumoursincomplexnetworks/MoezDraief,LaurentMassoulie´.
p. cm.–(LondonMathematicalSocietylecturenoteseries;369)
Includesbibliographicalreferencesandindex.
ISBN978-0-521-73443-1(pbk.)
1.Computersecurity–Mathematics. 2.Epidemics–Computersimulation.
3.Biologically-inspiredcomputing. 4.Graphtheory. 5.Probabilities.
I.Massoulie´,Laurent. II.Title. III.Series.
QA76.9.A25D78 2010
004.6–dc22 2009032956
ISBN978-0-521-73443-1Paperback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto
inthispublication,anddoesnotguaranteethatanycontentonsuch
websitesis,orwillremain,accurateorappropriate.
Contents
Introduction page1
PARTI SHAPELESSNETWORKS 5
1 Galton–Watsonbranchingprocesses 7
1.1 Introduction 7
1.2 Depth-firstexploration 9
1.3 Breadth-firstexploration 10
1.4 One-by-oneexploration 12
1.5 Chernoffboundsandtotalpopulationsize 15
1.6 Notes 17
1.7 Problems 17
2 Reed–FrostepidemicsandErdo˝s–Re´nyirandomgraphs 19
2.1 Introduction 19
2.2 Emergenceofthegiantcomponent 21
2.3 Thesubcriticalregime 23
2.4 Thesupercriticalregime 25
2.5 Thecriticalregime 29
2.6 Notes 33
3 ConnectivityandPoissonapproximation 35
3.1 Introduction 35
3.2 Variationdistance 35
3.3 TheStein–Chenmethod 37
3.4 EmergenceofconnectivityinE-Rgraphs 40
3.5 Notes 44
4 DiameterofErdo˝s–Re´nyigraphs 46
4.1 Introduction 46
vi Contents
4.2 DiameterofE-Rgraphs 48
4.3 Controlofneighbourhoodgrowth 49
4.4 Boundsforthediameter 51
4.5 Notes 53
5 Frommicroscopictomacroscopicdynamics 54
5.1 Introduction 54
5.2 Birthanddeathprocesses 55
5.3 Kurtz’stheorem 57
5.4 Notes 61
PARTII STRUCTUREDNETWORKS 63
6 Thesmall-worldphenomenon 65
6.1 Introduction 65
6.2 SmallworldaccordingtoStrogatzandWatts 66
6.3 SmallworldaccordingtoKleinberg 71
6.4 Notes 76
7 Powerlawsviapreferentialattachment 77
7.1 Introduction 77
7.2 Baraba´si-Albertrandomgraphs 78
7.3 Yuleprocess 83
7.4 Notes 86
8 Epidemicsongeneralgraphs 87
8.1 Introduction 87
8.2 Reed–Frostmodel 88
8.3 SISmodel 90
8.4 Epidemicsonspecificgraphs 100
8.5 Notes 107
9 Viralmarketingandoptimisedepidemics 108
9.1 Modelandmotivation 108
9.2 Algorithmichardness 109
9.3 Submodularstructureanditsconsequences 110
9.4 Viralmarketing 112
9.5 Notes 115
References 117
Index 122
Introduction
Theideaofmimickingthepropagationofbiologicalepidemicstoachievedif-
fusion of useful information was first proposed in the late 1980s, the decade
that also saw the appearance of computer viruses. Back then, these viruses
propagated by copies on floppy disks and caused much less harm than their
contemporary versions. But it was already noticed that they evolved and sur-
vived much as biological viruses do, a fact that prompted the idea of putting
thesefeaturestogood(ratherthanevil)use.Thefirstapplicationtobeconsid-
eredwassynchronisationofdistributeddatabases.
Interest in this paradigm received new impetus with the advent of peer-to-
peer systems, online social systems and wireless mobile ad hoc networks in
the early 2000s. All these scenarios feature a complex network with poten-
tially evolving connections. In such large-scale dynamic environments, epi-
demicdiffusionofinformationisespeciallyappealing:itisdecentralised,and
itreliesonrandomiseddecisionswhichcanproveasefficientascarefullymade
decisions.Detailedaccountsofepidemicalgorithmscanbefoundinpapersby
Birman et al. [12] and Eugster et al. [33]. Their applications are manifold.
They can be used to perform distributed computation of global statistics in a
spatially extended environment (e.g. mean temperature seen by a collection
ofsensors),toperform real-timedelivery ofvideo data streams(e.g.tousers
receivingliveTVviapeer-to-peersystemsovertheinternet)andtopropagate
updatesofdynamiccontent(e.g.tomobilephoneuserswhosephoneoperating
systemrequirespatchingagainstvulnerabilities).
Thisbookismeantasanintroductionforappliedmathematiciansandcom-
puterscientiststothestudyofepidemicpropagationovercomplexnetworks.
The book’s firstpurpose istoprovide the reader withan accessible introduc-
tiontotheelementarymodelsofepidemicpropagation,anddevelopanunder-
standing of the basic phase transition phenomena (also called threshold phe-
nomena) that are typical of epidemic behaviour. To this end it introduces the
2 Introduction
relevantgenericmodels,analyticaltools,andmathematicalresults.Thisbook
alsoaimstoexplaintheroleplayedbynetworktopologyinthepropagationof
information.
Thisbookcanbeusedbothtoaccompanycoursesforgraduatestudentsin
computer science and applied probability and to provide an overview of the
probabilistic techniques widely used to study random processes on complex
networks.ItevolvedfromagraduatecoursegivenbytheauthorsatCambridge
Universityfrom2005to2007.Themathematicalprerequisitesarereasonable
maturity in probability theory at undergraduate level. The main tools intro-
ducedinthetextconsistoflargedeviationsinequalities,couplingofstochastic
processesandPoissonapproximation.
Readerswhowanttodelvedeeperwillfindthoroughtreatmentsofrandom
graphsinBolloba´s[13]andJansonetal.[42].TherecentbookbyDurrett[28]
alsoexpandsonsomeofthematerialhere,inseveraldirections,inparticular
onthebehaviourofrandomwalksongraphs.AnderssonandBritton[3]deals
with epidemic modelling in the biological context. Specific suggestions for
furtherreadingarealsoprovidedattheendofeachchapter.
Atourof thebook
Thetextisdividedintotwodistinctbutrelatedparts:shapelessnetworksand
structurednetworks.Thefirstpart(Chapters1–5)presentstechniquesforan-
alysing “homogeneous mixing” epidemic processes. In this setting the large
population in which the disease spreads has no particular (or totally random,
symmetric)topologicalstructure;thatis,eachindividualisthe“neighbour”of
every other individual. The infection can therefore spread from any infected
individual to any healthy one. In particular, this part of the book introduces
branching processes, Erdo˝s–Re´nyi random graphs and so-called Reed–Frost
epidemics. Issues such as ultimate outreach and time to global infection are
considered.
The second part (Chapters 6–9) covers recent results on the spread of epi-
demics in structured networks wherein individuals interact with a limited set
of neighbours, and where the corresponding topology can exhibit rich struc-
ture.Itintroducesmodelsofsuchtopologicallystructurednetworks,including
power-lawrandomgraphsandnavigablesmall-worldgraphs.Itgivesexplana-
tory models for the emergence of such structures, and addresses the impact
ofstructureonthebehaviour ofepidemics.Italsotouchesuponthealgorith-
mic issue of maximising outbreak as a function of the initial infectives. In
whatfollowswedescribethecontentofthebookinmoredetailandgivesome
Introduction 3
perspectiveontherelevanceofeachtopictotheanalysisofepidemicprocesses
onnetworks.
Chapter1reviewselementaryresultsontheGalton–Watsonbranchingpro-
cess. Branching processes arise naturally in the study of epidemics, as they
provideanaccuratedescriptionofepidemicbehaviourinlarge,shapelesspop-
ulations, at least in the early stages of dissemination. They are arguably the
simplest class of model that exhibits a phase transition: the phenomenon by
which an infinitesimal variation of microscopic characteristics (here, the off-
spring distribution) can lead to a macroscopic change in system behaviour
(here,infinitesurvivaloftheepidemic).
TheresultsofChapter1arethenexploitedtoestablishasimilarphasetransi-
tioninaclassicalepidemicmodel,namelytheReed–Frostmodel.Inparticular,
Chapters 2–4 exploit a parallel between the Reed–Frost epidemic model and
Erdo˝s–Re´nyi random graphs to gain insight into the behaviour of the former.
Chapter 2 describes a phase transition appearing in the fraction of ultimately
infectedindividuals.Chapter3identifiesunderwhichparameterrangesanepi-
demic starting from one infected individual eventually spreads to the whole
population.InChapter4wederiveanupperboundonthetimeneededforthe
epidemictoreachthewholepopulation.Chapter5describesasettinginwhich
the microscopic behaviour of the epidemic (how it spreads randomly in the
population) canbeapproximated byasetofdifferentialequationsdescribing
themacroscopicormean-fielddynamicsofthesystem.Italsointroducessome
classicalmodelsofepidemicspread.
The chapters of the second part of the book cover more advanced topics.
Theseusesometechniquesintroducedinthefirstpart,inparticularlargedevia-
tionsinequalities,butareotherwiselargelyself-contained.Chapter6proposes
two models and corresponding analyses of the small-world phenomenon, in-
troducingthenotionofnavigabilityofagraph.Chapter7focusesonanother
phenomenon observed in many real-world networks, namely the power-law
distribution of the degree sequence (or number of neighbours) of the graph.
The most salient feature of a sequence with a power-law distribution is that
ittypicallycontains sampleswithvery highvalues.Incontrast,classicalnet-
work models such as the Erdo˝s–Re´nyi graph have degree sequences highly
concentratedaroundtheirmean.Chapter7describesprocessesforgenerating
such power laws together with their analysis. Chapter 8 covers recent results
on the threshold behaviour of classical models of epidemics on general net-
works,identifyingthresholdswithgraphpropertiesoftheunderlyingnetwork
topology.Finally,Chapter9approachesthealgorithmicoptimisationproblem
of maximising the spread of an epidemic on a general network by targeting
nodesthatarelikelytoyieldalargecascadeofinfections.