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RandomGraphsandNetworks:AFirstCourse Networkssurroundus,fromsocialnetworkstoprotein–proteininteractionnetworks within the cells of our bodies. The theory of random graphs provides a necessary frameworkforunderstandingtheirstructureanddevelopment. This text provides an accessible introduction to this rapidly expanding subject. It coversallthebasicfeaturesofrandomgraphs–componentstructure,matchingsand Hamiltoncycles,connectivity,andchromaticnumber–beforediscussingmodelsof real-world networks, including intersection graphs, preferential attachment graphs, andsmall-worldmodels. Basedontheauthors’ownteachingexperience,RandomGraphsandNetworks:A FirstCoursecanbeusedasatextbookforaone-semestercourseonrandomgraphs andnetworksatadvancedundergraduateorgraduatelevel.Thetextincludesnumer- ous exercises, with a particular focus on developing students’ skills in asymptotic analysis. More challenging problems are accompanied by hints or suggestions for furtherreading. Alan Frieze is Professor in the Department of Mathematical Sciences at Carnegie MellonUniversity.Hehasauthoredalmost400publicationsintopjournalsandwas aplenaryspeakeratthe2014InternationalCongressofMathematicians. MichałKarońskiisProfessorEmeritusintheFacultyofMathematicsandComputer ScienceatAdamMickiewiczUniversity,wherehefoundedtheDiscreteMathematics group. He served as Editor-in-Chief of Random Structures and Algorithms for 30 years. Published online by Cambridge University Press Published online by Cambridge University Press Random Graphs and Networks: A First Course ALAN FRIEZE CarnegieMellonUniversity MICHAŁ KAROŃSKI AdamMickiewiczUniversity Published online by Cambridge University Press ShaftesburyRoad,CambridgeCB28EA,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre, NewDelhi–110025,India 103PenangRoad,#05–06/07,VisioncrestCommercial,Singapore238467 CambridgeUniversityPressispartofCambridgeUniversityPress&Assessment, adepartmentoftheUniversityofCambridge. WesharetheUniversity’smissiontocontributetosocietythroughthepursuitof education,learningandresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781009260282 DOI:10.1017/9781009260268 © AlanFriezeandMichałKaroński2023 Thispublicationisincopyright.Subjecttostatutoryexceptionandtotheprovisions ofrelevantcollectivelicensingagreements,noreproductionofanypartmaytake placewithoutthewrittenpermissionofCambridgeUniversityPress&Assessment. Firstpublished2023 AcataloguerecordforthispublicationisavailablefromtheBritishLibrary ACataloging-in-PublicationdatarecordforthisbookisavailablefromtheLibraryofCongress ISBN978-1-009-26028-2Hardback ISBN978-1-009-26030-5Paperback CambridgeUniversityPress&Assessmenthasnoresponsibilityforthepersistence oraccuracyofURLsforexternalorthird-partyinternetwebsitesreferredtointhis publicationanddoesnotguaranteethatanycontentonsuchwebsitesis,orwill remain,accurateorappropriate. Published online by Cambridge University Press To our grandchildren Published online by Cambridge University Press Published online by Cambridge University Press Contents Preface pageix Acknowledgments x Conventions/Notations xi PartI Preliminaries 1 1 Introduction 3 1.1 CourseTopics 3 1.2 CourseOutline 4 2 BasicTools 8 2.1 Asymptotics 8 2.2 Binomials 10 2.3 TailBounds 16 PartII Erdős–Rényi–GilbertModel 27 3 UniformandBinomialRandomGraphs 29 3.1 ModelsandRelationships 29 3.2 Thresholds 35 4 Evolution 45 4.1 SubcriticalPhase 45 4.2 SupercriticalPhase 54 4.3 PhaseTransition 58 5 VertexDegrees 64 5.1 DegreesofSparseRandomGraphs 64 5.2 DegreesofDenseRandomGraphs 70 6 Connectivity 78 6.1 Connectivity 78 6.2 𝑘-Connectivity 82 Published online by Cambridge University Press viii Contents 7 SmallSubgraphs 85 7.1 Thresholds 85 7.2 AsymptoticDistributions 89 8 LargeSubgraphs 93 8.1 PerfectMatchings 93 8.2 LongPathsandCycles 100 8.3 HamiltonCycles 102 8.4 SpanningSubgraphs 106 9 ExtremeCharacteristics 111 9.1 Diameter 111 9.2 LargestIndependentSets 114 9.3 ChromaticNumber 120 PartIII ModelingComplexNetworks 125 10 InhomogeneousGraphs 127 10.1 GeneralizedBinomialGraph 127 10.2 ExpectedDegreeSequence 134 10.3 FixedDegreeSequence 140 11 SmallWorld 154 11.1 Watts–StrogatzModel 154 11.2 Kleinberg’sModel 160 12 NetworkProcesses 163 12.1 PreferentialAttachment 163 12.2 SpatialPreferentialAttachment 171 13 IntersectionGraphs 178 13.1 BinomialRandomIntersectionGraphs 179 13.2 RandomGeometricGraphs 187 14 WeightedGraphs 197 14.1 MinimumWeightSpanningTree 198 14.2 ShortestPaths 200 14.3 MinimumWeightAssignment 205 References 210 AuthorIndex 216 SubjectIndex 218 Published online by Cambridge University Press Preface In2016,theCambridgeUniversityPresspublishedourbookentitledIntroductionto RandomGraphs(see[52]).Inthepreface,westatedthatourpurposeinwritingitwas ... toprovideagentleintroductiontoasubjectthatisenjoyingasurgeininterest.Webelieve thatthesubjectisfascinatinginitsownright,buttheincreaseininterestcanbeattributedto severalfactors.Onefactoristherealizationthatnetworksare“everywhere”.Fromsocial networkssuchasFacebook,theWorldWideWebandtheInternettothecomplexinteractions betweenproteinsinthecellsofourbodies,wefacethechallengeofunderstandingtheir structureanddevelopment.Byandlarge,naturalnetworksgrowunpredictably,andthisisoften modeledbyarandomconstruction.AnotherfactoristherealizationbyComputerScientists thatNP-hardproblemsaretypicallyeasiertosolvethantheirworst-casesuggests,andthatan analysisofrunningtimesonrandominstancescanbeinformative. AfterﬁveyearssincethecompletionofIntroductiontoRandomGraphs,wehave decided to prepare a slimmed down, reorganized version, at the same time supple- mentedwithsomenewmaterial.Afterhavingtaughtgraduatecoursesontopicsbased onmaterialfromouroriginalbookandafterhavingheardsuggestionsfromourcol- leagues,teachingsimilarcourses,wedecidedtoprepareanewversionwhichcouldbe usedasatextbook,supportingaone-semesterundergraduatecourseformathematics, computerscience,aswellasphysicsmajorsinterestedinrandomgraphsandnetwork science. Based on our teaching experience, the goal of this book is to give our potential readertheknowledgeofthebasicresultsofthetheoryofrandomgraphsandtoshow howithasevolvedtobuildﬁrmmathematicalfoundationsformodernnetworktheory, inparticular,intheanalysisofreal-worldnetworks.Wehavesupplementedtheoretical material with an extended description of the basic analytic tools used in the book, as well as with many exercises and problems. We sincerely hope that our text will encourageourpotentialreadertocontinuethestudyofrandomgraphsandnetworks onamoreadvancedlevelinthefuture. https://doi.org/10.1017/9781009260268.001 Published online by Cambridge University Press