Propagation of Trust and Distrust R - PDF document

Andrecursively,sinceatrustedacquaintancewillalsotrustthebeliefsofherfriends,trustsmaypropagate(withappro-priatediscounting)throughtherelationshipnetwork.Anapproachcenteredonrelationshipsoftrustprovidestwoprimarybenets.First,auserwishingtoassessalargenumberofreviews,judgments,orotherpiecesofinformationonthewebwillbenetfromtheabilityofaweboftrusttopresentaviewofthedatatailoredtotheindividualuser,andmediatedthroughthesourcestrustedbytheuser.Andsecond,userswhoaregloballywell-trustedmaycommandgreaterin uenceandhigherpricesforgoodsandservices.Suchasystemencouragesindividualstoactinatrustworthymanner[4],placingpositivepressureontheevolvingsocialconstructsoftheweb.Indeed,socialnetworktheoryandeconomicshaveconsideredavarietyoffacetsofthisgeneralsubject[1,2,3,6,25].1.2IntroducingdistrustRecentworks[14,21]giveamathematicalapproachtothepropagationoftrust,butdonotextendtothecaseinwhichusersmayalsoexpressdistrust.However,experiencewithreal-worldimplementedtrustsystemssuchasEpinionsandeBaysuggeststhatdistrustisatleastasimportantastrust.Intheabsenceoftreatmentofdistrustinpriorwork,itisunclearhowtomodelandpropagatedistrust.Forin-stance:doesatrustscoreof0translatetodistrustorto`noopinion'?Merelyshiftingalltrustscoressothatnonegativevaluesremain(andthenusingatrustpropagationmethodsuchas[14])willnotaddressthisfundamentalissue;sucha\shift"wouldbesensitivetooutliersandadditionallydistortthesemanticsofazeroscore.Modelingdistrustasnegativetrustraisesanumberofchallenges|bothalgorithmicandphilosophical.Forinstance,theprincipaleigenvectorofthematrixoftrustvaluesneednolongerbereal.ThisisabarriertoapproachesinwhichthetrustmatrixisturnedintoaMarkovchain(whatdonegativeprobabilitiesmean?)andtheprincipaleigenvec-torisinterpretedasanabsolutemeasureoftrustforeachnode.Infact,ourgoalisnotanabsolutemeasureoftrustforeachnode|rather,wewishtodetermineameasureoftrustfromanynodetoanyother.Anotherchallenge:whatdoesitmeantocombinedistruststhroughsuccessivepeopleinachain?Oneofthemaincontributionsofourpaperistoaddressthissituation.Wetrytodevelopanunderstandingofappropriatemodelsforthepropagationofdistrust(Sec-tion3.2.1andSection3.3).Oneofourndingsisthatevenasmallamountofinformationaboutdistrust(ratherthaninformationabouttrustalone)canprovidetangiblybetterjudgmentsabouthowmuchuserishouldtrustuserj.1.3SummaryofresultsTypicalwebsoftrusttendtoberelatively\sparse":virtu-allyeveryuserhasexpressedtrustvaluesforonlyahandfulotherusers.Afundamentalproblemisusingsuchwebsisthatofdeterminingtrustvaluesforthemajorityofuserpairsforwhomwehavenotexplicitlyreceivedatrustrating.Mechanismsforaddressingthisproblemhavebeenstud-iedineconomics,computerscienceandmarketing,albeittypicallywithoutacomputationalcomponent.Wepresentabroadtaxonomyofschemesforpropagationoftrustthroughanetworkofrelationships,andevaluate81combinationsoftrustanddistrustpropagationagainstalargecollectionsofexpressedtrustsprovidedbyEpinions.Toourknowledge,thisistherstempiricalstudyonalarge,real,deployedweboftrust.Werankdierentpropagationmechanismsmostlyfromtheperspectiveofpredictiveaccuracy,inthefollowingsense:ourexperimentsinvolvemaskingaportionoftheknowntrustratingsandpredictingthesefromtheremainder|aleave-one-outcross-validation.Thehopeisthatabetterunderstandingofwhatiscorrectwillleadtobetterapprox-imationstoaccuracy.Theremainderofthepaperproceedsasfollows.Section2coversrelatedwork.Section3thendescribesouralgorithms,andthetaxonomyofmechanismsthattiesthemtogether.Section4coverstheweboftrustweanalyze.InSection5weprovideexperimentalresultscomparingthealgorithmsanddrawconclusionsabouttheeectivenessoftrustprop-agationonreal-worlddata.2.RELATEDWORKAnumberofdisciplineshavelookedatvariousissuesre-latedtotrust,includingtheincrementalvalueassignedbypeopletotransactingwithatrustedpartyandhowtrustaectspeople'sbeliefsanddecisionmaking.Kahnemanetal.[13]wereamongthersttostudythesephenomenainthecontextofdecisionmaking.Thereisalsoasubstantialbodyofworkonunderstandingtrustintheeldofpoliticalscience[9,18,23].Onecoulddrawanumberofusefullessonsfromtheseelds,especiallyinassigningsemanticstotruststatements;unfortunately,thatworkisnotcomputationalinnature.Therehasbeenconsiderableworkontrustincomputerscience,mostofitfocusedintheareaofsecurity.Formallogicalmodels[8,10]havebeenusedtointhecontextofcryptographyandauthentication.PGP[24]wasoneofrstpopularsystemstoexplicitlyusetheterm\WebofTrust",thoughitwasnotinthecontextofsearchorinformation ow.Webelievethatthesamekindoftrustrelationsbe-tweenagentscanbeusedforamuchwiderrangeofap-plicationsthanjustforbeliefinstatementsaboutidentity.Gladwell'spopularbook\TheTippingPoint"[11]studiesthewayinformation owsaremediatedbythenetworksofpeopleandtheirassociatedtrustrelations.Therehasbeensubstantialworkinthebusinessmanage-mentcommunityonthevalueoftrust.Ackerlof'sclassic[1]showedtheimportanceofinformationregardingthequalityofaproduct(orservice).Ackerlofshowedhowinformation,i.e.,knowledgeaboutthetrustworthinessofaseller,isvitalforthefunctioningofamarket.Trustisanimportantaspectofon-linecommunities.ArmstrongandHagel[2]posittheimportanceoftrustandcommunityforon-linecommerce.Recently,duetotheemergenceofe-commerce,therehasbeenworkintheareaofdevelopingcomputationalmodelsoftrust.Ba,Whinston,andZhang[5]provideagametheoreticapproachoftrustandconcludethatinthepresenceofanauthenticatingthirdparty,themostutilitariancourseofactionfora(market)useristobehavehonestly.TherehavebeenanumberofproposedmodelsandempiricalstudiesoftheeBaytrustmodel[12,16,17,19,20,22].However,thatlineofworkhasnotconsideredmodelsofpropagatingtrust.Inthelastfewyears,anumberofresearchershavestartedlookingattheproblemofpropagatingtrustthroughnet-works.YuandSingh[25]proposeaframeworkwhich,incontrasttoourwork,assumessymmetryandarbitrarytran-sitivity.Kamvar,Schlosser,andGarcia-Molina[14]consider Figure1:Exampleofbasiselements:Directprop-agationandco-citation.Thedottedlinesindicatetrustpropagation.trustcoupling,inwhichi'strustofjpropagatestokbecausejandktrustpeopleincommon.TheseatomicpropagationsaresummarizedinTable2. AtomicPropagation Operator Description Directpropagation B SeetopofFigure1. Co-citation BTB SeebottomofFig-ure1. Transposetrust BT Ifatrustsbthentrustingbshouldim-plytrustinga. Trustcoupling BBT a;btrustc,sotrust-ingashouldimplytrustingb. Table2:Atomicpropagations.Let=(1;2;3;4)beavectorrepresentingweightsforcombiningourfouratomicpropagationschemes.ThenwecancapturealltheatomicpropagationsintoasinglecombinedmatrixCB;basedonabeliefmatrixBandaweightvectorasfollows:CB;=1B+2BTB+3BT+4BBT:Wenowexplorehowthoseatomicpropagationsmaybechainedtogether.3.2PropagationoftrustanddistrustOurendgoalistoproduceanalmatrixFfromwhichwecanreadothecomputedtrustordistrustofanytwousers.Intheremainderofthissection,werstproposetwotechniquesforcomputingFfromCB;.Next,wecompletethespecicationofhowtheoriginaltrustTanddistrustDmatricescanbecombinedtogiveB.Wethendescribesomedetailsofhowtheiterationitselfisperformedtocapturetwodistinctviewsofhowdistrustshouldpropagate.Finally,wedescribesomealternativesregardinghowthenalresultsshouldbeinterpreted.3.2.1PropagationofdistrustAsdescribedabove,letCB;beamatrixwhoseij-thentrydescribeshowbeliefsshould owfromitojviaanatomicpropagationstep;iftheentryis0,thennothingcanbeconcludedinanatomicstepabouti'sviewsonj.LetkbeapositiveintegerandletP(k)beamatrixwhoseij-thentryrepresentsthepropagationfromitojafterkatomicpropagations.Inotherwords,beginningwithabeliefmatrixB,wewillarriveatanewbeliefmatrixafterksteps.Thus,therepeatedpropagationoftrustisexpressedasamatrixpoweringoperation.WegivethreemodelstodeneB(thebeliefmatrix)andP(k)forthepropagationoftrustanddistrust,giveninitialtrustanddistrustmatricesTandDrespectively:(1)Trustonly:Inthiscase,weignoredistrustcom-pletely,andsimplypropagatetrustscores.ThedeningmatricesthenbecomeB=T;P(k)=CkB;:(2)One-stepdistrust:Assumethatwhenauserdis-trustssomebody,theyalsodiscountalljudgmentsmadebythatperson;thus,distrustpropagatesonlyasinglestep,whiletrustmaypropagaterepeatedly.Inthiscase,wehaveB=T;P(k)=CkB;
(T�D):(3)Propagateddistrust:Assumethattrustanddis-trustbothpropagatetogether,andthattheycanbetreatedastwoendsofacontinuum.Inthiscase,wetakeB=T�D;P(k)=CkB;:3.2.2IterativepropagationWecannowcomputenewbeliefsbasedonkstepsofatomicpropagations.WenowwishtodeneF,thenalmatrixrepresentingtheconclusionsanyusershoulddrawaboutanyotheruser.ButthematrixP(k)forsmallervaluesofkmaybemorereliable,sincetherehavebeenfewerprop-agationsteps;whilelargervaluesofkmaybringinmoreoutsideinformation.Weconsidertwonaturalapproachestoinferringnaltrustscoresfromoursequencesofpropa-gations.(1)Eigenvaluepropagation(EIG):LetKbeasuit-ablychosen(discussedlater)integer.Then,inthismodel,thenalmatrixFisgivenbyF=P(K):(2)Weightedlinearcombinations(WLC):Let beaconstant(thatissmallerthanthelargesteigenvalueofCB;)andletKbeasuitablychoseninteger.( isadis-countfactortopenalizelengthypropagationsteps.)Underthismodel,FisgivenbyF=KXk=1 k
P(k):3.2.3RoundingFinally,theresultvaluesofFmustbeinterpretedaseithertrustordistrust.Whilecontinuous-valued(ratherthandiscrete-valued)trustsaremathematicallyclean[21],weworkontheassumptionthatfromthestandpointofus-abilitymostreal-worldsystemswillinfactusediscreteval-uesatwhichoneusercanrateanother.Whileourmath-ematicaldevelopment(likepreviouswork)hasbeeninthecontinuousdomain,wenowconsiderthe\rounding"prob-lemofconvertingcontinuousbeliefvaluesfromanarbitraryrangeintodiscreteones(suchas1).Thiscorrespondsto +++--+++++?++-+-------jFigure2:Predictionofjbasedonthemajorityoflabelsofneighborsofi(+meanstrustand-meansdistrust)sortedbythetrustscores.Here,thepre-dictionwouldbe+.applicationsthatdemandaBooleanyes/nojudgmenttothequestion\Shoulditrustj?"(SuchBooleanroundingisalsonecessaryforourcross-validationexperimentsinSection5.)ThisistantamounttoroundingtheentriesinmatrixFtoeithertrustordistrust.Wediscussthreewaysthisroundingcanbeaccomplished.(1)Globalrounding:ThisroundingtriestoaligntheratiooftrusttodistrustvaluesinFtothatintheinputM.ConsidertherowvectorFi.WejudgethatitrustsjifandonlyifFijiswithinthetopfractionofentriesofthevectorFi,underthestandardordering.Thethresholdischosenbasedontheoverallrelativefractionsoftrustanddistrustinthe(sparse)input.(2)Localrounding:Here,wetakeintoaccountthetrust/distrustbehaviorofi.Asbefore,wejudgethatitrustsjifandonlyifFijiswithinthetopfractionofentriesofthevectorFi,underthestandardordering.Thethresholdischosenbasedontherelativefractionoftrustvs.distrustjudgmentsmadebyi.(3)Majorityrounding:Themotivationbehindthisroundingistocapturethelocalstructureoftheoriginaltrustanddistrustmatrix.ConsiderthesetJofusersonwhomihasexpressedeithertrustordistrust.ThinkofJasasetoflabeledexamplesusingwhichwearetopredictthelabelofauserj;j=2J.WeorderJalongwithjac-cordingtotheentriesFij0wherej02J[fjg.Attheendofthis,wehaveanorderedsequenceoftrustanddistrustla-belswiththeunknownlabelforjembeddedinthesequenceatauniquelocation(seeFigure2).Wenowpredictlabelofjtobethatofthemajorityofthelabelsinthesmall-estlocalneighborhoodsurroundingitwherethemajorityiswell-dened.Moresophisticatednotionsofroundingarepossible.No-ticeabovethatlocalroundingandmajorityroundingare\i-centric".Aj-centricdenitionispossibleinasimilarman-ner.Alsonotethatournotionofmajorityroundingtriestoexploitclusteringproperties.Itispossibletoderiveim-provedroundingalgorithmsbyusingbetterone-dimensionalclusteringalgorithms.Ourresultsshowthattheroundingalgorithmisofsignif-icantimportanceinthepredictivenessofthesystem.3.3OnthetransitivityofdistrustItseemsclearthatifitrustsj,andjtrustsk,thenishouldhaveasomewhatmorepositiveviewofkbasedonthisknowledge.Intherealmofdistrust,however,thistran-sitivitymightnothold.Assumeidistrustsj,whodistrustsk.Perhapsiisexpressingtheviewthatj'sentirevaluemodelissomisalignedwithi'sthatanyonejdistrustsismorelikelytobetrustedbyi(\theenemyofyourenemyisyourfriend").Alternately,however,perhapsihasconcludedthatj'sjudgmentsaresimplyinferiortoi'sown,andjhasconcludedthesameaboutk|inthiscase,ishouldstronglydistrustk(\don'trespectsomeonenotrespectedbysomeoneyoudon'trespect").Wecalltheformernotionmultiplicativeandthelatteradditivedistrustpropagation.Multiplicativetrustpropagationhassomeunexpectedside-eects:adirectedcyclearoundwhichthetrust/distrustvalueshaveanegativeproductimplythatiteratedprop-agationwillleadausertodistrusthimself!Moreover,suchiteratedpropagationwillovertimegenerateanalbeliefthatnegatesandoverwhelmstheuser'sexplicitlyexpressedbelief.Nevertheless,wecannotignoremultiplicativetrustpropagationbecauseithassomephilosophicaldefensibility.Thisproblemresultsbecausetrustanddistrustarecom-plexmeasuresrepresentingpeople'smulti-dimensionalutil-ityfunctions,andweseekheretorepresentthemasasinglevalue.Ratherthanproposethatoneanswerismorelikelytobecorrect,onecandenetwocorrespondingalgebraicnotionsofdistrustpropagationthatmaybeappropriatefordierentapplications.Noticethatbyvirtueofmatrixmul-tiplication,allourearlierdenitionsimplementthemulti-plicativenotion,ifweusethetrust/distrustvaluesperse.OnewaytoimplementtheadditivedistrustnotioninourframeworkisbytransformingthematrixMtoM0beforeapplyingtheiteration,asfollows:m0ij=exp(mij)mij6=0;0otherwise.4.EXPERIMENTALDATAWebeginwithadiscussionofEpinions,theproviderofourdata,andwecovertheproblemsthatmotivatedthemtodevelopandmaintainaweboftrustbetweenindividuals.Wethendigintothestructureofthegraphitself.4.1Datasource:EpinionsEpinionsisawebsitewhereuserscanwritereviewsaboutavarietyoftopics,rangingfromconsumerdurables(suchascarsandtoasters)tomediaobjects(suchasmusicandmovies)tocollegestovacationspots.Usersmayauthorreviews,ratethereviewsofotherauthors,andmostimpor-tantlyforourpurposes,mayindicatetrustordistrustforanotheruser.Amazon(amazon.com),Slashdot(slashdot.org),andsomeotherwebsiteshavesimilarconcepts,thoughtheyusedierentterminologies.Trustinformationperformstwokeyfunctions.First,manyusersvisitaproductcategoryratherthanaspecicproduct,andmustbeshowncertainitemsfromthecategory;trustinformationisemployedtoselectappropriateitems.Second,onceaparticularproductistobeshown,somereviewsmustalsobeselected.Mostob-jectsaccumulatemorereviewsthananyusercanread,andthereisawidevariationinthequalityofreviews.Trustinformationisusedtoprovideauser-specicselectionofparticularreviews,basedonthetrustrelationshipbetweentheuserandtheratersandauthorsofthevariousreviews.ReviewersatEpinionsarepaidroyaltiesbasedonhowmanytimestheirreviewsareread.Thisresultsinmanyeortsto\game"thesystem.Distrustwasintroducedaboutsixmonthsaftertheinitiallaunch,inparttodealwiththisproblem.Theresultingweboftrustisanimportantandsuccessfulmechanisminthepopularityofthesite,andthehighqualityofreviewsthatareselected.Ourexperimentsareperformedonthisdata,whichwenowdescribeinmoredetail. Iteration  Propagation Globalround. Localround. Maj.round. S S S Trustonly 0.1530.500 0.1230.399 0.0770.175 e1 One-stepdistrust 0.1190.251 0.1080.223 0.0670.162 Prop.distrust 0.3650.452 0.3680.430 0.0840.206 Trustonly 0.1530.500 0.1140.365 0.0800.190 EIG e2 One-stepdistrust 0.0970.259 0.0870.234 0.0660.159 Prop.distrust 0.1490.380 0.1210.279 0.0800.187 Trustonly 0.1530.500 0.1070.336 0.0770.180 e One-stepdistrust 0.0960.253 0.0860.220 0.0640.147 Prop.distrust 0.1100.284 0.1010.238 0.0790.180 Trustonly 0.1530.500 0.1230.390 0.1890.163 e1 One-stepdistrust 0.0930.231 0.0830.205 0.0980.205 Prop.distrust 0.1020.221 0.0980.199 0.1210.295 Trustonly 0.1530.500 0.1130.354 0.0740.174 WLC; =0:5 e2 One-stepdistrust 0.0880.254 0.0800.231 0.0930.187 Prop.distrust 0.1260.336 0.1000.252 0.0760.177 Trustonly 0.1530.500 0.1080.340 0.0780.159 e One-stepdistrust 0.0860.247 0.0760.217 0.0920.190 Prop.distrust 0.0870.237 0.0790.203 0.0740.162 Trustonly 0.1530.500 0.1230.391 0.1320.152 e1 One-stepdistrust 0.1020.241 0.0920.216 0.0690.171 Prop.distrust 0.1110.238 0.1060.211 0.1010.227 Trustonly 0.1530.500 0.1130.356 0.0780.184 WLC; =0:9 e2 One-stepdistrust 0.0920.260 0.0820.235 0.0710.173 Prop.distrust 0.1340.355 0.1060.261 0.0780.188 Trustonly 0.1530.500 0.1070.337 0.0750.169 e One-stepdistrust 0.0910.253 0.0820.222 0.0720.171 Prop.distrust 0.0910.254 0.0810.209 0.0780.177 Table3:Predictionofvariousalgorithms.Here,e=(0:4;0:4;0:1;0:1),K=20.5.1ResultsFromTable3,weseethatweachievepredictionerrorsaslowas6:4%ontheentiresetof3250trialsanderrorsaslowas14:7%onthesubsetS.Thebestperformanceisachievedfortheone-stepdistrustpropagationschemewithEIGiterationand=(0:4;0:4;0:1;0:1).5.1.1BasiselementsItwasourexpectationinundertakingtheseexperimentsthatdirectpropagationwouldbethemethodofchoice,andthattheotherbasiselementswouldprovidelimitedvalue.However,thevalueofco-citationhasbeenprovenforwebpagesbythesuccessoftheHITSalgorithm[15],sowein-cludeditandtheotherbasiselements.Theresults,showninFigure4,werequitesurprising:propagationbasedonlyonco-citationalone(basisvector=e2inthegure)per-formedquitewell.Noticethatinthismodel,simpleedgetransitivityintheunderlyingtrustgraphdoesnotapply:justbecauseitrustsjandjtrustsk,wecanconcludenoth-ingabouti'sviewofk.Soitisquitesurprisingthatthismethodperformswell.Overallcasesinourlargetable,eisthebestoverallperformer.Thissuggeststhatthereisacertainresiliencetovariationsinthedatabyadoptingmanydierentmechanismstoinfertrustrelationships.Werecom-mendthisschemeinenvironmentswhereitisaordable.5.1.2IncorporationofdistrustOne-stepdistrustpropagationisthebestperformerwiththeEIGtypeofiterationforeachoftheninecases(three Figure4:Resultsfordierentvaluesof,majorityrounding,againstresultscoreS. Figure5:ResultsfortheWLCiteration, 2f0:5;0:9g,showingiterationmethodsandbasisvec-torsagainstresultscoreS.roundingmethodsandthreebasisvectors).Wecancon-sistentlyrecommendone-stepdistrustinthiscase.WiththeWLCtypeofiteration,distrustisclearlyhelpful,butdependingonthebasisvector,eitherone-steporprop-agateddistrustmayperformbetter,asshowninFigure5.The =0:9case,whichfavorslongpaths,performsworseforone-stepdistrustthanthe =0:5case.Forotherdis-trustmodels,though,theresultsaremixed.Themoststrik-ingresultofFigure5isthatdirectpropagation(thee1case)istheonlysituationinwhichdistrustactuallyhurts,some-timesquitesubstantially;4inallothercaseswerecommendusingone-stepdistrustasrobust,eective,andeasytocom-pute.Directpropagation(=e1)intree-structurednet-worksthathavenoself-loopsandnoshortcyclesmayresultinlocalinformationhavinglittleimpactonthetrustscores,whichcouldbeundesirable.RecallthattheEIGiterationdoesnotintroduceany\restart"probability;thiswouldbeeasytoadd,andwouldresultinanalgorithmmoresimilartotheWLCiteration.5.1.3RoundingTheresultsforroundingarebrokenoutinFigure6.ThegurecomparesroundingalgorithmsforthebestsettingfortheEIGiteration(one-stepdistrustwith=e)andthebestsettingfortheWLCiteration(propagateddistrust, =0:5;=e).Inallcases,majorityclusteringbeatslocalrounding,whichinturnbeatsglobalrounding.Tooursurprise,thispartofthealgorithmturnedouttobequitecriticalbothingettinggoodresults,andinprovidingstrongperformanceacrossallthedierentcases.Werecommendusingadecisionmethodlikemajorityrounding.5.1.4IterationmodelsFigure7restrictsattentiontothegenerallybestbasisvector(=e)andthebestroundingmethod(majorityrounding),andcomparesresultsforEIG,andWLCwith =f0:5;0:9g.ThebestresultsareattainedwithEIGwithone-stepdistrust. 4SeeSection3.3foradiscussionofthedicultissuesthatariseindirectpropagationofdistrust. Figure6:ResultsforroundingusingthebestoverallsettingsfortheEIGandtheWLCiterationagainstresultscoreS. Figure7:Resultsforalliterationmethodswith=e,majorityrounding,againstresultscoreS. Iter. Trustonly One-stepdistrust Prop.distrust =e1 =e =e S S S 1 0.1200.300 0.0960.209 0.0800.209 2 0.1890.216 0.0860.197 0.0820.191 3 0.1770.184 0.0880.203 0.0740.184 4 0.1570.153 0.0910.206 0.0840.188 5 0.1500.156 0.0860.200 0.0820.197 6 0.1410.153 0.0860.203 0.0800.197 7 0.1350.156 0.0820.197 0.0810.194 Table4:EectofnumberofiterationsonandSforclusterrounding.TheiterationtypeisEIGwith =0:9andthenumberofsamplesis1000.5.1.5Theeffectofthenumberofiterations,KThefollowingtable(Table4)showstheeectofthenum-berofiterationsforthreeselectedsettingsofparameters.Fortrustonlypropagationwith=e1,meaningonlydi-rectpropagationallowed,increasingthenumberofiterationshasamoredramaticeectonimprovingthepredictioner-rorthanforotherpropagationmethods.Thisisasexpectedasdirectpropagationoccursalongthedirectededgesofthegraph.Incontrast,theotherpropagationmethods,assistedby=e=(0:4;0:4;0:1;0:1),donotenjoysimilardramaticimprovementswithincreasingthenumberofiterations.Inpart,thisisbecausetheshortestpathbetweenmosttestpairshaslengthtwo,solongeriterationsmayfailtohelp.6.CONCLUSIONSOverthelastfewyears,anumberofe-commercerelatedsiteshavemadeatrustnetworkoneoftheircornerstones.Propagationoftrustisafundamentalproblemthatneedstobesolvedinthecontextofsuchsystems.Inthispa-per,wedevelopaformalframeworkoftrustpropagationschemes,introducingtheformalandcomputationaltreat-mentofdistrustpropagation.Wealsodevelopatreatmentof\rounding"computedcontinuous-valuedtruststoderivethediscretevaluesmorecommoninapplications.Eachofourmethodsmaybeappropriateincertaincircumstances;weevaluatetheschemesonalarge,realworld,workingtrustnetworkfromtheEpinionswebsite.Weshowthatasmallnumberofexpressedtrustsperindividualallowsthesystemtopredicttrustbetweenanytwopeopleinthesystemwithhighaccuracy.Weshowhowdistrust,roundingandothersuchphenomenonhavesignicanteectsonhowtrustispropagated.7.ACKNOWLEDGMENTSTheauthorswouldliketothankEpinionsforgraciouslymakingavailablethedataforthisstudy.Wewouldinpar-ticularliketothankNiravToliaandJoelTruherforalltheirhelp.TherstauthorwouldalsoliketothankNavalRavikant,BenchmarkCapital,andAugustCapitalforhelp-ingcreateEpinions.8.REFERENCES[1]G.Ackerlof.Themarketforlemons:Qualityuncertaintyandthemarketmechanism.QuarterlyJournalofEconomics,84:488{500,1970.[2]A.ArmstrongandJ.HagelIII.Therealvalueofonlinecommunities.HarvardBusinessReview,pages134{141,1996.[3]C.Avery,P.Resnick,andR.Zeckhauser.Themarketforevaluations.TheAmericanEconomicReview,89:564{584,1999.[4]S.BaandP.Pavlou.Evidenceoftheeectoftrustbuildingtechnologyinelectronicmarkets:Pricepremiumsandbuyerbehavior.MISQuarterly,26(3):243{268,2002.[5]S.Ba,A.B.Whinston,andH.Zhang.Buildingtrustinonlineauctionmarketsthroughaneconomicincentivemechanism.DecisionSupportSystems,35(3):273{286,2002.[6]T.Beth,M.Borcherding,andB.Klein.Valuationoftrustinopennetworks.In3rdEuropeanSymposiumonResearchinComputerSecurity,pages3{19,1994.[7]A.Z.Broder,R.Kumar,F.Maghoul,P.Raghavan,S.Rajagopalan,R.Stata,A.Tomkins,andJ.Wiener.Graphstructureintheweb.WWW9/ComputerNetworks,33(1{6):309{320,2000.[8]M.Burrows,M.Abadi,andR.Needham.Alogicofauthentication.ACMTransactionsonComputerSystems,8(1):18{36,1990.[9]J.Coleman.FoundationsofSocialTheory.HarvardUniversityPress,1990.[10]U.Frendrup,H.Huttel,andJ.N.Jensen.Modallogicsforcryptographicprocesses.ElectronicNotesinTheoreticalComputerScience,68(1),2002.[11]M.Gladwell.TheTippingPoint,HowLittleThingsCanMakeaBigDierence.LittleBrown,2000.[12]D.HouserandJ.Wooders.Reputationinauctions:Theory,andevidencefromeBay.Technicalreport,UniversityofArizona,2000.[13]D.Kahneman,P.Slovic,andA.Tversky.JudgmentUnderUncertainty:HeuristicsandBiases.CambridgeUniversityPress,1982.[14]S.D.Kamvar,M.T.Schlosser,andH.Garcia-Molina.TheeigentrustalgorithmforreputationmanagementinP2Pnetworks.InProceedingsofthe12thInternationalWorldWideWebConference,pages640{651,2003.[15]J.M.Kleinberg.Authoritativesourcesinahyperlinkedenvironment.JournaloftheACM,46(5):604{632,1999.[16]P.Kollock.Theproductionoftrustinonlinemarkets.InE.J.LawleramdM.Macy,S.Thyne,andH.A.Walker,editors,AdvancesinGroupProcesses,volume16,pages99{123.JAIPress,1999.[17]C.G.McDonaldandV.C.SlawsonJr.Reputationinaninternetauctionmodel.Technicalreport,UniversityofMissouri-Columbia,2000.[18]B.Misztal.TrustinModernSocieties:TheSearchfortheBasesofSocialOrder.PolityPress,1996.[19]P.ResnickandR.Zeckhauser.Trustamongstrangersininternettransactions:EmpiricalanalysisofeBay'sreputationsystem.Technicalreport,UniversityofMichigan,2001.[20]P.Resnick,R.Zeckhauser,E.Friedman,andK.Kuwabara.Reputationsystems.CommunicationsoftheACM,43:45{8,2000.