-
OntheMarginalUtility of Network TopologyMeasurements
PaulBarford,AzerBestavros,JohnByers,Mark Crovella
Abstract—The cost and complexity of deploying measurement
in-
frastructur e in the Inter net for the purpose of analyzingits
structur e and behavior is considerable. Basic questionsabout the
utility of increasingthe number of measurementsand measurement
siteshave not yet beenaddressedwhichhasled to a “mor eis better”
approachto wide-areameasure-ment studies. In this paper, we step
toward a more quan-tifiable understanding of the marginal utility
of performingwide-areameasurementsin the context of Inter net
topologydiscovery. Wecharacterizethe observable topology in termsof
nodes,links, nodedegreedistrib ution, and distrib ution
ofend-to-endflows using statistical and
information-theoretictechniques. We classify nodesdiscovered on the
routesbe-tweena setof 8 sourcesand 1277destinationsto differ
entiatenodeswhich make up the so called “backbone” fr om thosewhich
border the backboneand thoseon links betweentheborder nodesand
destination nodes. This processincludesreducing nodesthat advertise
multiple interfaces to singleIP addresses.We show that the utility
of adding sourcesbe-yond the secondsourcequickly diminishesfr om
the perspec-tive of interface, node, link and node degreediscovery.
Wealso show that the utility of adding destinationsis constantfor
interfaces,nodes,links and node degreeindicating thatit is more
important to add destinationsthan sources.
Keywords— Network measurement, traceroute, topologydiscovery,
Inter net tomography
I . INTRODUCTION
An emergingstrategy to gain insightinto
theconditionsandconfigurationinsidethe Internetis the useof
end-to-endmeasurementsfrom a setof distributedmeasurementpoints.A
designgoalof theInternethasbeento emphasize
This work was partially supportedby NSF
researchgrantsCCR-9706685andANIR-9986397.Thedatausedin this
researchwascol-lectedaspartof CAIDA’s skitter
initiative,http://www.caida.org. Sup-port for skitteris providedby
DARPA, NSF, andCAIDA membership.
P. Barford is with the University of Wisconsin, Madison.
A.Bestavros, J. Byers and M. Crovella are with the ComputerSci-ence
Departmentat Boston University. E-mail:
[email protected],�best,byers,crovella� @cs.bu.edu.
simplicity in its internalcomponents;for this
reason,mea-surementsmadeatnetwork
endpointsareespeciallyattrac-tive. An exampleof this approachis
theuseof tracer-oute [17] for thediscovery of network connectivity
androuting.
While tracerouteis remarkablyflexible andinformative,it is
anopenquestionhow usefultracerouteis for uncov-ering
topologicalinformation aboutthe Internet. In thispaperwe study the
useof tracerouteas a tool for Inter-net topology discovery. We
considerthe commoncase,in
whichactivemeasurementsites(traceroutesources)arerelatively
scarce,while passive measurementsites(tracer-oute targets) are
plentiful. In such experiments,eachtraceroutesourceis ableto
discover a directedgraph,in-ducedby IP routing,from itself to all
of thedestinations.1
We are interestedin the propertiesof the graph that isformedby
theunionof theseindividual views.
In orderto find theunionof theseviews it is necessaryto identify
routersthat advertisemultiple interfaces,andto
associateeachadvertisedinterfacewith a router. Ourfirst
contribution is to discussour experiencesin solvingthis
problemandassesstheimportanceof this issuewhenmergingmultiple
traceroutesinto asinglegraph.
Our main contribution is to show how studying
thisgraphhelpsclarify how end-to-endpathspassthroughtheInternet.A
principalobservation is that themarginal util-ity of
addingadditionalactive measurementsitesdeclinesrapidly after the
secondor third site. This motivatesarough model for the routing
graphdiscoveredby tracer-outeasarichly-connected“switchingcore”
fedby ingressandegresspaths(“feeders”). Our work indicatesthat
thecoreconsistsof arelatively smallfractionof nodesandweshow that
almostall pathsin our datapassthroughthiscore.
If thesource-destinationpairsin ourdataarerepresenta-tiveof
typicalendpointpairsfor IP flows, thentheswitch-ing coreis commonto
mostend-to-endpathstakenin theInternet. Thus the propertiesof the
core are especially�
Wemakethesimplifying assumptionthatIP routingpathsarestableover
the timeframeof individual tracerouteexecutions;while this
as-sumptionis sometimesincorrect,it freesus to focuson a
differentsetof questions.Notethatthisassumptiondoesnotimply
thattheresultingdirectedgraphfrom a sourceis a tree.
-
Leaf StubBorder Backbone
Fig. 1. Classificationof Internetnodes
interestingfor understandingInternetperformance.
Wenotethat,comparedto thesetof all switchingcorenodespresentin
ourdataset,themajority arevisible from only
asinglemeasurementsource.Thatis, setsof IP flows origi-natingfrom
differentlocationstendto passthroughsimilarsetsof switchingnodeson
their way to commondestina-tions. This makesit relatively
lessproductive to discovernew switchingnodesby
addingsources,evenwhenthesetof measurementdestinationsis large.
To assistusin ourtask,wehaveleverageddetailedrout-ing
tracesgatheredby CAIDA (Cooperative Associationfor Internet Data
Analysis) for the Skitter project [11].Thesetracesspanthousandsof
routesbetween8 sourcesand1277destinationstakenrepeatedlyover
thecourseofseveral months. While we canprovide no
guaranteethattheCAIDA measurementsiteswerechosenin a
represen-tative way, the locationof
thesitesaregeographicallydi-verse,spanningNorth
America,EuropeandAsia. Com-piling togetherall nodesandedgesof
thegraphvisitedbyroutesin thesetraces,we built up a partial
pictureof theway the Internetbackboneappearedin May 2000.
Then,usingthis pictureasourbaseline,we go backto thetracesto
observewhich paths,or collectionsof paths,weremostproductive in
generatingtheoverallmap.
To understandthetopologydiscoveryprocessin greaterdetail, we
employ a nodeclassificationtechniquewhichorganizesnodesinto oneof
four types:leafs,stubs,borderandbackboneillustratedin Figure1. For
the graphthatwe evaluate(after resolvingroutersthat
advertisemulti-ple interfacesto a singlenode)over half of
thenodesdis-coveredareclassifiedasbackbonenodeswhile
lessthan10%arebordernodes,giving a pictureof thecollectedIProutesas
consistingof a large backbonewith somewhatlimited
ingressandegress.Much of our analysisfocuseson marginal utility
with respectto thediscovery andchar-acterizationof
backbonenodes.
The restof the paperis organizedas follows. In Sec-tion II we
describerelatedanalyticalwork in evaluatingtheeffectivenessof
deploying wide-areameasurementin-frastructurewith a focuson
topologymapping.In SectionIII, weestablishbasicdefinitionsfor
thenetworkdiscoveryproblemswe considerandoutlinehow to
casttheseprob-
lemsin amarginalutility framework. In SectionIV, wede-scribeour
dataset,our graphclassificationprocedureandthe limitations of our
approach.We presentour statisti-calresultsfor
interfacedisambiguation,nodeclassificationandmarginal utility in
SectionV. We defineinformation-theoretictoolsandresultsfrom
theirapplicationto thedatain SectionVI.
Wesummarize,concludeanddiscussfuturework in SectionVII.
I I . RELATED WORK
A numberof researchgroupshave generatedmapsofthe Internetusing
route tracing tools suchastracer-oute [8], [11] andhavebuilt
repositoriesof Internetmap-ping information.We now survey
themostcloselyrelatedof thoseworksto ours.
Work by Govindan [24], [14] outlinesheuristic tech-niquesfor
generatingcompletedomainmaps.Oneof thechallengesin this areagoesfar
beyond thecapabilitiesoftraceroute,and lies in mappingout the
nooksand cran-nies of regions in autonomoussystems(AS’s) which
donot transit a substantialamountof data. This work
alsodiscussestheproblemof alias resolutionin detail,whichis
thesameasour interfacedisambiguationproblem.Theyemploy
thesametechniquesaswe do to resolve multipleinterfacesat
asinglenode.
Jaminetal [18] studyalgorithmsfor effectiveplacementof
Internetinstrumentationin thecontext of their IDMapsproject,a
projectwhich seeksto provide
anInternet-widedistanceestimationservice,following
thearchitecturede-signedin [12]. Themajorityof theirwork
focusesonalgo-rithmic approachesfor placinga fixedsetof
measurementsitesongeneratedtopologies,andmeasurementsontheef-fectivenessof
the placement.While their work mentionsdiminishingreturnsin
thecontext of infrastructureplace-ment,it
doesnotprovideanalyticalresultsin thisarea.
PansiotandGrad[19] reporton the topologyresultingfrom a
detailedcollectionof end-to-endroutesthey col-lectedin 1995 with
the goal of constructingrepresenta-tive multicasttrees. Using
traceroute,they tracedroutesto
5000geographicallydistributedhostschosenfrom theirnetwork
accountingdatabase.Thenthey choseasubsetof11 of thesehoststo be
additionalsourcesof routes,andran traceroutefrom these11 hoststo
eachof the original5000hosts(with theassistanceof
theLooseSourceRout-ing option). In the topologyrevealedby this
experiment,they found that the routesfrom any subsetof six
sourcescontainednearly90% of the nodesandedgesultimatelydiscovered.
They alsoprovided a classificationof nodessimilar to theonewe
provide andpresentthedistributionof the degreeof nodesof the
graphthey discover, a dis-tribution which clearly follows a power
law. (This power
-
law andevidenceof otherpower laws in this dataset,aswell asin
otherdatasets,werereportedin [10]). However,they provide no
qualitative discussionof thecharacteriza-tion of
thetopologythatthey obtain,nordothey
attempttoquantifythemarginalvalueof
informationgainedasmea-surementsareadded.
Broido andClaffy [5] alsoleveragetraceroutedatasetsfrom CAIDA to
build up andstudy the aggregation of asetof treetopologiesinducedby
IP routing. While theireffort doesprovide useful
characterizationsand insightsinto thesetopologies,it doesnot
focuson thequestionsofmarginalutility whichwestudyhere.
Paxson[20], [21], [22] deployeda “network probedae-mon” (NPD) at
37 sites in the wide-area,which usedtraceroute to
investigateend-to-endroutingbehaviorandlater, performanceof
transportprotocolsbetweenallpairs of sitesover several weeks. His
work emphasizedthe importanceof exploring a large numberof
pathstoobserve rare and occasionallyanomalousrouting behav-ior.
Paxsonalsostudiedtheissueof interfacedisambigua-tion in [21] from
the perspective of resolving nodestogeographiclocationsandnot
necessarilyspecificrouters.Wide-areameasurementandanalysiscontinuesto
bea fo-cusof many researchandindustrygroupsincludingNIMI[2], WAWM
[3] andSurveyor [25]. Anotherpieceof gen-erally relatedwork
aretheInternetweatherreportssuchas[27], [26].
Thesearegeneralcompilationsof the packetlossandroundtrip
timemeasurementsfrom Internetmon-itoring boxesdeployedin
thewide-area.
Finally, other recentstudieshave
usedmeasurement-basedapproachesto studyaspectsof
theInternettopology,albeit usingdifferent tools.
Someresearchershave usedlogs collectedin the wide-areaby
BGP-capableroutersto study the effectsof policy-basedrouting, with
an em-phasison quantifying the inflation in route lengths[16],[31],
[28]. At a higherlevel of abstraction,therehasbeenconsiderablework
onunderstandingAS-level connectivity[13], [4] includingwork which
leveragestraceroutemea-surementsandBGProutesto helpinfer AS-Level
connec-tivity [7], [6], Thesepiecesof work, like
ours,emphasizetheimportanceof incorporatingsnapshotstakenfrom
mul-tiple vantagepointsto providing themostcompletereflec-tion of
theoverall topology.
I I I . DEFINITIONS AND OBJECTIVES
The network discovery problemswe considerhave
anaturalgraph-theoreticformulation, study of which maybe of
independentinterest both to theoristsand to re-searcherswho wish to
bettercharacterizenetwork topolo-gies.Consideranetwork
topologyrepresentedby anundi-rectedgraph �������
��� in which � ������� . The central
questionwhich we studyis theextent to which theunder-lying
topologycanaccuratelybecharacterizedasthenum-berof
end-to-endobservationpointsgrows. In practice,weassumethat �
sourcesand � destinationsarechosenuni-formly at randomfrom
thevertex setof this graph. Thenwe considerthefractionof thevertex
setandedgesetthatis coveredby thesetof routingpathsfrom
thesourcestoeachof thedestinations,usingthefollowing
terminology.
Definition1: Givenagraph������������ andasubgraph��������
�!��"�#� of � , thenodecoverage of � by ��� is theratio $
%'&($$ %)$ . Similarly, theedge coverage of � by ��� is
theratio $ *+&($$ *�$ .
Definition2: Take a setof sourcevertices ,�-.� anda set of
destinationvertices / -0� . Also assumethatwe have a
routingalgorithm 1 which selectsfixed routesbetweenall pairs 2 34,
, 5637/ . Wedefinetheunionof thesetof �28959� pathsin � to
bethesubgraphof � inducedby1 onall pairsof routesfrom , to / .
The subgraphinducedby a routing algorithm corre-spondsto
overlaysof “projections”from multiplesources,i.e. theunionof
individual directedgraphsrootedat thesevantagepointsto thesetof
destinations.Thefunctionsde-finedbelow describehow
coverageincreasesasthenum-ber of endpointsusedto generatethe
inducedsubgraphgrows.
Definition3: For a graph � with routesinducedby 1andfor
parameters� and � , let :9�7� denotethe ex-pectednodecoverageof �
by thesubgraphinducedby arandomlychosensetof sources, of
cardinality � , a ran-domlychosensetof destinations/ of cardinality
� . Sim-ilarly, let ?@;=��AB�7� denotetheexpectededgecoverageof� by
sucha subgraph.
The rateat which :
-
caseariseswhen �X�Y� ).2 A relateddirectionof futureinterestlies
in the characterizationand understandingofthoseregionsof
theInternettopologywhich arerelativelydifficult to uncover
usingtraceroute.Sucha studycouldconceivably leadto a
betterunderstandingon theconnec-tion
betweentopologyandroutingbehavior or providefur-ther insight into
relationshipsbetweentopologyandpeer-ing agreements.
We focus specificallyon marginal utility, i.e. the
in-crementalbenefitobtainedby conductingoneor moread-ditional
measurements.For edgecoverage,we definethemarginal utility of
addingtargetsasfollows (relateddefi-nitionsaresimilar):
Definition4: The marginal utility of
conductingedgecoveragemeasurementZ6[.D on graph ��� from a set of�
sourcesis ? � ; ���>9ZA[WD\�^]_? �; ���>9Z`� .
This andrelatedquantitieswill betheprimary focusofthe restof
thepaper. We first studymarginal utility froma
purelyempiricalperspective, focusingon
thedistinctionbetweenthecoreof thenetwork andfeedernetworks.
Wethenreturnto theproblemfrom a
theoreticalperspective,developingandstudyingan
information-theoreticformal-ismof marginalutility in
thiscontext.
IV. EXPERIMENTAL METHODOLOGY
Wenow presenttheexperimentalmethodologyweusedto
investigatescalingbehavior in
theInternet.Thetracer-outedatasetsweusein thissectiondeviatein
severalwaysfrom theidealtheoreticalframework weprescribedin
Sec-tion III, anda significantportionof this sectionis devotedto a
discussionof additionalassumptionswhich we madeandadescriptionof
mechanismsfor post-processingof ac-tualdatasets.
A. InternetTraceData
Thetopologydatausedin thiswork wassuppliedby theSkitter
projectat CAIDA. The Skitter projecthasa num-ber of goalsincluding
Internetmapping,routecharacter-istic
analysisandperformanceanalysis. At the time theprimary datasetfor
this studywascollected(May 2000),theSkitter
infrastructureconsistedof 16 sourcenodesde-ployedaroundtheworld;
wereceiveddatafrom 8 of thosenodes.
Eachsourcenodesendstraceroute-like
probestodestinationnodeslocatedworld-wide. All of thedestina-tion
nodesareWebservers.Our primarydatasetcontainsresultsfrom tracesrun
to 1277 destinations;The
sourcenodesandthecorrespondingupstreamproviders(listedina
While our work is primarily experimentalin nature,we believe
thatthe theoreticalstudyof thesepropertieson graphsof
interest(suchaspower-law graphsvs. randomgraphs)with
idealizedroutingalgorithms(suchasuseof
shortest-pathroutes)mayhelpprovidedeeperinsight.
parentheses)werelocatedin Hamilton,NZ
(UniversityofWaikato);Tokyo, Japan(APAN),
Singapore,SG(providerunknown); San Jose,CA, USA (Worldcom); San
Jose,CA, USA (ABOVENet); Ottawa, CA (CANET); London,UK (RIPE);
andWashingtonDC, USA(QWest). On av-erage,probesaresentto
eachdestinationonceevery 30minutes. While it is not
clearpreciselyhow destinationsfor destinationsare selectedin
Skitter, the Skitter website statesthat
destinationsarerandomlysampledfrom a“crawl of IP addressspace”[11].
We alsoincluderesultsfrom a larger datasetwith 12 sourcesand over
300,000destinations. This datasetincludesthe eight sites
listedabove, with the exceptionof Singapore,plus Marina DelRey, CA,
USA (ISI); Moffett Field, CA, USA (NASA),Palo Alto, CA, USA;
SanDiego,CA, USA (CAIDA) andLondon,UK ()
B. NodeandEdgeClassification
Using theexperimentalresultswe gathered,it
wasim-mediatelyapparentthatthenetwork graphunderobserva-tion wasnot
a randomnetwork, but consistedof two con-stituentcomponents:1) a
centralrouting core, and 2) aset of “feeder” links which feed into
the backbone.Wethenfocusedonhow
successfullytraceroutecouldbeusedwhenappliedto identifying thesetwo
constituentcompo-nents,which hadevidently differentproperties.A
centralchallengeto doing so is to develop an automatedproce-dure
which classifiesnodes(and edges)into thesecate-gories. Using the
terminologyof Zeguraet al [33] to de-scribetheir GT-ITM
topologygenerator, we assumethatthereis a
naturalandidentifiableseparationbetweentran-sit domains,which
comprisethe Internetbackbone,andstubdomains,which only
transittraffic eitheroriginatingor terminatingin their domain. In
this model, the setoftransitdomainstypically forms a highly
connectedback-bone,with a number(at leasttwo andoften many more)of
node-disjointpathsbetweenany two transit domains,while
stubdomainstypically consistof treeswith a singleconnectionto
thetransitdomainbackbone.
The objective of our classificationalgorithmis to takeour
observationsof a topologyanddeterminethe bound-ary betweenwherethe
backboneendsandstubdomainsbegin basedon the availableevidence.
Therearea num-ber of reasonswhy our classificationproceduremay
failto classifynodescorrectly– in future work, we
intendtoconductvalidation trials to measurethe effectivenessofour
classificationmethodsfrom traceroutemeasurements.Routesto
destinationswhichdid not respondto
thetracer-outerequestswerediscarded,but routesin which
interme-diate hostsfailed to respondto ICMP requestswere
in-cluded.Evenusinga relatively smallnumberof measure-
-
mentsites,a cleardistinctionbetweenbackbonelinks andstub links
in this subgraph��� emerged(we will demon-stratethisandquantifyhow
mucherrorwasremovedfromour classificationprocessasthenumberof
measurementsincreased).
Given this subgraph,our classificationprocedurenowamountsto a
labelling of the nodesandedgesof ��� . Tothis end,nodeswhich
correspondto routersandInternethostsare classifiedas core routers,
border routers, stubrouters andleaf nodes. Our
nodeclassificationprocedureis performedasfollows. First,
leafnodesareidentifiedandlabelledassuch,andedgesadjoiningleaf
nodesareclas-sifiedasstublinks. Then,in a
bottom-upfashion,internalnodeswhich adjoina setof edgesall but
oneof which arestublinks, areclassifiedasstubrouters.
Upon completionof this procedure,the logical treesforming the
visible portion of stubdomainsin � �
arees-tablished.Theremainingunclassifiednodesall
satisfythepropertythatat leasttwo of their
incidentedgesareunla-beled– thatentireunlabeledportionof thegraph
��� is thenetwork backbone,andwe classifyit assuch.Within
thenetwork backbone,unlabellednodeswhich adjoinat leastonestub link
areclassifiedasborderrouters,all remain-ing nodesare classifiedas
core routers,and thoselinkswhich arenot yet
classifiedarebackbonelinks. Figure1providesa simplediagramof
theresultsof a classificationprocedure.
C. Coveragevs.Marginal Coverage
In theexampleswehavedescribedsofar, ourclassifica-tion
procedurelabelsthesubsetof Internetnodesandlinksvisible in oneor
moreof theend-to-endmeasurementsinourstudy. Sinceweareprimarily
interestedin characteriz-ing the Internetbackbone,andsincewe have
no expecta-tion of completelymappingstubdomains,we would ide-ally
like to measurethecoverage of theInternetbackboneachieved by our
experiments,using the definitions pre-sentedin SectionIII. However,
thisapproachis infeasible,as the exact topology of the graphwhich
comprisesthisbackboneis notknown aposteriori.While
wecannotmea-suretotal coveragedirectly, we
canmeasurethemarginalimprovementin coverageaswe
conductadditionalmea-surements.To quantify this approach,we take
theaggre-gatedinformation from all of the collectedtracesas
thebaselinegraphfor our study, andmeasurehow well smallsubsetsof
themeasurementsmanageto coverthatbaselinegraph. This point
highlightsan importantdistinctionbe-tweenmarginal
coverageandoverall coverage— thefactthat additionalmeasurementsmay
provide low marginalcoveragedoesnot necessarilyimply that the
overall cov-erageobtainedis high — it maybe thecasethat thecov-
erageis poor, but
theadditionalmeasurementschosenarenotproductive.3
D. InterfaceDisambiguation
One of the unfortunateissuesaboutbuilding networkmapsbasedon
tracerouteis the existenceof routerswithmultiple
interfaces,eachwith differentnetwork addresses.This issueis
pervasive– in ourstudywe foundthatnearlytwentypercentof all
thenodeswe classifiedasbackbonenodesusedmultiple interfaceswith
distinct IP addressesto transmitpackets. Clearly, studieswhich
disregard thisissue,by treatingeachdistinctInternetaddressasif it
wereadistinctnode,generateinaccuratemaps.
The techniquewe employed to disambiguatemultipleinterfacesat a
single nodeusesthe samebasicprincipleastheoneoriginally usedby
PansiotandGrad[19]. Thekey to this techniqueis that
whentransmittingan ICMPmessage,a routerwill typically transmitthat
packet witha sourceaddressequalto that of the outboundinterfaceon
which the packet is sent. Therefore,if one suspectsthat a
routerhastwo interfacesbdc and b\e , onecantrans-mit a UDP packet
to an unusedport at eachof thosein-terfacesfrom a commonsource. If
the interfacesare infact on the samerouter, the routerwill
respondwith twoICMP PortUnreachablemessages,bothof whichwill
havethe samesourceaddressbgf , possiblyequal to bdc or b\e .By
performingpost-hocprobesof this form from a com-mon
source(BostonUniversity) to all potentiallydistinctinterfaces,we
areable to detectand collapsehostswithduplicateinterfaces.
Unfortunately, this techniqueis notinfallible. First,
approximately10% of the core routersneverrespondedto
UDPmessagestransmittedto
unknownports;othersrespondextremelysporadically– we
conjec-turethatthelikelihoodof responsemaybecorrelatedwiththe load
on the router. For thoserouters,disambiguationappearsto
beimpossiblewith this currenttechnique.Sec-ond, our
techniquereliesuponroutersrespondingwith asourceaddressequalto
theoutboundinterface. If routersinsteadrespondwith a
sourceaddressequalto the UDPdestinationaddress,our techniquewould
berendereduse-less. We have no way of estimatingthe likelihoodof
thisevent;however, thefactthatwefrequentlyobserverouterswhich
respondwith addresseswhichdiffer from thetargetaddressgives us
someinformal level of confidencethatroutersdo in
factbehaveaccordingto specification.h
An
analogoussituationariseswhenchoosingblack-boxtestcasestoprovidecoverageof
codepathsin asoftwaremodule.
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Stub InterfacesBorder Interfaces
Backbone InterfacesStub Nodes
Border NodesBackbone Nodes
Fig. 2. Classof
nodesandinterfacesdiscoveredassourcesareadded(greedily)whenclassificationis
notknown apriori.
E. Accuracyof Classification
Onecentralaspectof nodeclassificationis theaccuracywith which we
performclassification.With a smallnum-ber of sources(lessthanfive),
many backbonenodesaremisclassifiedaseitherstubnodesor bordernodesby
virtueof the fact that the observable Internetis the union of
asmallnumberof trees.Figure2depictstherelativeclassifi-cationof
nodesandlinks assourcesareincreased.In someplotsin thispaper,
theorderin whichsourcesareaddedhasasignificantimpacton theoverall
results.A greedyorder-ing addsthesourcesin
theorderwhichmaximizesateachstepthe total numberof distinct
nodesobserved. A ran-domorderingaveragesoverasetof trials in
whichsourcesareaddedpurelyatrandom(withoutreplacement)for
eachtrial. In thecontext of accuracy of classification,behaviorof
greedyandrandomorderingsweresimilar; thegreedyorderingis
depicted.
As we increasethe numberof sources,our classifica-tion
procedureincreasesin accuracy. For example,oncewe have
amassedsufficient evidenceto classifya nodeasa backboneor
borderrouter, no setof additionalmeasure-mentswill
reversethatclassificationdecision.Ontheotherhand,nodeswhichweinitally
classifyaspartof astubdo-mainmayin
factbebackbonenodes,andwemayuncoverevidenceto that effect with
additionalmeasurements.Ingeneral,we expectto underestimatethe
fraction of back-bonenodesandoverestimatethefractionof
stubnodesinour classification.Thediagramin Figure2
quantifiesthatintuition whenthenumberof measurementsitesis
small,but it is alsointerestingto notethatfor
thisdataset,classi-ficationstabilizesafteronly aboutfive
measurementsites(vantagepoints)areused.
F. Limitationsof theApproach
Themetricsweproposearedifficult to usedirectly,
firstbecausethegraphwhich comprisesthe Internetis neitherfixed nor
given in advance. Moreover, even if the graph
# of Interfaces 1 2 3 4 5 6 7 10# of Routers 4892 602 169 54 29
13 3 1
Fig.3. Distributionof observedinterfacedensityacrossrouters.
comprisingtheInternetwereknown in advance,our mea-suresof
coveragemayfluctuate,sincethebehavior of theroutingalgorithmsin
theInternetis non-deterministic,dueto theeffectof
routingpolicies[28], [32]. Also, while onemight hopeto quantify
topology scalinglaws on certainclassesof graphs(suchason power-law
randomgraphs)whenshortest-pathrouting is in effect,
policy-basedrout-ing at the level of AS’s skews (or “inflates”)
routes,mak-ing theproblemof
accuratelymodelingthesescalinglawsmuchmoredifficult. We notethat
factorsrangingfrom awidevarietyof
routingmetricsandprotocols,variability innetwork
load,andpolicy-basedeconomicagreementsbe-tweenautonomoussystemscausethe
routeschosento bequitedifferentthananobserver with accessonly to
topol-ogy informationmight expect.
V. RESULTS
Theresultsin thissectionaredividedinto fiveparts:(1)the
resultsof interfacedisambiguatonrun on all nodesinthe primary
dataset, (2) a quantitative evaluationof thenumberof nodesandlinks
discoveredin thebackboneasthe numberof sourcesanddestinationsvary,
(3) an eval-uationof the estimateddistribution of nodedegreein
thebackboneasthenumberof sourcesanddestinationsvary,(4) fitting the
evidenceof theseevaluationsto statisticalmodelsand(5)
assessingtheaccuracy of thenodeclassifi-cationprocedureitself.
A. Resultsof runningthedisambiguationprocedure
Approximatelythreeweeksafterthetraceroutedatawascollectedby
CAIDA, we ran our interface disambigua-tion tool to all network
interfaceswhich we had classi-fied as part of the network backbone.
An early lessonwelearnedin ourpreliminaryexperimentswith
thedisam-biguationsoftwarewasthatasubstantialfractionof
routersrespondedto ourprobeswith very low frequency. In anef-fort
to elicit responsesfrom asmany respondinginterfacesaspossible,we
transmittedfive ICMP messagesto eachinterfaceevery twentyminutesfor
12successivehours.
Of the 7451 interfaceson our list, 6510 respondedtooneor moreof
our probesandtheremaining941(12.6%)never responded.We
recordedpairsof the form [TargetAddress,ResponseAddress]and
recorded6709 distinctpairsfrom the 6510targetedinterfaceswhich
responded.Wesuspectthatthisslight(3%)discrepancy is dueto
routefluctuationaffectingthefirst hopof thereturnpathto B.U.
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Dis
cove
red
iSources
InterfacesNodes
Fig. 4. Number of nodesdiscovered as sourcesare
added(greedily)
anddoesnot representanomalousbehavior. Thenext stepwe took was
to representthe set of addressespresentinour list of pairsasnodesin
agraph.Wedrew acorrespon-dencebetweeneachconnectedcomponentof this
graphandasinglerouter, wherethenodesof thecomponentcor-respondto
distinct addressesfor interfacesof the router.Using this strategy,
the 6510 targetedinterfacesmappedto 5763distinct routers. The
distribution of multiple in-terfaceswe observed is depictedin
Figure 3. Using theresultsin this table,weobservedanincidencerateof
mul-tiple interfacesof Rj ck jBl f �mD\npoMDrq .B.
Estimatingthesetof nodesandlinks in theInternet
In the resultsbelow, we have the goal of taking
mea-surementsover a setof pathswhich cover at least � dis-tinct
nodes(resp.links) in theInternet.Our first setof
ex-perimentsdemonstratesrapidly diminishingmarginal re-turns as
sourcesare addedto traceroutesto the full setof 1277
destinations,while our secondset demonstratesnearlyconstantmarginal
returnsasdestinationsareincre-mentallyaddedto a
destinationsettargetedby thefull setof 8 sources.
In Figures4 and5, we demonstratehow thenodecov-erageandlink
coveragein theInternetimproveassourcesareadded.In bothof
theseplots, thereis pronouncedev-idenceof
diminishingreturnsassourcesareadded,whichis highly
evidentevenwhenrunningtraceroutebetweenasmallnumberof sources(8)
anda muchlargernumberofdestinations(1277).In
eachfigurewealsodemonstratetheeffectof nodeandlink
discoverybeforeandafterinterfacedisambiguation.
In Figures6 and7, we demonstratehow thenodecov-erageandlink
coveragein the Internetimprove asdesti-nationsareadded. In both of
theseplots, thereis a rela-tively
constantadditionasdestinationsareadded.A
sim-pleslopecalculationshows
thatafter200destinations,ap-proximately3 new nodesarediscoveredand4
new linksare discoveredwhen a new destinationis added. Each
0
2000
4000
6000
8000
10000
12000
14000
16000
1 2 3 4 5 6 7 8
Cum
ulat
ive
Uni
que
Link
s D
isco
vere
d
sSources
Links with interfacesLinks with nodes
Fig.5. Numberof links discoveredassourcesareadded(greed-ily)
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0 200 400 600 800 1000 1200 1400C
umul
ativ
e U
niqu
e N
odes
Dis
cove
red
Destinations
Fig. 6. Numberof
nodesdiscoveredasdestinationsareadded(randomly).Eachline is for a
singlesource
of thesefiguresshows effectsafter interfacedisambigua-tion.
Resultsfor interfacediscoveryareapproximatelythesame.
Next, we breakdown nodediscovery by nodeclassifi-cation. In
Figure8 we show how
nodesandinterfacesarediscoveredassourcesareaddedwhenthenodeclassifica-tion
is known apriori. This resultshows thatweprimarilydiscover new
backbonenodesandinterfacesasadditionalsourcesareadded,but
backbonediscoveryshow diminish-ing marginalutility.
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0 200 400 600 800 1000 1200 1400
Cum
ulat
ive
Uni
que
Link
s D
isco
vere
d
sDestinations
Fig. 7. Numberof links
discoveredasdestinationsareadded(randomly).Eachline is for a
singlesource
-
0
1000
2000
3000
4000
5000
6000
7000
0 1 2 3 4 5 6 7Cum
. Uni
que
Nod
es/In
terf
aces
Dis
cove
red
iSources
Stub InterfacesBorder Interfaces
Backbone InterfacesStub Nodes
Border NodesBackbone Nodes
Fig. 8. Classof
nodesandinterfacesdiscoveredassourcesareadded(greedily)whenclassificationis
known apriori.
C. ContourPlots
Thefollowing diagramsplot thescalingbehavior of
thesubgraphsinducedby IP routing for the topologiesob-served via
the CAIDA tracedata,assumingthat eachofthe CAIDA
sourcesanddestinationsreflectsa randomlychosenvertex in thegraph.In
particular, we studythebe-havior of the function :9�7� as � and �
vary. Thevaluesof � and � areplottedalongthe t and u
-axes,re-spectively. Eachlabelledcontour, or
isoline,representsthediscoveryof afixedconstantnumberof
nodes,suchthatallsetsof measurementscorrespondingto apoint �!t�Buv�
alongacontourhaveanequalvalueof :
-
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25 30 35
P[X
<=
x]
Node Degree
1 Server2 Servers3 Servers4 Servers5 Servers6 Servers7 Servers8
Servers
Fig. 10. CDF of
backbonenodedegreeassourcesareadded(randomly)
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
log1
0(P
[X >
x])
log10(Node Degree)
1 Server2 Servers3 Servers4 Servers5 Servers6 Servers7 Servers8
Servers
Fig. 11. Tail of CDF of
backbonenodedegreeassourcesareadded(randomly)
tribution staysrelatively constantasdestinationnodesareadded,the
tail weight increasesas destinationnodesareadded.
E. Comparisonto Larger Dataset
The resultsso far provide considerableinsight into
IProutingpatternsbut thelimited sizeof thenodesetcoveredmakesit
hardto extendourconclusionsto theInternetasawhole. To addressthis
we examinea muchlargerdatasetto seewhetherit shows similar
patternsof diminishingre-turnswhenaddingmeasurementsites.
Theseconddatasetsconsistsof 12sourcesand313,709
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
1 2 3 4 5 6 7 8 9Roo
t Mea
n S
quar
e D
iffer
ence
in D
istr
ibut
ions
Added Server Number
Fig. 12. Rootmeansquarederrordifferencein
backbonenodedegreedistributions
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30 35
P[X
<=
x]
Node Degree
100 Destns200 Destns300 Destns400 Destns500 Destns600 Destns700
Destns800 Destns900 Destns
1000 Destns1100 Destns1200 Destns
Fig.13. CDFof
backbonenodedegreeasdestinationsareadded(randomly)
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2lo
g10(
P[X
> x
])log10(Node Degree)
100 Destns200 Destns300 Destns400 Destns500 Destns600 Destns700
Destns800 Destns900 Destns
1000 Destns1100 Destns1200 Destns
Fig. 14. Tail of CDF of
backbonenodedegreeasdestinationsareadded(randomly)
destinations;thusit is morethan10 timesthe sizeof thefirst
dataset. This datasetwasgatheredin October, 2000.Sourcelocationsfor
this datasetwereHamilton,NZ; SanJose,CA, USA; London,UK; Marinadel
Rey, CA, USA;Palo Alto, CA, USA; Tokyo, JP; Ottawa, CA; London,UK;
Moffett Field,CA, USA; Washington,DC, USA; SanJose,CA, USA;
andSanDiego,CA, USA.
Unlikethefirst dataset,in thiscasesourcesdid nottraceroutesto
acommonsetof destinations.In fact,nodestina-tion in this setis
commonto all sources.Furthermore,theconsiderablesizeof
thisnodesetmakesit muchmoredif-ficult to
disambiguateinterfaces,soourresultsarein termsof
interfacesratherthannodes(routers).
In Figure15weshow how thenumberof interfacesdis-coveredgrows
aswe addsourcesgreedily. In this case,addinga sourcemeansthat we
addall the measurementpathsoriginatingfrom that source.The line
labelled“in-terfaces”denotesthenumberof
interfacesthatwouldhavebeendiscoveredhadeachsourcebeenusedindependentlyfrom
the others. In Figure16 we show how the numberof
interface-interface links discovered grows as we addsources.
Presumablyeachindividual interface-interfacelink correspondsto a
router-routerlink, sofor thisplot
thedistinctionbetweennodesandinterfacesis lessimportant.
Thesefigures show a declining slope as sourcesare
-
0
100000
200000
300000
400000
500000
600000
700000
1 3 5 7 9 11
Uni
que
Inte
rfac
es D
isco
vere
d
Sources
Cumulative InterfacesInterfaces
Fig. 15. Numberof nodesdiscoveredas sourcesare
added(greedily)
0
100000
200000
300000
400000
500000
600000
700000
800000
1 3 5 7 9 11
Uni
que
Link
s D
isco
vere
dSources
Cumulative LinksLinks
Fig. 16. Number of links discovered as sourcesare
added(greedily)
added,similar in generalshapeto Figures4 and5. Whilethe lack of
identicalexperimentalsetup(i.e., the absenceof
commondestinations)makes it impossibleto directlycomparethetwo
pairsof figures,thesimilarity is sugges-tivethataphenomenonof
diminishingreturnsasmeasure-mentsitesareaddedis presentin
themuchlargerdatasetaswell.
VI . AN INFORMATION THEORETIC MEASURE OFMARGINAL UTIL ITY
Two elementarymetrics which we definedearlier tocomparea graphto
oneof its subgraphsarethenodeandedgecoverage,andmarginalutility of
additionalmeasure-mentsreflectsthe increasein thesemetrics. We now
re-turn to anotherquestionclosely to thoseposedin Sec-tion III: If
we run additionaltraceroutesto provide fur-therrefinementto
anexisting topologysnapshot,how canwe quantitavely specify the
information gainedby thesemeasurements.We provide a
morepreciseformulationininformation-theoreticterms.
A. Theory
The information content (measuredin bits) revealedfrom the
outcome 2 of an experiment , is definedas]
-
utility of experiment,^ before conductingany
additionalexperiments,�¦ , �§V� .
An alternative formulationof marginal utility
evaluateseachexperimentonanex
postfactobasis,measuringeachexperiment’s usefulnessoffline afterall
experimentshavebeenconducted. While this evaluation cannotbe
per-formedonline, it providesan estimateof marginal
utilitywhichisnotbiasedby theorderingin
whichmeasurementsareconducted.
Definition7: Theoffline marginal utility of experiment, is
definedto be ¢©¨£��, � , which is givenby:¢ ¨ ��, �¤� ���2 ¨ �¥
-
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
20 40 60 80 100 120 140 160
Ma
rgin
al U
tilityª
Node DiscoveryLink Discovery
Fig. 19. Utility of additionaldestinations(on-line)
tancemetricseemto decreasemonotonically.Comparative Utility of
Adding SourcesversusAddingDestinations: Oneof
theattractiveaspectsof informationtheoreticmeasuresof marginal
utility is that they enablecomparisonof marginal utility (1)
acrossmultiple distri-butions(e.g.link
vsnodevsdegreediscoveryaswasdonein Figure17) and(2) acrossmultiple
experimentalsetups(e.g. addingnew sourcesvs addingnew
destinations).Toexemplify the latter of thesecases,considerthe
questionof comparingtheutility of addingtraceroutesourcesto
theutility of
addingtraceroutedestinations.Comparingthere-sultsillustratedin
Figure17 to thoseillustratedin Figure18 revealsthat doublingthe
numberof destinationsfrom80 to 160while holding thenumberof
sourcesfixedat 8(a total of
640additionaltracerouteexperiments)yieldsamarginalutility thatis
approximatelyequivalentto thatre-sultingfrom increasingthenumberof
sourcesfrom 7 to 8while runningtracerouteto all
1277destinations(atotalof1,277additionaltracerouteexperiments).
VI I . CONCLUSIONS AND FUTURE WORK
In principle, it shouldbe possibleto gain
considerableinsightinto theconditionsandconfigurationsin
thecoreoftheInternet,givenasufficientarrayof
measurementpointslocatedin
endsystems.Thisconcepthasbeencalled“net-work tomography”
becauseeachmeasurementpoint seesa “projection” of theInternet’s
resourcesin a mannerspe-cific to its location.
While the conceptof network tomography is attrac-tive andin
keepingwith thedesignphilosophy of
keepingnetwork-internalcomponentsassimpleaspossible,sofarit hasnot
beenclearhow extensive a measurementinfras-tructureis neededin
order to seea large fraction of thenetwork from its edges.In
theabsenceof preciseknowl-edge,theprevailing wisdomin
Internetmeasurementhasseemedto be“more is better.” In thispaperwe
have takena steptowarddevelopinga morerefinedunderstandingofthis
problem. We have concentratedon the problemof
discoveryof basicInternetcomponents— links
andnodes(endsystemsandrouters).Weassumedthecommonmea-surementsituationin
which active measurementsitesarescarce,but passive targetsfor
measurementprobesarerel-atively plentiful.
Ourpreliminaryresultsindicatethatthemarginalutilityof
additionalmeasurementsitesdeclinesrapidlyevenafterthefirst two
sites.Thisis evidentin thediscoveryof nodes,of links, andof
nodedegreedistribution. We consideredtheaggregationof all
datasetsto bethemostcompletepic-tureavailable;in
eachcase(nodes,links, andnodedegreedistribution) a vastmajority of
the informationpresentintheaggregateddatasetwaspresentin thefirst
two or threedatasetsalone. On the
otherhand,conductingadditionalmeasurementsinvariably provided a
more completepic-tureof theentiretopology.
Our conclusionsareunavoidablysensitive to thepartic-ular
choiceof measurementsitesto which we
hadaccess,andwebelievethatfurthermeasurementsarewarrantedtoreinforceour
conclusions.However we believe that theseresultsshedlight on how
typical IP routespassthroughthe Internet,and show that a majority
of routestend topassthrough a relatively well-defined“switching
core.”We also note that traceroute measurementsare justone
techniquefor studyingthe marginal utility andscal-ing
questionsweposehere;numerousotherdatasetsmightalsoapplywell,
albeitwith differentprosandcons.
Finally, discoveryof nodesandlinks in aninternetworkprovides
only the most basic topographicalinformationaboutthenetwork.
Questionsaboutmarginal utility couldbeframedin thecontext of
richernetwork characteristics,suchas studyingthe marginal utility
of additionalmea-surementsto characterizethedistributionof packet
lossinthe network. We hopethat this paper, which we believeto
bethefirst to rigorouslyquantifythemarginalutility ofnetwork
measurements,will eventuallyseebroadapplica-tion to a rangeof
importantproblemdomainsin networkmeasurement.
VI I I . ACKNOWLEDGEMENTS
The datausedin this researchwascollectedaspart ofCAIDA’s skitter
initiative, http://www.caida.org. Supportfor skitteris providedby
DARPA, NSF, andCAIDA mem-bership. The authorswould like to thankkc
claffy, AmyBlanchardandEdouardLagachefrom CAIDA for
makingSkittertracedataavailablefor this study.
The authorswould alsolike to thankJenniferRexfordfor
hervaluablehelpshepherdingthispaperandtheanony-mousIMW reviewersfor
their suggestionsfor improvingthepaper.
-
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