The robust yet fragile nature of the Internet.docx
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The robust yet fragile nature of the Internet.docx
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TherobustyetfragilenatureoftheInternet
Abstract
Thesearchforunifyingpropertiesofcomplexnetworksispopular,challenging,andimportant.Formodelingapproachesthatfocusonrobustnessandfragilityasunifyingconcepts,theInternetisanespeciallyattractivecasestudy,mainlybecauseitsapplicationsareubiquitousandpervasive,andwidelyavailableexpositionsexistateverylevelofdetail.Nevertheless,alternativeapproachestomodelingtheInternetoftenmakeextremelydifferentassumptionsandderiveoppositeconclusionsaboutfundamentalpropertiesofoneandthesamesystem.Fortunately,adetailedunderstandingofInternettechnologycombinedwithauniqueabilitytomeasurethenetworkmeansthatthesedifferencescanbeunderstoodthoroughlyandresolvedunambiguously.ThisarticleaimstomakerecentresultsofthisprocessaccessiblebeyondInternetspecialiststothebroaderscientificcommunityandtoclarifyseveralsourcesofbasicmethodologicaldifferencesthatarerelevantbeyondeithertheInternetorthetwospecificapproachesfocusedonhere(i.e.,scale-freenetworksandhighlyoptimizedtolerancenetworks).
ApopularcasestudyforcomplexnetworkshasbeentheInternet,withacentralissuebeingtheextenttowhichitsdesignandevolutionhavemadeit“robustyetfragile”(RYF),thatis,unaffectedbyrandomcomponentfailuresbutvulnerabletotargetedattacksonitskeycomponents.OnelineofresearchportraystheInternetas“scale-free”(SF)witha“hub-like”corestructurethatmakesthenetworksimultaneouslyrobusttorandomlossesofnodesyetfragiletotargetedattacksonthehighlyconnectednodesor“hubs”(1–3).Theresultingerrortolerancewithattackvulnerabilityhasbeenproposedasapreviouslyoverlooked“Achilles'heel”oftheInternet.Theappealofsuchasurprisingdiscoveryisunderstandable,becauseSFmethodsarequitegeneralanddonotdependonanydetailsofInternettechnology,economics,orengineering(4,5).
OnepurposeofthisarticleistoexplorehowthisSFdepictioncompareswiththerealInternetandexplainthenatureandoriginofsomeimportantdiscrepancies.AnotherpurposeistosuggestthatamorecoherentperspectiveontheInternetasacomplexnetwork,andinparticularitsRYFnature,ispossibleinawaythatisfullyconsistentwithInternettechnology,economics,andengineering.Acompleteexpositionreliesonthemathematicsofrandomgraphsandstatisticalphysics(6),whichunderlietheSFtheory,aswellasontheverydetailsoftheInternetignoredintheSFformulation(7).Nevertheless,weaimtoshowherethattheessentialissuescanbereadilyunderstood,ifnotrigorouslyproven,byusinglesstechnicaldetail,andthelessonslearnedarerelevantwellbeyondeithertheInternetorSF-networkmodels(8–10).
PowerLawsandSFModels
Onewidespreadfocusofattentionhasbeenon“powerlaws”(or“scaling”)ingraphvertexconnectivity.Foragraphhavingnvertices,letdidenotethedegreeofvertexi,1≤i≤n.WecallD={d1,d2,...,dn}thedegreesequenceofthegraph,assumedwithoutlossofgeneralityalwaystobeorderedd1≥d2≥···≥dn.LetG(D)denotethesetofallconnectedsimplegraphs(i.e.,noself-loopsorparalleledges)havingthesamegraphdegreeD.Wewillsaythatgraphsg∈G(D)havescaling-degreesequenceD(orDisscaling)ifforall1≤k≤ns≤n,Dsatisfiesapower-lawrank-sizerelationshipoftheform
where0 ThemostsignificantSFclaimsfortheInternetarethattheroutergraphhaspower-lawdegreesequencesthatgiverisetohubs,whichbySFdefinitionarehighlyconnectedverticesthatarecrucialtotheglobalconnectivityofthenetworkandthroughwhichmosttrafficmustpass(3).TheSFassertion(laterformalizedinref.12)isthatsuchhubsholdthenetworktogether,givingit“errortolerance”torandomvertexfailures,becausemostverticeshavelowconnectivity(i.e.,arenonhubs)butalsohave“attackvulnerability”totargetedhubremoval,apreviouslyoverlookedAchilles'heel.TherationaleforthisclaimcanbeillustratedbyusingthetoynetworksshowninFig.1,allofwhichhavetheidenticalscaling-degreesequenceDshowninFig.1e.Fig.1ashowsagraph(sizeissuesnotwithstanding)thatisrepresentativeofthetypeofstructuretypicallyfoundingraphsgeneratedbySFmodels,inthiscasepreferentialattachment(PA).Thisgraphisdrawnintwoways: theleftandrightvisualizationsemphasizethegrowthprocessandInternetproperties,respectively.Clearly,thehighest-degreenodesareessentialforgraphconnectivity,andthisfeaturecanbeseenevenmoreclearlyforthemoreidealizedSFgraphshowninFig.1b.Thus,theSFclaimswouldcertainlyholdiftheInternetlookedatalllikeFigs.1aandb.Aswewillsee,theInternetlooksnothinglikethesegraphsandismuchclosertoFig.1d,whichhasthesamedegreesequenceDbutisotherwisecompletelydifferent,withhigh-degreeverticesattheperipheryofthenetwork,wheretheirremovalwouldhaveonlylocaleffects.Thus,althoughscaling-degreesequencesimplythepresenceofhigh-degreevertices,theydonotimplythatsuchnodesformnecessarily“crucialhubs”intheSFsense. Viewlargerversion: Inthispage Inanewwindow DownloadPPT Fig.1. DiversityamonggraphshavingthesamedegreesequenceD.(a)RNDnet: anetworkconsistentwithconstructionbyPA.Thetwonetworksrepresentthesamegraph,butthefigureontherightisredrawntoemphasizetherolethathigh-degreehubsplayinoverallnetworkconnectivity.(b)SFnet: agraphhavingthemostpreferentialconnectivity,againdrawnbothasanincrementalgrowthtypeofnetworkandinaformthatemphasizestheimportanceofhigh-degreenodes.(c)BADNet: apoorlydesignednetworkwithoverallconnectivityconstructedfromachainofvertices.(d)HOTnet: agraphconstructedtobeasimplifiedversionoftheAbilenenetworkshowninFig.2.(e)Power-lawdegreesequenceDfornetworksshownina–d.Onlydi>1isshown. ThedeeperoriginsoftheclaimsinvolvingpowerlawsandhubsarisefromtheSFmodels'rootsinstatisticalphysics,inwhichanyparticulargraphisinterpretedasanelementfromalargerstatisticalensembleofgraphs,withprobabilityweightsthattypicallyariseeitherimplicitlythroughsomeunderlyingstochasticgenerationprocessorbyamechanismthatexplicitlyassignsaweighttoeachelementoftheensemble(13,14).Althoughthereexistavarietyofmethodsforgeneratingensemblesofgraphshavingscaling-degreesequences,includingPA,generalizedrandomgraph,powerlawrandomgraph(15),andrandomdegree-preservingrewiring(16),theresultingmodelsarewidelyconjecturedtobeasymptoticallyequivalent(e.g.,seeref.6andreferencestherein). Inparticular,foragraphghavingdegreesequenceD,wedefinethepurelygraph-theoreticquantitys(g)=Σ(i,j)∈E(g)didj,whereE(g)isthesetofedgesinthegraph.Itiseasytocheckthathighs(g)requireshigh-degreeverticestoconnecttootherhigh-degreevertices.Normalizingagainstsmax=max{s(g): g∈G(D)},wedefinethemeasure0≤S(g)≤1ofthegraphgasS(g)=s(g)/smax.Althoughs(g)andS(g)canbecomputedforanygraphanddonotdependonanyparticularconstructionmechanism,theyhaveaspecialmeaninginthecontextofensemblesofgraphs.Specifically,S(g)hasadirectinterpretationastherelativelog-likelihoodofagraphresultingfromthegeneralizedrandom-graphconstruction(17);thus,alloftheSF-model–generationmechanismsgenerateessentiallyonlyhighSgraphs.TheS-metricalsopotentiallyunifiesotheraspectsofSFgraphs,becauseitiscloselyrelatedtobetweenness,degreecorrelation(6),andgraphassortativity(18)andcapturesseveralnotionsofself-similarityrelatedtographtrimming,coarsegraining,andrandomrewiring(6). Thefocusonensemble-basedmethodsmeansthattheanalysisinSFmodelshasimplicitlyignoredthosegraphsthatareunlikelytoresultfromsuchconstructions,inparticulargraphswithsmallS.Thus,althoughpower-lawdegreedistributionsareunlikelyundersometraditionalrandomgraphconstructions[e.g.,Erdös–Renyírandomgraphs(19)],thereareamultitudeofothermodel-generationmechanismsthatgiverisetopowerlaws(20).TheSF-generatingmechanismsareonlyonekind,buttheytendtogenerateonlyhighSgraphs,whichleavesunexploredanenormousdiversityoflowSgraphs,asseeninFig.1.ThegraphsinFig.1aandbarerelativelylikelytoresultfromprobabilisticconstruction,whereasthegraphsinFig.1canddarevanishinglyunlikely.ThePA-typegraphshowninFig.1ahasS(ga)=0.61andistypicalofthegraphsthatarelikelyunderavarietyofrandom-generationmethods.ThegraphshowninFig.1bisthesmaxgraphandthusbydefinitionhasS(gb)=1.0.Itcanbethoughtofbothasthemostlikelygraphandalso(uniquely)asthemost“perfectly”SFgraphwiththisdegreesequence.Ofcourse,thesheerenormityofthenumberofdifferenthighSgraphsmeansthatanyparticularonegraph,eventherelativelymostlikely,isactuallyunlikelyinabsolutetermstobeselected.ThegraphsinFig.1canddhavethevaluesS(gc)=0.33andS(gd)=0.34,respectively;furthermore,therearerelativelyfewgraphswithSvaluesthislow,andthusanygraphssimilartothesearevanishinglyunlikelytoariseatrandom(6).Theremainde
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