Chapter Graphs Fromthebook NetworksCrowdsandMarketsReasoningaboutaHighlyConnectedWorld ByDavidEasleyandJonKleinbergmCambridgeUniversityPresskqopom Completepreprintonllineathttpynnwwwmcsmcornellmedunh

Chapter Graphs Fromthebook NetworksCrowdsandMarketsReasoningaboutaHighlyConnectedWorld ByDavidEasleyandJonKleinbergmCambridgeUniversityPresskqopom Completepreprintonllineathttpynnwwwmcsmcornellmedunh - Description

GRAPHS gah Agraphon4nodes gbh Adirectedgraphon4nodes qigureYSWeTwographseMaNanundirectedgraphQandMbNadirectedgraphS willbeundirectedunlessnotedotherwiseS GraphsasModelsofNetworksw rraphsareusefulbecausetheyserveasmathematical modelsofnetworkstructure ID: 24670 Download Pdf

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Chapter Graphs Fromthebook NetworksCrowdsandMarketsReasoningaboutaHighlyConnectedWorld ByDavidEasleyandJonKleinbergmCambridgeUniversityPresskqopom Completepreprintonllineathttpynnwwwmcsmcornellmedunh

GRAPHS gah Agraphon4nodes gbh Adirectedgraphon4nodes qigureYSWeTwographseMaNanundirectedgraphQandMbNadirectedgraphS willbeundirectedunlessnotedotherwiseS GraphsasModelsofNetworksw rraphsareusefulbecausetheyserveasmathematical modelsofnetworkstructure

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Chapter Graphs Fromthebook NetworksCrowdsandMarketsReasoningaboutaHighlyConnectedWorld ByDavidEasleyandJonKleinbergmCambridgeUniversityPresskqopom Completepreprintonllineathttpynnwwwmcsmcornellmedunh




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Presentation on theme: "Chapter Graphs Fromthebook NetworksCrowdsandMarketsReasoningaboutaHighlyConnectedWorld ByDavidEasleyandJonKleinbergmCambridgeUniversityPresskqopom Completepreprintonllineathttpynnwwwmcsmcornellmedunh"— Presentation transcript:


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Chapter2 Graphs Fromthebook Networks,Crowds,andMarkets:ReasoningaboutaHighlyConnectedWorld ByDavidEasleyandJonKleinbergmCambridgeUniversityPresskqopom Completepreprintonllineathttpynnwwwmcsmcornellmedunhomenkleinbernnetworkslbookn tnthisfirstpartofthebookwedevelopsomeofthebasicideasbehindgraphtheoryQ thestudyofnetworkstructureSThiswillallowustoformulatebasicnetworkpropertiesina unifyinglanguageSThecentraldefinitionsherearesimpleenoughthatwecandescribethem relativelyquicklyattheoutsetffollowingthisQweconsidersomefundamentalapplications ofthedefinitionsS

2w¨BasicDefinitions Graphs:NodesandEdgesw graph isawayofspecifyingrelationshipsamongacollecR tionofitemsSlgraphconsistsofasetofobjectsQcalled nodes Qwithcertainpairsofthese objectsconnectedbylinkscalled edges SqorexampleQthegraphinqigureYSWMaNconsists of“nodeslabeled Qand Qwith connectedtoeachoftheotherthreenodesby edgesQand and connectedbyanedgeaswellSWesaythattwonodesare neighbors if theyareconnectedbyanedgeSqigureYSWshowsthetypicalwayonedrawsagraph—with littlecirclesrepresentingthenodesQandalineconnectingeachpairofnodesthatarelinked byanedgeS

tnqigureYSWMaNQyoushouldthinkoftherelationshipbetweenthetwoendsofanedgeas beingsymmetricftheedgesimplyconnectsthemtoeachotherStnmanysettingsQhoweverQ wewanttoexpressasymmetricrelationships—forexampleQthat pointsto butnot viceversaSqorthispurposeQwedefinea directedgraph toconsistofasetofnodesQas beforeQtogetherwithasetof directededges feachdirectededgeisalinkfromonenode toanotherQwiththedirectionbeingimportantSoirectedgraphsaregenerallydrawnasin qigureYSWMbNQwithedgesrepresentedbyarrowsSWhenwewanttoemphasizethatagraph isnotdirectedQwecanrefertoitasan undirectedgraph

fbutingeneralthegraphswediscuss DraftversionyJunepokqopo Y[
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Y CHAPTER2.GRAPHS gah Agraphon4nodes. gbh Adirectedgraphon4nodes. qigureYSWeTwographseMaNanundirectedgraphQandMbNadirectedgraphS willbeundirectedunlessnotedotherwiseS GraphsasModelsofNetworksw rraphsareusefulbecausetheyserveasmathematical modelsofnetworkstructuresSWiththisinmindQitisusefulbeforegoingfurthertoreplace thetoyexamplesinqigureYSWwitharealexampleSqigureYSYdepictsthenetworkstructure ofthetnternet—thencalledthelrpanet—inoecemberWdbU[YW“]QwhenithadonlyW[

sitesSyodesrepresentcomputinghostsQandthereisanedgejoiningtwonodesinthispicture ifthereisadirectcommunicationlinkbetweenthemStgnoringthesuperimposedmapofthe USSSMandthecirclesindicatingblownRupregionsinxassachusettsandSouthernnaliforniaNQ therestoftheimageissimplyadepictionofthisW[RnodegraphusingthesamedotsRandRlines stylethatwesawinqigureYSWSyotethatforshowingthepatternofconnectionsQtheactual placementorlayoutofthenodesisimmaterialfallthatmattersiswhichnodesarelinked towhichothersSThusQqigureYS[showsadi erentdrawingofthesameW[Rnodelrpanet graphS

rraphsappearinmanydomainsQwheneveritisusefultorepresenthowthingsareeither physicallyorlogicallylinkedtooneanotherinanetworkstructureSTheW[Rnodelrpanetin qiguresYSYandYS[isanexampleofa communicationnetwork Qinwhichnodesarecomputers orotherdevicesthatcanrelaymessagesQandtheedgesrepresentdirectlinksalongwhich messagescanbetransmittedStnnhapterWQwesawexamplesfromtwootherbroadclassesof graphstructurese socialnetworks QinwhichnodesarepeopleorgroupsofpeopleQandedges representsomekindofsocialinteractionfand informationnetworks Qinwhichthenodes

areinformationresourcessuchasWebpagesordocumentsQandedgesrepresentlogical
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2.2.PATHSANDCONNECTIVITY Y] qigureYSYelnetworkdepictingthesitesonthetnternetQthenknownasthelrpanetQin oecemberWdbUSMtmagefromqSseartQlSxcvenzieQuSxc¨uillianQandoSWalden[YW“]f onRlineathttpeTTsomScsudhSeduTcisTlpressThistoryTarpamapsTSN connectionssuchashyperlinksQcitationsQorcrossRreferencesSThelistofareasinwhich graphsplayaroleisofcoursemuchbroaderthanwhatwecanenumerateherefqigureYS givesafewfurtherexamplesQandalsoshowsthatmanyimagesweencounteronaregular basishavegraphsembeddedinthemS

2w2PathsandConnectivity WenowturntosomeofthefundamentalconceptsanddefinitionssurroundinggraphsS—erR hapsbecausegraphsaresosimpletodefineandworkwithQanenormousrangeofgraphR theoreticnotionshavebeenstudiedfthesocialscientistuohnmarnesoncedescribedgraph theoryasa“terminologicaljungleQinwhichanynewcomermayplantatree”[“]]SqortuR natelyQforourpurposesQwewillbeabletogetunderwaywithjustabriefdiscussionofsome ofthemostcentralconceptsS
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Ya CHAPTER2.GRAPHS qigureYS[elnalternatedrawingoftheW[RnodetnternetgraphfromoecemberWdbUS Pathsw

llthoughwe’vebeendiscussingexamplesofgraphsinmanydi erentareasQthere areclearlysomecommonthemesintheuseofgraphsacrosstheseareasS—erhapsforemost amongtheseistheideathatthingsoftentravelacrosstheedgesofagraphQmovingfrom nodetonodeinsequence—thiscouldbeapassengertakingasequenceofairlineflightsQa pieceofinformationbeingpassedfrompersontopersoninasocialnetworkQoracomputer userorpieceofsoftwarevisitingasequenceofWebpagesbyfollowinglinksS Thisideamotivatesthedefinitionofa path inagrapheapathissimplyasequenceof nodeswiththepropertythateachconsecutivepairinthesequenceisconnectedbyanedgeS

Sometimesitisalsousefultothinkofthepathascontainingnotjustthenodesbutalsothe sequenceofedgeslinkingthesenodesSqorexampleQthesequenceofnodes mit, bbn, rand, ucla isapathinthetnternetgraphfromqiguresYSYandYS[Qasisthesequence case, lincoln, mit, utah, sri, ucsb SlswehavedefinedithereQapathcanrepeatnodesefor exampleQ sri, stan, ucla, sri, utah, mit isapathSmutmostpathsweconsiderwillnot dothisfifwewanttoemphasizethatthepathwearediscussingdoesnotrepeatnodesQwe canrefertoitasa simplepath Cyclesw lparticularlyimportantkindofnonRsimplepathisa cycle Qwhichinformallyisa

“ring”structuresuchasthesequenceofnodes linc, case, carn, harv, bbn, mit, linc ontherightRhandRsideofqigureYS[SxorepreciselyQacycleisapathwithatleastthree edgesQinwhichthefirstandlastnodesarethesameQbutotherwiseallnodesaredistinctS TherearemanycyclesinqigureYS[e sri, stan, ucla, sri isasshortanexampleaspossible accordingtoourdefinitionMsinceithasexactlythreeedgesNQwhile sri, stan, ucla, rand, bbn, mit, utah, sri isasignificantlylongerexampleS tnfactQeveryedgeintheWdbUlrpanetbelongstoacycleQandthiswasbydesigneitmeans

thatifanyedgeweretofailMeSgSaconstructioncrewaccidentallycutthroughthecableNQ therewouldstillbeawaytogetfromanynodetoanyothernodeSxoregenerallyQcycles
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2.2.PATHSANDCONNECTIVITY Yb gah Airlineroutes gbh Subwaymap gch Flowchartofcollegecourses gdh TankStreetBridgeinBrisbane qigureYS“e Imagesofgraphsarisingindi erentdomainsmThedepictionsofairlineandsubwaysystems ingahandgbhareexamplesof transportationnetworks kinwhichnodesaredestinationsandedgesrepresent directconnectionsmMuchoftheterminologysurroundinggraphsderivesfrommetaphorsbasedontransportal

tionthroughanetworkofroadskraillineskorairlineflightsmTheprerequisitesamongcollegecoursesingchis anexampleofa dependencynetwork kinwhichnodesaretasksanddirectededgesindicatethatonetaskmust beperformedbeforeanothermThedesignofcomplexsoftwaresystemsandindustrialprocessesoftenrequires theanalysisofenormousdependencynetworkskwithimportantconsequencesfore cientschedulinginthese settingsmTheTankStreetBridgefromBrisbanekAustraliashowningdhisanexampleofa structuralnetwork withjointsasnodesandphysicallinkagesasedgesmTheinternalframeworksofmechanicalstructuressuchas

buildingskvehicleskorhumanbodiesarebasedonsuchnetworkskandtheareaof rigiditytheory kattheinterl sectionofgeometryandmechanicalengineeringkstudiesthestabilityofsuchstructuresfromagraphlbased perspective[rww]mgImagesygahwwwmairlineroutemapsmcomnUSAnNorthwest Airlines asia pacificmshtmlkgbh wwwmwmatamcomnmetrorailnsystemmapmcfmkgchwwwmcsmcornellmedunugradnflowchartmhtmmh
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Yc CHAPTER2.GRAPHS qigureYS]elgraphwiththreeconnectedcomponentsS incommunicationandtransportationnetworksareoftenpresenttoallowforredundancy

theyprovideforalternateroutingsthatgothe“otherway”aroundthecycleStnthesocial networkoffriendshipstooQweoftennoticecyclesineverydaylifeQevenifwedon’treferto themassuchSWhenyoudiscoverQforexampleQthatyourwife’scousin’sclosefriendfrom highschoolisinfactsomeonewhoworkswithyourbrotherQthisisacycle—consisting ofyouQyourwifeQhercousinQhishighRschoolRfriendQhiscoRworkerMiSeSyourbrotherNQand finallybacktoyouS Connectivityw rivenagraphQitisnaturaltoaskwhethereverynodecanreachevery othernodebyapathSWiththisinmindQwesaythatagraphis connected ifforeverypairof

nodesQthereisapathbetweenthemSqorexampleQtheW[Rnodelrpanetgraphisconnectedf andmoregenerallyQoneexpectsmostcommunicationandtransportationnetworkstobe connected—oratleastaspiretobeconnected—sincetheirgoalistomovetra cfrom onenodetoanotherS zntheotherhandQthereisno apriori reasontoexpectgraphsinothersettingstobe connected—forexampleQinasocialnetworkQyoucouldimaginethattheremightexisttwo peopleforwhichit’snotpossibletoconstructapathfromonetotheotherSqiguresYS] andYSagiveexamplesofdisconnectedgraphsSThefirstisatoyexampleQwhilethesecond

isbuiltfromthecollaborationgraphatabiologicalresearchcenter[W[“]enodesrepresent
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2.2.PATHSANDCONNECTIVITY Yd qigureYSaeThecollaborationgraphofthebiologicalresearchcenter StructuralGenomicsof PathogenicProtozoawSGPPx [W[“]QwhichconsistsofthreedistinctconnectedcomponentsS Thisgraphwaspartofacomparativestudyofthecollaborationpatternsgraphsofnine researchcenterssupportedbyyts’s—roteinStructuretnitiativefSr——wasanintermediate casebetweencenterswhosecollaborationgraphwasconnectedandthoseforwhichitwas fragmentedintomanysmallcomponentsS

researchersQandthereisanedgebetweentwonodesiftheresearchersappearjointlyona coRauthoredpublicationSMThustheedgesinthissecondfigurerepresentaparticularformal definitionofcollaboration—jointauthorshipofapublishedpaper—anddonotattemptto capturethenetworkofmoreinformalinteractionsthatpresumablytakeplaceattheresearch centerSN Componentsw qiguresYS]andYSamakevisuallyapparentabasicfactaboutdisconnected graphseifagraphisnotconnectedQthenitbreaksapartnaturallyintoasetofconnected “piecesQ”groupsofnodessothateachgroupisconnectedwhenconsideredasagraphin

isolationQandsothatnotwogroupsoverlapStnqigureYS]Qweseethatthegraphconsists ofthreesuchpieceseoneconsistingofnodes and Qoneconsistingofnodes Qand andoneconsistingoftherestofthenodesSThenetworkinqigureYSaalsoconsistsofthree pieceseoneonthreenodesQoneonfournodesQandonethatismuchlargerS TomakethisnotionpreciseQwewesaythata connectedcomponent ofagraphMoften shortenedjusttotheterm“component”NisasubsetofthenodessuchthateMiNeverynode inthesubsethasapathtoeveryotherfandMiiNthesubsetisnotpartofsomelargerset withthepropertythateverynodecanreacheveryotherSyoticehowbothMiNandMiiN
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[U

CHAPTER2.GRAPHS arenecessarytoformalizetheintuitivedefinitioneMiNsaysthatthecomponentisindeed internallyconnectedQandMiiNsaysthatitreallyisafreeRstanding“piece”ofthegraphQnot aconnectedpartofalargerpieceSMqorexampleQwewouldnotthinkofthesetofnodes Qand inqigureYS]asformingacomponentQbecausethissetviolatespartMiiNofthe definitionealthoughtherearepathsamongallpairsofnodesinthesetQitbelongstothe largersetconsistingof QinwhichallpairsarealsolinkedbypathsSN oividingagraphintoitscomponentsisofcourseonlyafirstQglobalwayofdescribing

itsstructureSWithinagivencomponentQtheremaybericherinternalstructurethatis importanttoone’sinterpretationofthenetworkSqorexampleQthinkingaboutthelargest componentfromqigureYSainlightofthecollaborationsthatitrepresentsQonenoticescertain suggestivefeaturesofthestructureeaprominentnodeatthecenterQandtightlyRknitgroups linkedtothisnodebutnottoeachotherSznewaytoformalizetheroleoftheprominent centralnodeistoobservethatthelargestconnectedcomponentwouldbreakapartintothree distinctcomponentsifthisnodewereremovedSlnalyzingagraphthiswayQintermsofits

denselyRconnectedregionsandtheboundariesbetweenthemQisapowerfulwayofthinking aboutnetworkstructureQanditwillbeacentraltopicinnhapter[S GiantComponentsw Thereturnsouttobeausefulqualitativewayofthinkingabout theconnectedcomponentsoftypicallargenetworksQandforthisithelpstobeginwiththe followingthoughtexperimentSnonsiderthesocialnetworkoftheentireworldQwithalink betweentwopeopleiftheyarefriendsSyowQofcourseQthisisagraphthatwedon’tactually haveexplicitlyrecordedanywhereQbutitisonewherewecanuseourgeneralintuitionsto answersomebasicquestionsS

qirstQisthisglobalfriendshipnetworkconnectedj—resumablynotSlfterallQconnecR tivityisafairlybrittlepropertyQinthatthebehaviorofasinglenodeMorasmallsetof nodesNcannegateitSqorexampleQasinglepersonwithnolivingfriendswouldconstitute aoneRnodecomponentintheglobalfriendshipnetworkQandhencethegraphwouldnotbe connectedSzrthecanonical“remotetropicalislandQ”consistingofpeoplewhohavehad nocontactwiththeoutsideworldQwouldalsobeasmallcomponentinthenetworkQagain showingthatitisnotconnectedS mutthereissomethingmoregoingonhereStfyou’reatypicalreaderofthisbookQthen

youhavefriendswhogrewupinothercountriesSYou’reinthesamecomponentasallthese friendsQsinceyouhaveapathMcontainingasingleedgeNtoeachofthemSyowQifyouconsiderQ sayQtheparentsofthesefriendsQyourfriends’parents’friendsQtheirfriendsanddescendantsQ thenallofthesepeopleareinthesamecomponentaswell—andbynowQwe’retalking aboutpeoplewhohaveneverheardofyouQmaywellnotsharealanguagewithyouQmay havenevertraveledanywherenearwhereyouliveQandmayhavehadenormouslydi erent lifeexperiencesSSoeventhoughtheglobalfriendshipnetworkmaynotbeconnectedQthe
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2.2.PATHSANDCONNECTIVITY [W

componentyouinhabitseemsverylargeindeed—itreachesintomostpartsoftheworldQ includespeoplefrommanydi erentbackgroundsQandseemsinfactlikelytocontaina significantfractionoftheworld’spopulationS Thisisinfacttruewhenonelooksacrossarangeofnetworkdatasets—largeQcomplex networksoftenhavewhatiscalleda giantcomponent Qadeliberatelyinformaltermfora connectedcomponentthatcontainsasignificantfractionofallthenodesSxoreoverQwhen anetworkcontainsagiantcomponentQitalmostalwayscontainsonlyoneSToseewhyQlet’s gobacktotheexampleoftheglobalfriendshipnetworkandtryimaginingthattherewere

twogiantcomponentsQeachwithhundredsofmillionsofpeopleSlllitwouldtakeisasingle edgefromsomeoneinthefirstofthesecomponentstosomeoneinthesecondQandthetwo giantcomponentswouldmergeintoasinglecomponentSuustasingleedge—inmostcasesQ it’sessentiallyinconceivablethatsomesuchedgewouldn’tformQandhencetwocoRexisting giantcomponentsaresomethingonealmostneverseesinrealnetworksSWhenthereisa giantcomponentQitisthusgenerallyuniqueQdistinguishableasacomponentthatdwarfsall othersS tnfactQinsomeoftherarecaseswhentwogiantcomponentshavecoRexistedforalong

timeinarealnetworkQtheirmerginghasbeensuddenQdramaticQandultimatelycatastrophicS qorexampleQuaredoiamond’sbook Guns¨Germs¨andSteel [W[U]devotesmuchofits attentiontothecataclysmthatbefellthecivilizationsoftheWesternhemispherewhen puropeanexplorersbeganarrivinginitroughlyhalfamilleniumagoSznecanviewthis developmentfromanetworkperspectiveasfollowsefivethousandyearsagoQtheglobal socialnetworklikelycontainedtwogiantcomponents—oneinthelmericasQandonein thepuropeRlsialandmassSmecauseofthisQtechnologyevolvedindependentlyinthetwo

componentsQandperhapsevenworseQhumandiseasesevolvedindependentlyfandsowhen thetwocomponentsfinallycameincontactQthetechnologyanddiseasesofonequicklyand disastrouslyoverwhelmedtheotherS Thenotionofgiantcomponentsisusefulforreasoningaboutnetworksonmuchsmaller scalesaswellSThecollaborationnetworkinqigureYSaisonesimpleexamplefanother interestingexampleisdepictedinqigureYSbQwhichshowstheromanticrelationshipsinan lmericanhighschooloveranWcRmonthperiod[“d]SMTheseedgeswerenotallpresentat oncefratherQthereisanedgebetweentwopeopleiftheywereromanticallyinvolvedatany

pointduringthetimeperiodSNThefactthatthisgraphcontainssuchalargecomponentis significantwhenonethinksaboutthespreadofsexuallytransmitteddiseasesQafocusofthe researchersperformingthestudySlhighRschoolstudentmayhavehadasinglepartnerover thistimeperiodandnevertheless—withoutrealizingit—bepartofthislargecomponent andhencepartofmanypathsofpotentialtransmissionSlsmearmanQxoodyQandStovel noteinthepaperwheretheyanalyzethisnetworkQ“Thesestructuresreflectrelationships thatmaybelongoverQandtheylinkindividualstogetherinchainsfartoolongtobe
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[Y CHAPTER2.GRAPHS

qigureYSbelnetworkinwhichthenodesarestudentsinalargelmericanhighschoolQand anedgejoinstwowhohadaromanticrelationshipatsomepointduringtheWcRmonthperiod inwhichthestudywasconducted[“d]S thesubjectofeventhemostintensegossipandscrutinySyeverthelessQtheyarerealelike socialfactsQtheyareinvisibleyetconsequentialmacrostructuresthatariseastheproductof individualagencyS 2w3DistanceandBreadthvFirstSearch tnadditiontosimplyaskingwhethertwonodesareconnectedbyapathQitisalsointeresting inmostsettingstoaskhow long suchapathis—intransportationQtnternetcommunicationQ

orthespreadofnewsanddiseasesQitisoftenimportantwhethersomethingflowingthrough anetworkhastotraveljustafewhopsormanyS TobeabletotalkaboutthisnotionpreciselyQwedefinethe length ofapathtobethe numberofstepsitcontainsfrombeginningtoend—inotherwordsQthenumberofedges inthesequencethatcomprisesitSThusQforexampleQthepath mit, bbn, rand, ucla in qigureYS[haslengththreeQwhilethepath mit, utah haslengthoneSUsingthenotionof
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2.3.DISTANCEANDBREADTH-FIRSTSEARCH [[ qigureYScemreadthRfirstsearchdiscoversdistancestonodesone“layer”atatimefeachlayer

isbuiltofnodesthathaveanedgetoatleastonenodeinthepreviouslayerS apath’slengthQwecantalkaboutwhethertwonodesareclosetogetherorfarapartina graphewedefinethe distance betweentwonodesinagraphtobethelengthoftheshortest pathbetweenthemSqorexampleQthedistancebetween linc and sri isthreeQthoughto believethisyouhavetofirstconvinceyourselfthatthereisnolengthRWorlengthRYpath betweenthemS BreadthvFirstSearchw qoragraphliketheoneinqigureYS[Qwecangenerallyfigure outthedistancebetweentwonodesbyeyeballingthepicturefbutforgraphsthatareeven

abitmorecomplicatedQweneedsomekindofasystematicmethodtodeterminedistancesS Themostnaturalwaytodothis—andalsothemoste cientwaytocalculatedistances foralargenetworkdatasetusingacomputer—isthewayyouwouldprobablydoitifyou
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[ CHAPTER2.GRAPHS qigureYSdeThelayersarisingfromabreadthRfirstoftheoecemberWdbUlrpanetQstarting atthenode mit reallyneededtotraceoutdistancesintheglobalfriendshipnetworkMandhadtheunlimited patienceandcooperationofeveryoneintheworldNSThisispicturedinqigureYSce MWN YoufirstdeclareallofyouractualfriendstobeatdistanceWS MYN Youthenfindallof

theirfriends MnotcountingpeoplewhoarealreadyfriendsofyoursNQ anddeclarethesetobeatdistanceYS M[N Thenyoufindallof their friendsMagainQnotcountingpeoplewhoyou’vealreadyfound atdistancesWandYNanddeclarethesetobeatdistance[S MSSSN nontinuinginthiswayQyousearchinsuccessivelayersQeachrepresentingthenext distanceoutSpachnewlayerisbuiltfromallthosenodesthatMiNhavenotalready beendiscoveredinearlierlayersQandthatMiiNhaveanedgetosomenodeintheprevious layerS Thistechniqueiscalled breadth-firstsearch QsinceitsearchesthegraphoutwardfromastartR

ingnodeQreachingtheclosestnodesfirstStnadditiontoprovidingamethodofdetermining distancesQitcanalsoserveasausefulconceptualframeworktoorganizethestructureofa graphQarrangingthenodesbasedontheirdistancesfromafixedstartingpointS
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2.3.DISTANCEANDBREADTH-FIRSTSEARCH [] zfcourseQdespitethesocialRnetworkmetaphorweusedtodescribebreadthRfirstsearchQ theprocesscanbeappliedtoanygrapheonejustkeepsdiscoveringnodeslayerRbyRlayerQ buildingeachnewlayerfromthenodesthatareconnectedtoatleastonenodeintheprevious layerSqorexampleQqigureYSdshowshowtodiscoveralldistancesfromthenode

mit inthe W[RnodelrpanetgraphfromqigureYS[S TheSmallvWorldPhenomenonw lswithourdiscussionoftheconnectedcomponents inagraphQthereissomethingqualitativewecansayQbeyondtheformaldefinitionsQabout distancesintypicallargenetworksStfwegobacktoourthoughtexperimentsontheglobal friendshipnetworkQweseethattheargumentexplainingwhyyoubelongtoagiantcompoR nentinfactassertssomethingstrongerenotonlydoyouhavepathsoffriendsconnecting youtoalargefractionoftheworld’spopulationQbutthesepathsaresurprisingly short Taketheexampleofafriendwhogrewupinanothercountryefollowingapaththroughthis

friendQtohisorherparentsQtotheirfriendsQyou’vefollowedonlythreestepsandendedup inadi erentpartoftheworldQinadi erentgenerationQwithpeoplewhohaveverylittlein commonwithyouS Thisideahasbeentermedthe small-worldphenomenon —theideathattheworldlooks “small”whenyouthinkofhowshortapathoffriendsittakestogetfromyoutoalmost anyoneelseStt’salsoknownQperhapsmorememorablyQasthe sixdegreesofseparation fthis phrasecomesfromtheplayofthistitlebyuohnruare[YUU]Qandinparticularfromtheline utteredbyoneoftheplay’scharacterse“treadsomewherethateverybodyonthisplanetis

separatedbyonlysixotherpeopleSSixdegreesofseparationbetweenusandeveryoneelse onthisplanetS Thefirstexperimentalstudyofthisnotion—andtheoriginofthenumber“six”inthe popRculturalmantra—wasperformedbyStanleyxilgramandhiscolleaguesintheWdaUs [YdbQ[dW]SwackinganyofthemassivesocialRnetworkdatasetswehavetodayQandwitha budgetofonlyIacUQhesetouttotestthespeculativeideathatpeoplearereallyconnectedin theglobalfriendshipnetworkbyshortchainsoffriendsSTothisendQheaskedacollectionof Ydarandomlychosen“starters”totryforwardingalettertoa“target”personQastockbroker

wholivedinasuburbofmostonSThestarterswereeachgivensomepersonalinformation aboutthetargetMincludinghisaddressandoccupationNandwereaskedtoforwardthe lettertosomeonetheyknewonafirstRnamebasisQwiththesameinstructionsQinorderto eventuallyreachthetargetasquicklyaspossibleSpachletterthuspassedthroughthehands ofasequenceoffriendsinsuccessionQandeachtherebyformedachainofpeoplethatclosed inonthestockbrokeroutsidemostonS qigureYSWUshowsthedistributionofpathlengthsQamongthea“chainsthatsucceeded inreachingthetargetfthemedianlengthwassixQthenumberthatmadeitswaytwodecades

laterintothetitleofruare’splaySThatsomanylettersreachedtheirdestinationQandby
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[a CHAPTER2.GRAPHS qigureYSWUelhistogramfromTraversandxilgram’spaperontheirsmallRworldexperiment [[dW]SqoreachpossiblelengthMlabeled“numberofintermediaries”onthe RaxisNQtheplot showsthenumberofsuccessfullycompletedchainsofthatlengthStntotalQa“chainsreached thetargetpersonQwithamedianlengthofsixS suchshortpathsQwasastrikingfactwhenitwasfirstdiscoveredQanditremainssotodayS zfcourseQitisworthnotingafewcaveatsabouttheexperimentSqirstQitclearlydoesn’t

establishastatementquiteasboldas“sixdegreesofseparationbetweenusandeveryone elseonthisplanet”—thepathswerejusttoasingleQfairlya uenttargetfmanyletters nevergottherefandattemptstorecreatetheexperimenthavebeenproblematicduetolack ofparticipation[Y]]]SSecondQonecanaskhowusefultheseshortpathsreallyaretopeople insocietyeevenifyoucanreachsomeonethroughashortchainoffriendsQisthisusefulto youjooesitmeanyou’retrulysocially“close”tothemjxilgramhimselfmusedaboutthis inhisoriginalpaper[Ydb]fhisobservationQparaphrasedslightlyQwasthatifwethinkofeach

personasthecenteroftheirownsocial“worldQ”then“sixshortsteps”becomes“sixworlds apart”—achangeinperspectivethatmakessixsoundlikeamuchlargernumberS oespitethesecaveatsQtheexperimentandthephenomenathatithintsathaveformed acrucialaspectinourunderstandingofsocialnetworksStntheyearssincetheinitial experimentQtheoverallconclusionhasbeenacceptedinabroadsenseesocialnetworkstend tohaveveryshortpathsbetweenessentiallyarbitrarypairsofpeopleSlndevenifyoursixR
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2.3.DISTANCEANDBREADTH-FIRSTSEARCH [b qigureYSWWeThedistributionofdistancesinthegraphofallactivexicrosofttnstantxesR

sengeruseraccountsQwithanedgejoiningtwousersiftheycommunicatedatleastonce duringamonthRlongobservationperiod[Yb[]S stepconnectionstonpzsandpoliticalleadersdon’tyieldimmediatepayo sonaneveryday basisQtheexistenceofalltheseshortpathshassubstantialconsequencesforthepotential speedwithwhichinformationQdiseasesQandotherkindsofcontagioncanspreadthrough societyQaswellasforthepotentialaccessthatthesocialnetworkprovidestoopportunities andtopeoplewithverydi erentcharacteristicsfromone’sownSllltheseissues—and theirimplicationsfortheprocessesthattakeplaceinsocialnetworks—arerichenough

thatwewilldevotenhapterYUtoamoredetailedstudyofthesmallRworldphenomenonand itsconsequencesS InstantMessaginguPaulErdosuandKevinBaconw znereasonforthecurrentemR piricalconsensusthatsocialnetworksgenerallyare“smallworlds”isthatthishasbeen increasinglyconfirmedinsettingswherewedohavefulldataonthenetworkstructureSxilR gramwasforcedtoresorttoanexperimentinwhichlettersservedas“tracers”througha globalfriendshipnetworkthathehadnohopeoffullymappingonhisownfbutforother kindsofsocialnetworkdatawherethefullgraphstructureisknownQonecanjustloadit

intoacomputerandperformthebreadthRfirstsearchproceduretodeterminewhattypical
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[c CHAPTER2.GRAPHS qigureYSWYeRonrraham’shandRdrawnpictureofapartofthemathematicscollaboration graphQcenteredon—aulprdos[Wcd]SMtmagefromhttpeTTwwwSoaklandSeduTenpTcgraphSjpgN distanceslooklikeS zneofthelargestsuchcomputationalstudieswasperformedbyuureweskovecandpric sorvitz[Yb[]STheyanalyzedtheY“Umillionactiveuseraccountsonxicrosofttnstant xessengerQbuildingagraphinwhicheachnodecorrespondstoauserQandthereisan edgebetweentwousersiftheyengagedinatwoRwayconversationatanypointduringa

monthRlongobservationperiodSlsemployeesofxicrosoftatthetimeQtheyhadaccessto acompletesnapshotofthesystemforthemonthunderstudyQsotherewerenoconcerns aboutmissingdataSThisgraphturnedouttohaveagiantcomponentcontainingalmost allofthenodesQandthedistanceswithinthisgiantcomponentwereverysmallStndeedQ thedistancesinthetnstantxessengernetworkcloselycorrespondedtothenumbersfrom xilgram’sexperimentQwithanestimatedaveragedistanceofaSaQandanestimatedmedian
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2.3.DISTANCEANDBREADTH-FIRSTSEARCH [d ofsevenSqigureYSWWshowsthedistributionofdistancesaveragedoverarandomsample

ofWUUUusersebreadthRfirstsearchwasperformedseparatelyfromeachoftheseWUUUusersQ andtheresultsfromtheseWUUUnodeswerecombinedtoproducetheplotinthefigureS Thereasonforthisestimationbysamplingusersisacomputationaloneethegraphwas solargethatperformingbreadthRfirstsearchfromeverysinglenodewouldhavetakenan astronomicalamountoftimeS—roducingplotslikethise cientlyformassivegraphsisan interestingresearchtopicinitself[[[c]S tnasenseQtheplotinqigureYSWWstartstoapproximateQinastrikingwayQwhatxilgram andhiscolleaguesweretryingtounderstand—thedistributionofhowfarapartweallare

inthefullglobalfriendshipnetworkSltthesametimeQreconcilingthestructureofsuch massivedatasetswiththeunderlyingnetworkstheyaretryingtomeasureisanissuethat comesuphereQasitwillmanytimesthroughoutthebookStnthiscaseQenormousasthe xicrosofttxstudywasQitremainssomedistanceawayfromxilgram’sgoaleitonlytracks peoplewhoaretechnologicallyRendowedenoughtohaveaccesstoinstantmessagingQand ratherthanbasingthegraphonwhoistrulyfriendswithwhomQitcanonlyobservewho talkstowhomduringanobservationperiodS Turningtoasmallerscale—atthelevelofhundredofthousandsofpeopleratherthan

hundredsofmillions—researchershavealsodiscoveredveryshortpathsinthecollaboration networkswithinprofessionalcommunitiesStnthedomainofmathematicsQforexampleQ peopleoftenspeakoftheitinerantmathematician—aulprdos—whopublishedroughly W]UUpapersoverhiscareer—asacentralfigureinthecollaborativestructureofthefieldS TomakethispreciseQwecandefineacollaborationgraphaswedidforqigureYSaQinthis casewithnodescorrespondingtomathematiciansQandedgesconnectingpairswhohave jointlyauthoredapaperSMWhileqigureYSaconcernedasingleresearchlabQwearenow

talkingaboutcollaborationwithintheentirefieldofmathematicsSNqigureYSWYshowsa smallhandRdrawnpieceofthecollaborationgraphQwithpathsleadingto—aulprdos[Wcd]S yowQamathematician’s Erdosnumber isthedistancefromhimorhertoprdosinthisgraph [Wdc]SThepointisthatmostmathematicianshaveprdosnumbersofatmost“or]Qand extendingthecollaborationgraphtoincludecoRauthorshipacrossallthesciences—most scientistsinotherfieldshaveprdosnumbersthatarecomparableoronlyslightlylargerf llbertpinstein’sisYQpnricoqermi’sis[Qyoamnhomsky’sandwinus—auling’sareeach“Q

qrancisnrick’sanduamesWatson’sare]andarespectivelySTheworldofscienceistruly asmalloneinthissenseS tnspiredbysomemixtureofthexilgramexperimentQuohnruare’splayQandacompelling beliefthatvevinmaconwasthecenterofthesollywooduniverseQthreestudentsatllbright nollegein—ennsylvaniasometimearoundWdd“adaptedtheideaofprdosnumberstothe collaborationgraphofmovieactorsandactressesenodesareperformersQanedgeconnects twoperformersifthey’veappearedtogetherinamovieQandaperformer’s Baconnumber is
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“U CHAPTER2.GRAPHS hisorherdistanceinthisgraphtovevinmacon[[bY]SUsingcastlistsfromthetnternet

xovieoatabaseMtxomNQitispossibletocomputemaconnumbersforallperformersvia breadthRfirstsearch—andaswithmathematicsQit’sasmallworldindeedSTheaverage maconnumberQoverallperformersinthetxomQisapproximatelyYSdQandit’sachallenge tofindonethat’slargerthan]StndeedQit’sfittingtoconcludewithanetworkRandRmovie enthusiast’sdescriptionofhislateRnightattemptstofindthelargestmaconnumberinthe txombyhande“WithmylifeRlongpassionformoviesQtcouldn’tresistspendingmany hoursprobingthedarkrecessesoffilmhistoryuntilQataboutWUlxonSundayQtfoundan

incrediblyobscureWdYcSovietpiratefilmQ PlennikiMorya Qstarring—SSavinwithamacon numberofbQandwhosesupportingcastofcappearednowhereelse”[Wdb]Szneisleftwith theimageofalongexplorationthatarrivesfinallyattheouteredgeofthemovieworld intheearlyhistoryoffilmQintheSovietUnion—andyetinanothersenseQonlycsteps fromwhereitstartedS 2w4NetworkDatasets:AnOverview TheexplosionofresearchonlargeRscalenetworksinrecentyearshasbeenfueledtoalarge extentbytheincreasingavailabilityoflargeQdetailednetworkdatasetsSWe’veseenexamples

ofsuchdatasetsthroughoutthesefirsttwochaptersQandit’susefulatthispointtostep backandthinkmoresystematicallyaboutwherepeoplehavebeengettingthedatathat theyemployinlargeRscalestudiesofnetworksS ToputthisinperspectiveQwenotefirstofallthatthereareseveraldistinctreasons whyyoumightstudyaparticularnetworkdatasetSzneisthatyoumaycareaboutthe actualdomainitcomesfromQsothatfineRgraineddetailsofthedataitselfarepotentially asinterestingasthebroadpictureSlnotheristhatyou’reusingthedatasetasaproxy forarelatednetworkthatmaybeimpossibletomeasure—asforexampleintheway

thexicrosofttxgraphfromqigureYSWWgaveusinformationaboutdistancesinasocial networkofascaleandcharacterthatbeginstoapproximatetheglobalfriendshipnetworkS lthirdpossibilityisthatyou’retryingtolookfornetworkpropertiesthatappeartobe commonacrossmanydi erentdomainsQandsofindingasimilare ectinunrelatedsettings cansuggestthatithasacertainuniversalnatureQwithpossibleexplanationsthatarenot tiedtothespecificsofanyoneofthedomainsS zfcourseQallthreeofthesemotivationsareoftenatworksimultaneouslyQtovarying degreesQinthesamepieceofresearchSqorexampleQtheanalysisofthexicrosofttxgraph

gaveusinsightintotheglobalfriendshipnetwork—butatamorespecificlevelQthereR searchersperformingthestudywerealsointerestedinthedynamicsofinstantmessaging inparticularfandatamoregenerallevelQtheresultofthetxgraphanalysisfitintothe broaderframeworkofsmallRworldphenomenathatspanmanydomainsS
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2.4.NETWORKDATASETS:ANOVERVIEW “W lsafinalpointQwe’reconcernedherewithsourcesofdataonnetworksthatare large tfonewantstostudyasocialnetworkonYUpeople—sayQwithinasmallcompanyQora fraternityorsororityQorakarateclubasinqigureWSW—thenonestrategyistointerview

allthepeopleinvolvedandaskthemwhotheirfriendsareSmutifwewanttostudythe interactionsamongYUQUUUpeopleQorYUQUUUindividualnodesofsomeotherkindQthenwe needtobemoreopportunisticinwherewelookfordataeexceptinunusualcasesQwecan’t simplygooutandcollecteverythingbyhandQandsoweneedtothinkaboutsettingsin whichthedatahasinsomeessentialwayalreadybeenmeasuredforusS WiththisinmindQlet’sconsidersomeofthemainsourcesoflargeRscalenetworkdata thatpeoplehaveusedforresearchSTheresultinglistisfarfromexhaustiveQnorarethe categoriestrulydistinct—asingledatasetcaneasilyexhibitcharacteristicsfromseveralS

CollaborationGraphs. nollaborationgraphsrecordwhoworkswithwhominaspecific settingfcoRauthorshipsamongscientistsandcoRappearanceinmoviesbyactorsand actressesaretwoexamplesofcollaborationgraphsthatwediscussedinSectionYS[S lnotherexamplethathasbeenextensivelystudiedbysociologistsisthegraphon highlyRplacedpeopleinthecorporateworldQwithanedgejoiningtwoiftheyhave servedtogetherontheboardofdirectorsofthesameqortune]UUcompany[[UW]SThe onRlineworldprovidesnewinstancesetheWikipediacollaborationgraphMconnecting twoWikipediaeditorsifthey’veevereditedthesamearticleN[WYYQY“a]andtheWorldR

ofRWarcraftcollaborationgraphMconnectingtwoWRoRWusersifthey’veevertakenpart togetherinthesameraidorotheractivityN[“Wd]arejusttwoexamplesS Sometimesacollaborationgraphisstudiedtolearnaboutthespecificdomainitcomes fromfforexampleQsociologistswhostudythebusinessworldhaveasubstantiveinR terestintherelationshipsamongcompaniesatthedirectorlevelQasexpressedvia coRmembershiponboardsSzntheotherhandQwhilethereisaresearchcommunity thatstudiesthesociologicalcontextofscientificresearchQabroadercommunityof peopleisinterestedinscientificcoRauthorshipnetworkspreciselybecausetheyform

detailedQpreRdigestedsnapshotsofarichformofsocialinteractionthatunfoldsovera longperiodoftime[[Wc]SmyusingonRlinebibliographicrecordsQonecanoftentrack thepatternsofcollaborationwithinafieldacrossacenturyormoreQandtherebyatR tempttoextrapolatehowthesocialstructureofcollaborationmayworkacrossarange ofharderRtoRmeasuresettingsaswellS Who-talks-to-WhomGraphs. ThexicrosofttxgraphisasnapshotofalargecommuR nityengagedinseveralbillionconversationsoverthecourseofamonthStnthiswayQ itcapturesthe“whoRtalksRtoRwhom”structureofthecommunitySSimilardatasets

havebeenconstructedfromtheeRmaillogswithinacompany[a]orauniversity[Y]d]Q aswellasfromrecordsofphonecallseresearchershavestudiedthestructureof call
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“Y CHAPTER2.GRAPHS graphs inwhicheachnodeisaphonenumberQandthereisanedgebetweentwoifthey engagedinaphonecalloveragivenobservationperiod[WQ[[“]Sznecanalsousethe factthatmobilephoneswithshortRrangewirelesstechnologycandetectothersimilar devicesnearbySmyequippingagroupofexperimentalsubjectswithsuchdevicesand studyingthetracestheyrecordQresearcherscantherebybuild“faceRtoRface”graphs

thatrecordphysicalproximityeanodeinsuchagraphisapersoncarryingoneofthe mobiledevicesQandthereisanedgejoiningtwopeopleiftheyweredetectedtobein closephysicalproximityoveragivenobservationperiod[W“WQW“Y]S tnalmostallofthesekindsofdatasetsQthenodesrepresentcustomersQemployeesQor studentsoftheorganizationthatmaintainsthedataSTheseindividualswillgenerally havestrongexpectationsofprivacyQnotnecessarilyevenappreciatinghoweasilyone canreconstructdetailsoftheirbehaviorfromthedigitaltracestheyleavebehind whencommunicatingbyeRmailQinstantmessagingQorphoneSlsaresultQthestyleof

researchperformedonthiskindofdataisgenerallyrestrictedinspecificwayssoasto protecttheprivacyoftheindividualsinthedataSSuchprivacyconsiderationshave alsobecomeatopicofsignificantdiscussioninsettingswherecompaniestrytousethis typeofdataformarketingQorwhengovernmentstrytouseitforintelligenceRgathering purposes[[W]]S Relatedtothiskindof“whoRtalksRtoRwhom”dataQeconomicnetworkmeasurements recordingthe“whoRtransactsRwithRwhom”structureofamarketorfinancialcommuR nityhasbeenusedtostudythewaysinwhichdi erentlevelsofaccesstomarket participantscanleadtodi erentlevelsofmarketpoweranddi

erentpricesforgoodsS Thisempiricalworkhasinturnmotivatedmoremathematicalinvestigationsofhow anetworkstructurelimitingaccessbetweenbuyersandsellerscana ectoutcomes [a[QWbaQY[YQYaW]QafocusofdiscussioninnhaptersWU—WYS InformationLinkageGraphs. SnapshotsoftheWebarecentralexamplesofnetwork datasetsfnodesareWebpagesanddirectededgesrepresentlinksfromonepageto anotherSWebdatastandsoutbothinitsscaleandinthediversityofwhatthenodes representebillionsoflittlepiecesofinformationQwithlinkswiringthemtogetherSlnd clearlyitisnotjusttheinformationthatisofinterestQbutthesocialandeconomic

structuresthatstandbehindtheinformationehundredsofmillionsofpersonalpageson socialRnetworkingandbloggingsitesQhundredsofmillionsmorerepresentingcompanies andgovernmentalorganizationstryingtoengineertheirexternalimagesinacrowded networkS lnetworkonthescaleofthefullWebcanbedauntingtoworkwithfsimplymanipuR latingthedatae ectivelycanbecomearesearchchallengeinitselfSlsaresultQmuch networkresearchhasbeendoneoninterestingQwellRdefinedsubsetsoftheWebQincludR
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2.4.NETWORKDATASETS:ANOVERVIEW “[ ingthelinkagesamongbloggers[Ya“]QamongpagesonWikipedia[“U“]Qamongpages

onsocialRnetworkingsitessuchasqacebookorxySpace[Wc]]Qoramongdiscussions andproductreviewsonshoppingsites[YUW]S ThestudyofinformationlinkagegraphssignificantlypredatestheWebethefieldof citationanalysis hasQsincetheearlypartoftheYUthcenturyQstudiedthenetwork structureofcitationsamongscientificpapersorpatentsQasawayoftrackingthe evolutionofscience[W“]]SnitationnetworksarestillpopularresearchdatasetstodayQ forthesamereasonthatscientificcoRauthorshipgraphsareeevenifyoudon’thavea substantiveinterestinthesocialprocessesbywhichsciencegetsdoneQcitationnetworks

areverycleandatasetsthatcaneasilyspanmanydecadesS TechnologicalNetworks. llthoughtheWebisbuiltonalotofsophisticatedtechnologyQ itwouldbeamistaketothinkofitprimarilyasatechnologicalnetworkeitisreallya projectionontoatechnologicalbackdropofideasQinformationQandsocialandeconomic structurecreatedbyhumansSmutaswenotedintheopeningchapterQtherehas clearlybeenaconvergenceofsocialandtechnologicalnetworksoverrecentyearsQand muchinterestingnetworkdatacomesfromthemoreovertlytechnologicalendofthe spectrum—withnodesrepresentingphysicaldevicesandedgesrepresentingphysical

connectionsbetweenthemSpxamplesincludetheinterconnectionsamongcomputers onthetnternet[W]]]oramonggeneratingstationsinapowergrid[“WW]S pvenphysicalnetworksliketheseareultimatelyeconomicnetworksaswellQrepresentR ingtheinteractionsamongthecompetingorganizationsQcompaniesQregulatorybodiesQ andothereconomicentitiesthatshapeitSznthetnternetQthisismadeparticularly explicitbyatwoRlevelviewofthenetworkSltthelowestlevelQnodesareindividual routersandcomputersQwithanedgemeaningthattwodevicesactuallyhaveaphysical connectiontoeachotherSmutatahigherlevelQthesenodesaregroupedintowhatare

essentiallylittle“nationRstates”termed autonomoussystems Qeachonecontrolledbya di erenttnternetserviceRprovidersSThereisthenawhoRtransactsRwithRwhomgraph ontheautonomoussystemsQknownasthe ASgraph Qthatrepresentsthedatatransfer agreementsthesetnternetserviceRprovidersmakewitheachotherS NetworksintheNaturalWorld. rraphstructuresalsoaboundinbiologyandthe othernaturalsciencesQandnetworkresearchhasdevotedparticularattentiontoseveral di erenttypesofbiologicalnetworksSserearethreeexamplesatthreedi erentscalesQ fromthepopulationleveldowntothemolecularlevelS lsafirstexampleQ foodwebs

representthewhoReatsRwhomrelationshipsamongspecies inanecosystem[W[b]ethereisanodeforeachspeciesQandadirectededgefromnode tonode indicatesthatmembersof consumemembersof SUnderstanding thestructureofafoodwebasagraphcanhelpinreasoningaboutissuessuchas
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CHAPTER2.GRAPHS cascadingextinctions eifcertainspeciesbecomeextinctQthenspeciesthatrelyonthem forfoodriskbecomingextinctaswellQiftheydonothavealternativefoodsourcesf theseextinctionscanpropagatethroughthefoodwebasachainreactionS lnotherheavilyRstudiednetworkinbiologyisthestructureofneuralconnectionswithin

anorganism’sbrainethenodesareneuronsQandanedgerepresentsaconnection betweentwoneurons[[cU]STheglobalbrainarchitectureforsimpleorganismslike C.Elegans Qwith[UYnodesandroughlybUUUedgesQhasessentiallybeencompletely mapped[[]fbutobtainingadetailednetworkpictureforbrainsof“higher”organismsis farbeyondthecurrentstateoftheartSsoweverQsignificantinsighthasbeengainedby studyingthestructureofspecificmoduleswithinacomplexbrainQandunderstanding howtheyrelatetooneanotherS lfinalexampleisthesetofnetworksthatmakeupacell’smetabolismSThereare

manywaystodefinethesenetworksQbutroughlyQthenodesarecompoundsthatplay aroleinametabolicprocessQandtheedgesrepresentchemicalinteractionsamong them[“[]SThereisconsiderablehopethatanalysisofthesenetworkscanshedlight onthecomplexreactionpathwaysandregulatoryfeedbackloopsthattakeplaceinside acellQandperhapssuggest“networkRcentric”attacksonpathogensthatdisrupttheir metabolismintargetedwaysS 2w5Exercises WS znereasonforgraphtheory’spowerasamodelingtoolisthefluiditywithwhich onecanformalizepropertiesoflargesystemsusingthelanguageofgraphsQandthen

systematicallyexploretheirconsequencesStnthisfirstsetofquestionsQwewillwork throughanexampleofthisprocessusingtheconceptofa pivotal nodeS qirstQrecallfromnhapterYthata shortestpath betweentwonodesisapathofthe minimumpossiblelengthSWesaythatanode is pivotal forapairofdistinctnodes and if liesoneveryshortestpathbetween and Mand isnotequalto either or NS qorexampleQinthegraphinqigureYSW[Qnode ispivotalfortwopairsethepair consistingof and Qandthepairconsistingof and SMyoticethat isnot pivotalforthepairconsistingof and sincetherearetwodi erentshortestpaths connecting and QoneofwhichMusing and

Ndoesn’tpassthrough SSo isnoton every shortestpathbetween and SNzntheotherhandQnode isnot pivotalforanypairsS MaN riveanexampleofagraphinwhich every nodeispivotalforatleastonepairof nodesSpxplainyouranswerS
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2.5.EXERCISES “] qigureYSW[etnthisexampleQnode ispivotalfortwopairsethepairconsistingof and Qandthepairconsistingof and SzntheotherhandQnode isnotpivotalforany pairsS MbN riveanexampleofagraphinwhich every nodeispivotalforatleasttwodi erent pairsofnodesSpxplainyouranswerS McN riveanexampleofagraphhavingatleastfournodesinwhichthereisasingle node thatispivotalfor every

pairofnodesMnotcountingpairsthatinclude NSpxplainyouranswerS YS tnthenextsetofquestionsQweconsiderarelatedclusterofdefinitionsQwhichseekto formalizetheideathatcertainnodescanplaya“gatekeeping”roleinanetworkSThe firstdefinitionisthefollowingewesaythatanode isa gatekeeper ifforsomeother twonodes and Qeverypathfrom to passesthrough SqorexampleQinthe graphinqigureYSW“Qnode isagatekeeperQsinceitliesforexampleoneverypath from to SMttalsoliesoneverypathbetweenotherpairsofnodes—forexampleQ thepair and QaswellasotherpairsSN

Thisdefinitionhasacertain“global”flavorQsinceitrequiresthatwethinkaboutpaths inthefullgraphinordertodecidewhetheraparticularnodeisagatekeeperSlmore “local”versionofthisdefinitionmightinvolveonlylookingattheneighborsofanodeS sere’sawaytomakethispreciseewesaythatanode isa localgatekeeper ifthere aretwoneighborsof Qsay and QthatarenotconnectedbyanedgeSMThatisQ for tobealocalgatekeeperQthereshouldbetwonodes and sothat and eachhaveedgesto QbutnottoeachotherSNSoforexampleQinqigureYSW“Qnode isalocalgatekeeperaswellasbeingagatekeeperfnode QontheotherhandQisa

localgatekeeperbutnotagatekeeperSMyode hasneighbors and thatarenot connectedbyanedgefhoweverQeverypairofnodes—including and —canbe connectedbyapaththatdoesnotgothrough SN Sowehavetwonewdefinitionse gatekeeper Qand localgatekeeper SWhenfacedwith
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“a CHAPTER2.GRAPHS qigureYSW“eyode isagatekeeperSyode isalocalgatekeeperbutnotagatekeeperS newmathematicaldefinitionsQastrategythatisoftenusefulistoexplorethemfirst throughexamplesQandthentoassessthematamoregenerallevelandtrytorelate themtootherideasanddefinitionsSwet’strythisinthenextfewquestionsS MaN

riveanexampleMtogetherwithanexplanationNofagraphinwhichmorethan halfofallnodesaregatekeepersS MbN riveanexampleMtogetherwithanexplanationNofagraphinwhichthereareno gatekeepersQbutinwhicheverynodeisalocalgatekeeperS [S Whenwethinkaboutasingleaggregatemeasuretosummarizethedistancesbetween thenodesinagivengraphQtherearetwonaturalquantitiesthatcometomindSzneis the diameter Qwhichwedefinetobethemaximumdistancebetweenanypairofnodes inthegraphSlnotheristhe averagedistance Qwhich—asthetermsuggests—isthe averagedistanceoverallpairsofnodesinthegraphS

tnmanygraphsQthesetwoquantitiesareclosetoeachotherinvalueSmutthereare graphswheretheycanbeverydi erentS MaN oescribeanexampleofagraphwherethediameterismorethanthreetimesas largeastheaveragedistanceS MbN oescribehowyoucouldextendyourconstructiontoproducegraphsinwhichthe diameterexceedstheaveragedistancebyaslargeafactorasyou’dlikeSMThatisQ foreverynumber Qcanyouproduceagraphinwhichthediameterismorethan timesaslargeastheaveragedistancejN