Random Graphs and Networks: A First Course

233 Pages·2023·12.558 MB·English

by  Alan Frieze, Michał Karoński

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
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Informationonthistitle:www.cambridge.org/9781009260282
DOI:10.1017/9781009260268
© AlanFriezeandMichałKaroński2023
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Firstpublished2023
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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
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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
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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