Complex networks and decentralized search algorithms Jon Kleinberg Abstract - PDF document

ComplexnetworksanddecentralizedsearchalgorithmsJonKleinbergThestudyofcomplexnetworkshasemergedoverthepastseveralyearsasathemespanningmanydisciplines,rangingfrommathematicsandcomputersciencetothesocialandbiologicalsciences.AsigniÞcantamountofrecentworkinthisareahasfocusedonthede-velopmentofrandomgraphmodelsthatcapturesomeofthequalitativepropertiesobservedinlarge-scalenetworkdata;suchmodelshavethepotentialtohelpusreason,atagenerallevel,aboutthewaysinwhichreal-worldnetworksareorganized.Wesurveyoneparticularlineofnetworkresearch,concernedwithsmall-worldphenomenaanddecentralizedsearchalgorithms,thatillustratesthisstyleofanalysis.Webeginbydescribingawell-knownexperimentthatprovidedtheÞrstempiricalbasisfortheÒsixdegreesofseparationÓphenomenoninsocialnetworks;wethendiscusssomeprobabilisticnetworkmodelsmotivatedbythiswork,illustratinghowthesemodelsleadtonovelalgorithmicandgraph-theoreticquestions,andhowtheyaresupportedbyrecentempiricalstudiesoflargesocialnetworks.MathematicsSubjectClassiÞcation(2000).Primary68R10;Secondary05C80,91D30.Keywords.Randomgraphs,complexnetworks,searchalgorithms,socialnetworkanalysis.1.IntroductionOverthepastdecade,thestudyofcomplexnetworkshasemergedasathemerun-ningthroughresearchinawiderangeofareas.ThegrowthoftheInternetandtheWorldWideWebhasledcomputerscientiststoseekwaystomanagethecomplexityofthesenetworks,andtohelpusersnavigatetheirvastinformationcontent.Socialscien-tistshavebeenconfrontedbysocialnetworkdataonascalepreviouslyunimagined:datasetsoncommunicationwithinorganizations,oncollaborationinprofessionalcommunities,andonrelationshipsinÞnancialdomains.BiologistshavedelvedintotheinteractionsthatdeÞnethepathwaysofacellÕsmetabolism,discoveringthatthenetworkstructureoftheseinteractionscanprovideinsightintofundamentalbiologi-calprocesses.ThedrivetounderstandalltheseissueshasresultedinwhatsomehavecalledaÒnewscienceofnetworksÓÐaphenomenologicalstudyofnetworksastheyariseinthephysicalworld,inthevirtualworld,andinsociety.Atamathematicallevel,muchofthisworkhasbeenrootedinthestudyofrandomgraphs[14],anareaattheintersectionofcombinatoricsanddiscreteprobabilitythat SupportedinpartbyaDavidandLucilePackardFoundationFellowship,aJohnD.andCatherineT.MacArthurFoundationFellowship,andNSFgrantsCCF-0325453,IIS-0329064,CNS-0403340,andBCS-ProceedingsoftheInternationalCongressofMathematicians,Madrid,Spain,2006©2006EuropeanMathematicalSociety JonKleinbergisconcernedwiththepropertiesofgraphsgeneratedbyrandomprocesses.WhilethishasbeenanactivetopicofstudysincetheworkofErdšsandRŽnyiinthe1950s[26],theappearanceofrich,large-scalenetworkdatainthe1990sstimulatedatremendousinßuxofresearchersfrommanydifferentcommunities.Muchofthisrecentcross-disciplinaryworkhassoughttodeveloprandomgraphmodelsthatmoretightlycapturethequalitativepropertiesfoundinlargesocial,technological,andinformationnetworks;inmanycases,thesemodelsarecloselyrelatedtoearlierworkintherandomgraphsliterature,buttheissuesarisinginthemotivatingapplicationsleadtonewtypesofmathematicalquestions.Forsurveyscoveringdifferentaspectsofthisgeneralarea,andinparticularreßectingthevarioustechniquesofsomeofthedifferentdisciplinesthathavecontributedtoit,wereferthereadertorecentreviewpapersbyAlbertandBarab‡si[4],Bollob‡s[15],KleinbergandLawrence[39],Newman[52],andStrogatz[60],thevolumeofarticleseditedbyBen-Naimetal.[10],andthemonographsbyDorogovtsevandMendes[23]andDurrett[25],aswellasbooksbyBarab‡si[8]andWatts[62]aimedatmoregeneralaudiences.Whatdoesonehopetoachievefromaprobabilisticmodelofacomplexnetworkarisinginthenaturalorsocialworld?AbasicstrategypursuedinmuchofthisresearchistodeÞneastylizednetworkmodel,producedbyarandommechanismthatreßectstheprocessesshapingtherealnetwork,andtoshowthatthisstylizedmodelreproducespropertiesobservedintherealnetwork.Clearlythefullrangeoffactorsthatcontributetotheobservedstructurewillbetoointricatetobefullycapturedbyanysimplemodel.ButaÞndingbasedonarandomgraphformulationcanhelparguethattheobservedpropertiesmayhaveasimpleunderlyingbasis,eveniftheirspeciÞcsareverycomplex.WhileitiscrucialtorealizethelimitationsofthistypeofactivityÐandnottoreadtoomuchintothedetailedconclusionsdrawnfromasimplemodelÐthedevelopmentofsuchmodelshasbeenavaluablemeansofproposingconcrete,mathematicallyprecisehypothesesaboutnetworkstructureandevolutionthatcanthenserveasstartingpointsforfurtherempiricalinvestigation.Andatitsmosteffective,thisprocessofmodelingviarandomgraphscansuggestnoveltypesofqualitativenetworkfeaturesÐstructuresthatpeoplehadnotthoughttodeÞnepreviously,andwhichbecomepatternstolookforinnewnetworkdatasets.Intheremainderofthepresentpaper,wesurveyonelineofwork,motivatedbytheÒsmall-worldphenomenonÓandsomerelatedsearchproblems,thatillustratesthisstyleofanalysis.WebeginwithastrikingexperimentbythesocialpsychologistStanleyMilgramthatframestheempiricalissuesveryclearly[50],[61];wedescribeasequenceofmodelsbasedonrandomgraphsthatcaptureaspectsofthisphenomenon[64],[36],[37],[38],[63];andwethendiscussrecentworkthathasidentiÞedsomeofthequalitativeaspectsofthesemodelsinlarge-scalenetworkdata[1],[43],[49].Weconcludewithsomefurtherextensionstotheserandomgraphmodels,discussingtheresultsandquestionsthattheyleadto. JonKleinberg3.Basicmodelsofsmall-worldnetworksWhyshouldsocialnetworksexhibitthistypeofasmall-worldproperty?EarlierwesuggestedthatinterestingempiricalÞndingsaboutnetworksoftenmotivatethedevelopmentofnewrandomgraphmodels,butwehavetobecarefulinframingtheissuehere:asimpleabundanceofshortpathsisinfactsomethingthatmostbasicmodelsofrandomgraphsalreadyÒgetright.ÓAsaparadigmaticexampleofsucharesult,considerthefollowingtheoremofBollob‡sanddelaVega[14],[17].Theorem3.13.1)Fixaconstant.Ifwechooseuniformlyatrandomfromthesetofall-nodegraphsinwhicheachnodehasdegreeexactly,thenwithhighprobabilityeverypairofnodeswillbejoinedbyapathoflength(Followingstandardnotationandterminology,wesaythatthedegreeofanodeisthenumberofedgesincidenttoit.WesaythatafunctionisO(f(n))ifthereisasothatforallsufÞcientlylarge,thefunctionisboundedbycf(n).)Infact,[17]statesamuchmoredetailedresultconcerningthedependenceon,butthiswillnotbecrucialforourpurposeshere.PathlengthsthatarelogarithmicinÐormoregenerallypolylogarithmic,boundedbyapolynomialfunctionoflogÐwillbeourÒgoldstandardÓinmostofthisdiscussion.WewillkeepthetermÒsmallworldÓitselfinformal;butwewillconsideragraphtobeasmallworld,roughly,whenall(ormost)pairsofnodesareconnectedbypathsoflengthpolylogarithmicin,sinceinsuchacasethepathlengthsareexponentiallysmallerthanthenumberofnodes.WattsandStrogatz[64]arguedthatthereissomethingcrucialmissingfromthepictureprovidedbyTheorem3.1.Astandardrandomgraph(forexample,asinTheorem3.1)islocallyverysparse;withreasonablyhighprobability,noneoftheneighborsofagivennodearethemselvesneighborsofoneanother.Butthisisfarfromtrueinmostnaturallyoccurringnetworks:inrealnetworkdata,manyofanodeÕsneighborsarejoinedtoeachotherbyedges.(Forexample,inasocialnetwork,manyofourfriendsknoweachother.)Indeed,atanimplicitlevel,thisisalargepartofwhatmakesthesmall-worldphenomenonsurprisingtomanypeoplewhentheyÞrsthearit:thesocialnetworkappearsfromthelocalperspectiveofanyonenodetobehighlyÒclustered,Óratherthanthekindofbranchingtree-likestructurethatwouldmoreobviouslyreachmanynodesalongveryshortpaths.Thus,WattsandStrogatzproposedthinkingaboutsmall-worldnetworksasakindofsuperposition:astructured,high-diameternetworkwitharelativelysmallnumberofÒrandomÓlinksaddedin.Asamodelforsocialnetworks,thestructuredunderlyingnetworkrepresentstheÒtypicalÓsociallinksthatweformwiththepeoplewholivenearus,orwhoworkwithus;theadditionalrandomlinksarethechance,long-rangeconnectionsthatplayalargeroleincreatingshortpathsthroughthenetworkasaThiskindofhybridrandomgraphmodelhadbeenstudiedearlierbyBollob‡sandChung[16];theyshowedthatasmalldensityofrandomlinkscanindeedproduce JonKleinbergtimeÐaprocessthatcouldwellhavefailedtoreachthetarget,evenifashortpathexisted.Thisalgorithmicaspectofthesmall-worldphenomenonraisesfundamentalques-tionsÐwhyshouldthesocialnetworkhavebeenstructuredsoastomakethistypeofdecentralizedroutingsoeffective?ClearlythenetworkcontainedsometypeofÒgradientÓthathelpedparticipantsguidemessagestowardthetarget,andthisissome-thingthatwecantrytomodel;thegoalwouldbetoseewhetherdecentralizedroutingcanbeprovedtoworkinasimplerandom-graphmodel,andifso,totryextractingfromthismodelsomequalitativepropertiesthatdistinguishnetworksinwhichthistypeofroutingcansucceed.ItisworthnotingthattheseissuesreachfarbeyondtheMilgramexperimentorevensocialnetworks;routingwithlimitedinformationissomethingthattakesplaceincommunicationnetworks,inbrowsingbehaviorontheWorldWideWeb,inneurologicalnetworks,andinanumberofothersettingsÐsoanunderstandingofthestructuralunderpinningsofefÞcientdecentralizedroutingisaquestionthatspansallthesedomains.Tobeginwith,weneedtobepreciseaboutwhatwemeanbyadecentralizedalgorithm.Inthecontextofthegrid-basedmodelintheprevioussection,wewillconsideralgorithmsthatseektopassamessagefromastartingnodetoatarget,byadvancingthemessagealongedges.Ineachstepofthisprocess,thecurrentmessage-holderhasknowledgeoftheunderlyinggridstructure,thelocationofthetargetonthegrid,anditsownlong-rangecontact.Thecrucialpointisthatitdoesnotknowthelong-rangecontactsofanyothernodes.(Optionally,wecanchoosetohaveknowthepathtakenbythemessagethusfar,butthiswillnotbecrucialinanyoftheresultstofollow.)Usingthisinformation,mustchooseoneofitsnetworkneighborstopassthemessageto;theprocessthencontinuesfrom.WewillevaluatedecentralizedalgorithmsaccordingtotheirdeliverytimeÐtheexpectednumberofstepsrequiredtoreachthetarget,overarandomlygeneratedsetoflong-rangecontacts,andrandomlychosenstartingandtargetnodes.OurgoalwillbetoÞndalgorithmswithdeliverytimesthatarepolylogarithmicinItisinterestingthatwhileWattsandStrogatzproposedtheirmodelwithoutthealgorithmicaspectinmind,itisremarkablyeffectiveasasimplesysteminwhichtostudytheeffectivenessofdecentralizedrouting.Indeed,tobeabletoposethequestioninanon-trivialway,onewantsanetworkthatispartiallyknowntothealgorithmandpartiallyunknownÐclearlyintheMilgramexperiment,aswellasinothersettings,individualnodesuseknowledgenotjustoftheirownlocalconnections,butalsoofcertainglobalÒreferenceframesÓ(comparabletothegridstructureinoursetting)inwhichthenetworkisembedded.Furthermore,fortheproblemtobeinteresting,theÒknownÓpartofthenetworkshouldbelikelytocontainnoshortpathfromthesourcetothetarget,butthereshouldbeashortpathinthefullnetwork.TheWatts-Strogatzmodelcombinesallthesefeaturesinaminimalway,andthusallowsustoconsiderhownodescanusewhattheyknowaboutthenetworkstructuretoconstructshortDespiteallthis,theÞrstresulthereisnegative. JonKleinbergcontactifthisgetsthemessageclosertothetargetonthegrid;otherwise,itusesalocalcontactinthedirectionofthetarget.)Theanalysisofthisalgorithmproceedsbyshowingthat,foraconstant0,thereisaprobabilityofatleasteverystepthatthegriddistancetothetargetwillbehalved.Itisalsoworthnotingthattheproofcanbedirectlyadaptedtoagridinanyconstantnumberofdimensions;ananalogoustrichotomyarises,withpolylogarithmicdeliverytimeachievableonlyisequaltothedimension.Atamoregenerallevel,theproofofTheorem4.2(b)showsthatthecrucialprop-ertyofexponent2isthefollowing:ratherthanproducinglong-rangecontactsthatareuniformlydistributedoverthegrid(asonegetsfromexponent0),itproduceslong-rangecontactsthatareapproximatelyuniformlydistributedoverÒdis-tancescalesÓ:theprobabilitythatthelong-rangecontactofisatagriddistancebetween2and2awayfromisapproximatelythesameforallvaluesoffrom1tologFromthisproperty,oneseesthatthereisareasonablechanceofhalvingthemessageÕsgriddistancetothetarget,independentofhowfarawayitcurrentlyis.ThepropertyalsohasanintuitivelynaturalmeaninginthecontextoftheoriginalMilgramexperiment;subjecttoalltheothersimpliÞcationsmadeinthegridmodel,itsaysveryroughlythatdecentralizedroutingcanbeeffectivewhenpeoplehaveapproximatelythesamedensityofacquaintancesatmanydifferentlevelsofdistanceresolution.AndÞnally,thisapproximateuniformityoverdistancescalesisthetypeofqualitativepropertythatwementionedasagoalattheoutset.ItissomethingthatwecansearchforinothermodelsandinrealnetworkdataÐtasksthatweundertakeinthenexttwosections.5.DecentralizedsearchinothermodelsHierarchicalmodels.AnaturalvariationonthemodeloftheprevioussectionistosupposethatthenetworkisembeddedinahierarchyratherthanagridÐinotherwords,thatthenodesresideattheleavesofacomplete-arytree,andtheunderlyingÒdistanceÓbetweentwonodesisbasedontheheightoftheirlowestcommonancestorinthistree.Thereareanumberofsettingswheresuchamodelsuggestsitself.Tobeginwith,follow-upworkontheMilgramexperimentfoundthatmostdecisionsmadebyparticipantsonhowtoforwardtheletterwerebasedononeoftwokindsofcues:geographicalandoccupational[35].Andifatwo-dimensionalgridisnaturalasasimpleabstractionfortheroleofgeography,thenahierarchyisareasonable,alsosimple,approximationofthewayinwhichpeoplecategorizeoccupations.AnotherdomaininwhichhierarchiesarisenaturallyisintherelationshipsamongWebpages:forexample,aWebpageaboutsequenceanalysisoftheyeastgenomecouldbeclassiÞedasbeingaboutgenetics,moregenerallyaboutbiology,andmoregenerallystillaboutscience,whileaWebpagereviewingperformancesofVerdiÕs JonKleinbergabove,separatelyineachofthetrees;theresultisthateachnodeofacquiresedgesindependentlythroughitsparticipationineachtree.(Thereareafewminordifferencesbetweentheirprocedurewithineachhierarchyandthehierarchicalmodeldescribedabove;inparticular,theymapmultiplenodestothesameleafineachhierarchy,andtheygenerateeachedgebychoosingthetailuniformlyatrandom,andthentheheadaccordingtothehierarchicalmodel.Theresultisthatnodeswillnotingeneralallhavethesameout-degree.)Preciselycharacterizingthepowerofdecentralizedsearchinthismodel,atananalyticallevel,isanopenquestion,butWattsetal.describeanumberofinterestingÞndingsobtainedthroughsimulation[63].Theystudywhatisperhapsthemostnaturalsearchalgorithm,inwhichthecurrentmessage-holderforwardsthemessagetoitsneighborwhoisclosest(inthesenseoftreedistance)tothetargetinanyofthehierarchies.UsinganempiricaldeÞnitionofefÞciencyonnetworksofseveralhundredthousandnodes,theyexaminedthesetof(,q)pairsforwhichthesearchalgorithmwasefÞcient;theyfoundthatthisÒsearchableregionÓwascenteredaroundvaluesof1(butrelativelycloseto1),andonsmallconstantvaluesof.(Settingto2or3yieldedthewidestrangeofforwhichefÞcientsearchwaspossible.)Theresultingclaim,ataqualitativelevel,isthatefÞcientsearchisfacilitatedbyhavingasmallnumberofdifferentwaystomeasureproximityofnodes,andbyhavingasmallbiastowardnearbynodesintheconstructionofrandomedges.Modelsbasedonsetsystems.Onecanimaginemanyotherwaystoconstructnet-worksinthisgeneralstyleÐforexample,placingnodesonbothahierarchyandalatticesimultaneouslyÐandsoitbecomesnaturaltoconsidermoregeneralframe-worksinwhicharangeoftheseboundsonsearchabilitymightfollowsimultaneouslyfromasingleresult.Onesuchapproachisbasedonconstructingarandomgraphfromanunderlyingsetsystem,followingtheintuitionthatindividualsinasocialnetworkoftenformconnectionsbecausetheyarebothmembersofthesamesmallgroup[38].Inotherwords,twopeoplemightbemorelikelytoformalinkbecausetheyliveinthesametown,workinthesameprofession,havethesamereligiousafÞliation,orfollowtheworkofthesameobscurenovelist.Concretely,westartwithasetofnodes,andacollectionofsubsets,whichwewillcallthesetofgroups.Itishardtosaymuchofinterestforarbitrarysetsystems,butwewouldlikeourframeworktoincludeatleastthecollectionofballsorsubsquaresinagrid,andthecollectionofrootedsub-treesinahierarchy.Thusweconsidersetsystemsthatsatisfysomesimplecombinatorialpropertiessharedbythesetwotypesofcollections.SpeciÞcally,forconstants1,weimposethefollowingthreeproperties.(i)Thefullsetisoneofthegroups.(ii)Ifisagroupofsize2containinganode,thenthereisagroupthatisstrictlysmallerthan,buthassizeatleastmin(g,g JonKleinbergÞle-sharingsystems;andtheyhavebeenfoundtocapturesomeofthelarge-scalestructureofhumansocialnetworksasreßectedinon-linedata.Peer-to-peersystemsandfocusedwebcrawling.Arecurringthemeinrecentworkoncomplexnetworksisthewayinwhichsimpleprobabilisticmodelscanrapidlybecomedesignprinciplesfornewtypesofnetworkedsystems.Inthecaseofsmall-worldnetworks,oneobservesthisphenomenoninthedevelopmentofprotocolsforpeer-to-peerÞlesharing.Thedesignofsuchprotocolshasbecomeanactivetopicofresearchintheareaofcomputersystems,motivatedinpartbytheexplosionofpopularinterestinpeer-to-peerapplicationsfollowingtheemergenceofNapsterandmusicÞle-sharingin1999.Thegoalofsuchapplicationswastoallowalargecollectionofuserstosharethecontentresidingontheirpersonalcomputers,andintheirinitialconception,thesystemssupportingtheseapplicationswerebasedonacentralizedindexthatsimplystored,inasingleplace,theÞlesthatalluserspossessed.Thisway,queriesforaparticularpieceofcontentcouldbecheckedagainstthisindex,androutedtothecomputercontainingtheappropriateÞle.Themusic-sharingapplicationofthesesystems,ofcourse,ranintosigniÞcantlegaldifÞculties;butindependentoftheeconomicandintellectualpropertyissuesraisedbythisparticularapplication,itisclearthatsystemsallowinglargeusercommunitiestosharecontenthaveamuchbroaderrangeofpotential,lesscontroversialuses,providedtheycanbestructuredinarobustandefÞcientway.Thishasstimulatedmuchsubsequentstudyintheresearchcommunity,focusingondecentralizedinwhichoneseeksÞle-sharingsolutionsthatdonotrelyonasinglecentralizedindexofallthecontent.Inthisdecentralizedversionoftheproblem,thecruxofthechallengeisclear:eachuserhascertainÞlesonhisorherowncomputer,butthereisnosingleplacethatcontainsagloballistofalltheseÞles;ifsomeoneposesaquerylookingforaspeciÞcpieceofcontent,howcanweefÞcientlydeterminewhichuser(ifany)possessesacopyofit?Withoutacentralindex,weareinasettingverymuchlikethatoftheMilgramexperiment:usersmustposethequerytoasubsetoftheirimmediatenetworkneighbors,whointurncanforwardthequerytosomeoftheirneighbors,andsoforth.Andthisiswheresmall-worldmodelshaveplayedarole:anumberofapproachestothisproblemhavetriedtoexplicitlysetupthenetworkonwhichtheprotocoloperatessothatitsstructuremakesefÞcientdecentralizedsearchpossible.WereferthereadertothesurveysbyAspnesandShah[6]andLuaetal.[44]forgeneralreviewsofthisbodyofwork,andtheworkofClarkeetal.(asdescribedin[32]),Zhangetal.[67],Malkhietal.[45],andMankuetal.[46]formorespeciÞcdiscussionsoftherelationshiptosmall-worldnetworks.ArelatedsetofissuescomesupinthedesignoffocusedWebcrawlers.WhereasstandardWebsearchenginesÞrstcompileanenormousindexofWebpages,andthenanswerqueriesbyreferringtothisindex,afocusedcrawlerattemptstolocatepagesonaspeciÞctopicbyfollowinghyperlinksfromonepagetoanother,withoutÞrstcompilinganindex.Again,theunderlyingissuehereisthedesignofdecentralized JonKleinbergtheopportunitytoinvestigate,overaverylargepopulation,howthedensityofsocialnetworklinksdecayswithdistance.Anon-trivialtechnicalchallengethatmustbeovercomeinordertorelatethisdatatotheearliermodelsisthatthepopulationdensityoftheU.S.isextremelynon-uniform,andthismakesitdifÞculttointerpretpredictionsbasedonamodelinwhichnodesaredistributeduniformlyoveragrid.Thegeneralizationtogroupstructuresintheprevioussectionisonewaytohandlenon-uniformity;Liben-Nowelletal.proposeanalternativegeneralization,rank-basedfriendships,thattheyarguemaybemoresuitabletothegeographicdatahere[43].Intherank-basedfriendshipmodel,onehasasetofpeopleassignedtolocationsonatwo-dimensionalgrid,whereeachgridnodemayhaveanarbitrarypositivenumberofpeopleassignedtoit.Byanalogywiththegrid-basedmodelfromSection4,eachpersonchoosesalocalcontactineachofthefourneighboringgridnodes,andthenchoosesanadditionallong-rangeasfollows.First,ranksallotherpeopleinorderoftheirdistancetoherself(breakingtiesinsomecanonicalway);weletrank(w)denotethepositionoforderedlist,andsaythatisatrankwithrespecttothenchoosesasherlong-rangecontactwithprobabilityproportionalto1(w)Notethatthismodelgeneralizesthegrid-basedmodelofSection4,inthesensethatthegrid-basedmodelwiththeinverse-squaredistributioncorrespondstorank-basedfriendshipinwhichthereisonepersonresidentateachgridnode.However,therank-basedfriendshipconstructioniswell-deÞnedforanypopulationdensity,andLiben-Nowelletal.provethatitsupportsefÞcientdecentralizedsearchingeneral.TheyanalyzeadecentralizedgreedyalgorithmthatalwaysforwardsthemessagetoagridnodeascloseaspossibletothetargetÕs;andtheydeÞnethedeliverytimeinthiscasetobetheexpectednumberofstepsneededtoreachthegridnodecontainingthetarget.(Sowecanimaginethatthetaskhereistoroutethemessagetothehometownofthetarget,ratherthanthetargethimself;thisisalsoconsistentwiththedataavailablefromLiveJournal,whichonlylocalizespeopletotheleveloftowns.)Theorem6.16.1)Foranarbitrarypopulationdensityonagrid,theexpecteddeliverytimeofthedecentralizedgreedyalgorithmintherank-basedfriendshipmodelOntheLiveJournaldata,Liben-Nowelletal.examinethefractionoffriendships(v,w)isatrankwithrespectto.TheyÞndthatthisfractionisveryclosetoinverselinearin,inclosealignmentwiththepredictionsoftherank-basedfriendshipmodel.ThisÞndingisnotableforseveralreasons.First,aswiththee-mailnetworkconsideredbyAdamicandAdar,thereisnoapriorireasontobelievethatalarge,apparentlyamorphoussocialnetworkshouldcorrespondsocloselytoadistributionpredictedbyasimplemodelforefÞcientdecentralizedsearch.Second,geographyisplayingastrongroleheredespitethefactthatLiveJournalisanon-linesysteminwhichtherearenoexplicitlimitationsonforminglinkswithpeoplearbitrarilyfaraway;asaresult,onemighthave(incorrectly)conjecturedthatitwouldbedifÞcult JonKleinberg-dimensionallattice,andineachtimestepeachnodepicksasingleothernodeandtellseverythingitcurrentlyknowsto;nodeisselectedastherecipientofthisinformationwithprobabilityproportionalto(v,w).Informationoriginatingatonenodethusspreadstoothernodes,relayedinanepidemicfashionovertime.Now,ifasinglenodeinitiallypossessesanewpieceofinformationattime0,howlongwillittakebeforeknowledgeofthisinformationhasspreadtoagivennode?Themainresultof[34]isthatthetimerequiredforthisispolylogarithmicin,ispolylogarithmicin(v,w)butindependentofd,andispolynomialin(v,w).Heretoothecaseisnotwellunderstood,whichisinterestingbecausethistransitionalvaluehasparticularimportanceinapplicationsofgossipalgorithmstodistributedcomputingsystems[54].(See[34]forpartialresultsconcerningForthespeciÞcgrid-basedmodeldescribedinSection4,MartelandNguyenshowedthatwithhighprobabilitythediameterisproportionaltolog,in-dimensionalcase[48].TheyalsoidentiÞedtransitionsatanalogoustothecaseoflong-rangepercolation[53].Inparticular,theirresultsshowthatwhiledecentralizedsearchcanconstructapathoflengththereinfactexistpathsthatareshorterbyalogarithmicfactor.(Notealsothecontrastwiththecorrespondingresultsforthelong-rangepercolationmodelwheninthegrid-basedmodel,theout-degreeofeachnodeisboundedbyaconstant,soadiameterproportionaltologisthesmallestonecouldhopefor;inthecaseoflong-rangepercolation,ontheotherhand,thenodedegreeswillbeunbounded,allowingforsmallerdiameters.)Decentralizedsearchwithadditionalinformation.Anumberofpapershavestud-iedthepowerofdecentralizedsearchalgorithmsthatareprovidedwithsmallamountsofadditionalinformation[28],[42],[47],[48],[66].Whereasthemodelofdecentral-izedalgorithmsinSection4chargedunitcosttothealgorithmforeachnodevisited,themodelsinthesesubsequentpapersmakethefollowingdistinction:anodemayÒconsultÓasmallnumberofnearbynodes,andthenbasedonwhatitlearnsfromthisconsultation,itchoosesanodetoforwardthemessagesto.Inboundingthenumberofstepstakenbythealgorithm,onlythemessage-forwardingoperationsarecounted,nottheconsultation.Inparticular,LebharandSchabanel[42]consideranalgorithminwhichthenodecurrentlyholdingthemessageconsultsasetofuptonodeswithinasmallnumberofstepsofit;afterthis,itforwardsthemessagealongapathtothenodethatisclosesttothetargetingriddistance.Theyshowthat,intotal,theexpectednumberofnodesconsultedbythisprocessis(asinthedecentralizedal-gorithmfromSection4),andthattheactualpathconstructedtothetargethasonlyloglogManku,Naor,andWieder[47]considerasimpleralgorithminthelong-rangepercolationmodelonthe-dimensionallattice.NotethatnodesherewillhaveunboundeddegreesÐproportionaltologinexpectation,rather JonKleinbergSlivkins[59]considersadifferentsetting,inwhichnodesareembeddedinanunderlyingmetricspace.Heshowsthatifthemetricis,inthesensethateveryballcanbecoveredbyaconstantnumberofballsofhalftheradius(seee.g.[7],[30]),thenthereisamodelsuchthateachnodegeneratesapolylogarithmicnumberoflong-rangecontactsfromspeciÞeddistributions,andadecentralizedalgorithmisthenabletoachieveapolylogarithmicdeliverytime.(SomeofthelogarithmicdependencehereisontheaspectratioofthemetricÐtheratioofthelargesttothesmallestdistanceÐbutitispossibletoavoidthisdependenceintheboundonthedeliverytime.See[59]forfurtherdetailsonthisissue.)Finally,otherworkhasstudiedsearchalgorithmsthatexploitdifferencesinnodedegrees.Thereareindicationsthatpeoplenavigatingsocialstructures,insettingssuchassmall-worldexperiments,takeintoaccountthefactthatcertainoftheirac-quaintancessimplyknowalargenumberofpeople[22].Similarly,inpeer-to-peernetworks,itisalsothecasethatcertainnodeshaveanunusuallylargenumberofneighbors,andmaythusbemoreusefulinhelpingtoforwardqueries.Adamicetal.[2]formalizetheseconsiderationsbystudyingarandomgraphmodelinwhichhigh-degreenodesarerelativelyabundant,anddecentralizedsearchalgorithmsonlyhaveaccesstoinformationaboutdegreesofneighboringnodes,nottoanyembed-dingofthegraph(spatialorotherwise).Throughsimulation,theyÞndthatforcertainmodels,knowledgeofdegreesprovidesanimprovementinsearchperformance.SimsekandJensen[58]consideramodelwhichcombinesspatialembeddingwithvariablenodedegrees.SpeciÞcally,theystudyavariantofthegrid-basedmodelfromSection4inwhichnodeshavewidelyvaryingdegrees,andadecentralizedalgorithmhasaccessbothtothelocationsofitsneighborsandtotheirdegrees.Throughsim-ulation,theyÞndthataheuristictakingboththesefactorsintoaccountcanperformmoreefÞcientlythandecentralizedalgorithmsusingonlyoneofthesesourcesofinformation.Findingtheoptimalwaytocombinelocationanddegreeinformationindecentralizedsearch,andunderstandingtherangeofnetworksthataresearchableundersuchoptimalstrategies,isaninterestingdirectionforfurtherresearch.8.ConclusionWehavefollowedaparticularstrandofresearchrunningthroughthetopicofcomplexnetworks,concernedwithshortpathsandtheabilityofdecentralizedalgorithmstoÞndthem.Assuggestedinitially,thesequenceofideashereischaracteristicoftheßavorofresearchinthisarea:anexperimentinthesocialsciencesthathighlightsafundamentalandnon-obviouspropertyofnetworks(efÞcientsearchability,inthiscase);asequenceofrandomgraphmodelsandaccompanyinganalysisthatseekstocapturethisnotioninasimpleandstylizedform;asetofmeasurementsonlarge-scalenetworkdatathatparallelsthepropertiesofthemodels,insomecasestoasurprisingextent;andarangeofconnectionstofurtherresultsandquestionsinalgorithms,graphtheory,anddiscreteprobability. JonKleinbergquestionalsomanifestsitselfinthegossipproblemdiscussedinSection7,wherewenotedthatthetransitionalvaluearisesindistributedcomputingapplications(seethediscussionin[34],[54]).3.Pathsoflogarithmiclength.Itwouldbeinterestingtoknowwhetherthereisadecentralizedalgorithminthe-dimensionalgrid-basedmodel,attheÒsearchableexponentÓ,thatcouldconstructpathsoflengthwhilevisitingonlyapolylogarithmicnumberofnodes.ThiswouldimprovetheresultofLebharandSchabanel[42]toanasymptoticallytightboundonpathlength.4.Small-worldnetworkswithanarbitrarybasegraph.Itwouldalsobeinterest-ingtoresolvetheopenproblemofFraigniaud[27]describedinSection7,formalizingthequestionofwhetheranygraphcanbeturnedintoanefÞcientlysearchablesmallworldbyappropriatelyaddinglong-rangelinks5.Extendingthegroup-basedmodel.Theorem5.2onthegroup-basedmodelcontainedapositiveresultgeneralizingtheonesforgridsandhierarchies,anditcontainedageneralnegativeresultforthecasewhenlong-rangeconnectionwereÒtoolong-rangeÓ(i.e.withexponent1).However,itdoesnotfullygeneralizetheresultsforgridsandhierarchies,becausetherearesetsystemssatisfyingconditions(i),(ii),and(iii)ofthetheoremforwhichefÞcientdecentralizedsearchispossibleevenforexponents1.ItwouldbeinterestingtoÞndavariationonthesethreepropertiesthatstillgeneralizesgridsandhierarchiesinanaturalway,andforwhich1istheuniqueexponentatwhichefÞcientdecentralizedsearchispossible.6.Multiplehierarchies.ObtainingprovableboundsfordecentralizedsearchintheÒmultiplehierarchiesÓmodelofWatts,Dodds,andNewman[63]isalsoanopenques-tion.SuchresultscouldformaninterestingparallelwiththeÞndingstheydiscoveredthroughsimulation.WithsomesmallmodiÞcationstothemodelofWattsetal.,onecancastitinthegroup-basedmodelofSection5,andsoitisentirelypossiblethatprogressonthisquestionandthepreviouscouldbecloselyconnected.7.Theevolutionofsearchablenetworks.Theremainingquestionshaveamoregeneralßavor,wheremuchofthechallengeistheformalizationoftheunderlyingissue.Tobeginwith,thecurrentmodelssupportingefÞcientdecentralizedsearchareessentially,inthattheydescribehowtheunderlyingnetworkisorganizedwithoutsuggestinghowitmighthaveevolvedintothisstate.Whatkindsofgrowthprocessesorselectivepressuresmightexisttocausenetworkstobecomemoreef-Þcientlysearchable?InterestingnetworkevolutionmodelsaddressingthisquestionhavebeenproposedbyClausetandMoore[19]andbySandberg[56],bothbasedonfeedbackmechanismsbywhichnodesrepeatedlyperformdecentralizedsearchesand Noteaddedinproof:Fraigniaud,Lebhar,andLotkerhaveveryrecentlyannouncedanegativeresolutionofthisquestion,constructingafamilyofgraphsthatcannotbeturnedintoefÞcientlysearchablesmallworldsbythisprocess. JonKleinbergOnecanformulatemanyspeciÞcquestionsofthisßavor.Forexample,givenanetworkknowntobegeneratedbythegrid-basedmodelwithagivenexponentcanweapproximatelyreconstructthepositionsofthenodesonthegrid?Whatifwearenottoldtheexponent?Canwedeterminewhetheragivennetworkwasmorelikelytohavebeengeneratedfromagrid-basedmodelwithexponent?Orwhatiftherearemultiplelong-rangecontactspernode,andweareonlyshownthelong-rangeedges,notthelocaledges?Aparallelsetofquestionscanbeaskedforthehierarchicalmodel.QuestionsofthistypehavebeenconsideredbySandberg[55],whoreportsontheresultsofcomputationalexperimentsbutleavesopentheproblemofobtainingprovableguarantees.BenjaminiandBerger[11]poserelatedquestions,includingtheproblemofreconstructingthedimensionoftheunderlyinglatticewhenpresentedwithagraphgeneratedbylong-rangepercolationonaÞnitepieceof10.Comparingnetworkdatasets.Aswesawearlier,themodelsproposedinSections4and5suggestageneralperspectivefromwhichtoanalyzenetworkdatasets,bystudyingthewayinwhichthedensityoflinksdecayswithincreasingdistanceorincreasinggroupsize(e.g.[1],[43]).OnecouldnaturallyusethisstyleofanalysistocomparerelatednetworkdatasetsÐforexampletakingthepatternsofcommunicationdifferentorganizations(asAdamicandAdardidforthecorporatelabtheystudied),anddeterminingexponentsforeachsuchthattheprobabilityofalinkbetweenindividualsinagroupofsizescalesapproximatelyasintheorganization.DifferencesamongtheseexponentswouldsuggeststructuraldifferencesbetweentheorganizationsatagloballevelÐcommunicationinsomeismorelong-range,whileinothersitismoreclusteredatthelowlevelsofthehierarchy.ItwouldbeinterestingtounderstandwhetherthesedifferencesinturnwerenaturallyreßectedinotheraspectsoftheorganizationsÕbehaviorandperformance.Moregenerally,large-scalesocial,technological,andinformationnetworksaresufÞcientlycomplexobjectsthattheguidingprinciplesprovidedbysimplemodelsseemcrucialforourunderstandingofthem.Theperspectivesuggestedherehasof-feredonesuchcollectionofprinciples,highlightinginparticularthewaysinwhichthesenetworksareintertwinedwiththespatialandorganizationalstructuresthattheyinhabit.Onecanhopethataswegatheranincreasingrangeofdifferentperspec-tives,ourunderstandingofcomplexnetworkswillcontinuetodeepenintoarichandinformativetheory.References[1]Adamic,L.,Adar,E.,Howtosearchasocialnetwork.SocialNetworks(3)(2005),[2]Adamic,L.A.,Lukose,R.M.,Puniyani,A.R.,Huberman,B.A.,SearchinPower-LawNetworks.Phys.Rev.E(2001),46135Ð46143. 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