DynamicsofTrustReciprocationinMulti-RelationalNetworks*AyushSinghal,Ka - PDF document

DynamicsofTrustReciprocationinMulti-RelationalNetworks*AyushSinghal,Ka
DynamicsofTrustReciprocationinMulti-RelationalNetworks*AyushSinghal,Ka

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DynamicsofTrustReciprocationinMulti-RelationalNetworks*AyushSinghal,KarthikSubbian,JaideepSrivastavaUniversityofMinnesota,Minneapolis,MN.ayush,karthik,srivastaTamaraG.Kolda,AliPinarSandiaNationalLaboratories,Livermore,CA.tgkolda,apinar@sandia.gov„Understandingthedynamicsofreciprocationisofgreatinterestinsociologyandcomputationalsocialscience.TherecentgrowthofMassivelyMulti-playerOnlineGames(MMOGs)hasprovidedunprecedentedaccesstolarge-scaledatawhichenablesustostudysuchcomplexhumanbehaviorinamoresystematicmanner.Inthispaper,weconsiderthreedifferentnetworksintheEverQuest2game:chat,trade,andtrust.Thechatnetworkhasthehighestlevelofreciprocation(33%)becausethereareessentiallynobarrierstoit.Thetradenetworkhasalowerrateofreciprocation(27%)becauseithas *fullversion:http://arxiv.org/abs/1303.6385https://www.everquest2.com/http://us.battle.net/wow/en/http://www.nickyee.com/daedalus/gateway demographics.html 0 5 10 15 20 25 0.55 0.6 0.65 0.7 0.75 0.8 ADDITIONAL DAYS OF TRADE INTERACTIONSAUC TRUST TRUST+TRADE TRUST+TRADE+HMPLY TRUST+HMPLY Fig.1:FigureshowsimprovementinAUCbyaddingtradeinteractionforpredictingtrustreciprocation.Dynamicsofthereciprocationvariesfromnetworktonetworkdependingonthelevelofbarrierforreciprocation.Thebarrierforreciprocatingatrustrelationshipcouldbelackofresourcesorhighriskinvolved.Needlesstosay,thesebarriersaffectthelevelsofreciprocationsigni“cantlyindifferentnetworks.Forinstance,inachatnetworkusershaveaverylowbarrierlevelfortrustingeachotherasthereisnocommitmentfromeithersidetoparticipateinanyinvolvedrelationshipor y sis and Minin g 661 A SONAM'13, August 25-29, 2013, Niagara, Ontario, CAN Copyright 2013 ACM 978-1-4503-2240-9 /13/08 ...$15.00 reciprocaledges,degree,andsizeofmodelednetwork.Duraketal.[]proposeanullmodelfordirectedgraphsthatcombinesthereciprocalandtheone-wayedgestogeneratedirectedgraphsthatmatchthein-,out-,andreciprocal-degreedistributions.Ahmadetal.[]usetheoriesofsocialexchangeasabasisforbuildingagenerativemodelGTPAformodelingtemporallyevolvingdirectednetworks.Thereareotherrecentworks,suchasGralaschelliandLoffredo[]thatusestatisticalmeasurestoconcludethatthereciprocationofanetworkisneveratrandomanditiseitherre-ciprocaloranti-reciprocal.Similarly,Zamora-Lopezetal.[proposeamethodtocomputetheexpectedreciprocationofthenetworkasafunctionofin-andout-degreedistributions.Reciprocationinweighteddirectednetworks,especiallyinlargescalemobilecommunicationnetworks,isdiscussedinin14],[].Szelletal.[]showthatnegativeinteractionshavealowerreciprocationcomparedtothepositiveinteractions.Reciprocationhasalsobeenusedforlinkprediction[],[[19].II.MMOGDATAInthissection,wedescribethenetworksusedinourexperiments.A.TrustNetworkInEQII,playersformteamsinordertocompletethegametasks.Astheplayersarelimitedbythenumberofitemstheycancarryatatime,playersbuyhousesasatemporarystoragetoretaintheirarmoryandotheraccessories.Throughthetrustnetwork,playerssharetheirhouseaccesswithotherplayers.Forthisreason,wealsorefertothisnetworkasthetrustnetwork.Wehave9monthsofdatafromJan-01-2006toSep-11-2006with63684nodesand140514edges.Eachnodeinthenetworkisaplayercharacterinthegame,andeachedgeisapermissionlevelgrantedbythecharactertoanothercharacter.Eachedgehasatimestampwhentheaccesswasgranted.Thetrustlevelsaredescribedasfollows.Trustee:Playercanstore,touch,move,add,andre-movethings,andhasalmostsameaccessastheowner.Friend:Playercanstore,touch,andmovethings.Visitor:Playercanenterthehouseandviewthings.:Playercanseethehouseexternallybutcannotenterit.B.TradeNetworkIntheEQIItradenetwork,playersexchangegoodsforcoinsorgoods.Inthisactivity,atradelinkisestablishedbetweentheseller(initiator)andthebuyer(acceptor)inthetradenetwork.Weanalyzedsuchatradenetworkcontaining295,055nodesand11,913,994edgesoveraperiodof9monthsfromJan-01-2006andSep-11-2006.C.ChatNetworkThechatnetworkisacommunicationmediumwhereplay-ersexchangeinstantmessages.Thenumberofnodesinthisnetworkis349,654,andthenumberofedgesis86,948,748,spanningoveraperiodofonemonthfromJul-29-2006toSep-D.NetworkPro“lesWepresentthedegreedistributionofthesenetworksin.Thedistributionswereconstructedusingsnapshotsofdifferentnetworksovertheentireobservationperiod.ThedistributionsseemtofollowthepowerlawwithexponentTABLEI:Statisticsofreciprocationintrade,chatandtrustnetworks. Network AllForward First Total Type(period) Edges Reciprocation Reciprocation Chat(1month) 1840492 441039(23.9%) 599548(32.6%) Trade(9months) 520861 74137(14.23%) 136809(26.3%) Trust(9months) 62674 8452(13.5%) 8083(14.0%) ofthepowerlawrangingfrom1to3.Theexponentwascalculatedusingtheslopeofaleastsquares“tinthelog-logIII.RECIPROCATIONINIFFERENTETWORKSThereciprocationofanetworkistheratioofforwardedges(sayfromplayer)thathaveacorrespondingbackwardedge(fromplayer),i.e.,theratioofmutualinteractions.A.BarriersofReciprocationBarriersofreciprocationcanbebroadlygroupedintoriskandutility.Theriskfactorsincludeloosinganasset,in-gamepoints,orin-gametime;andtheutilityinclude,immediategainsintermsofpointsandassetsandlongtermfutureprospects.Eachnetworkhasspeci“ccharacteristicsthatleadtolow,mediumorhighbarriersforreciprocation.Accordingly,trustnetworkhasthelevelofbarrier.Inthecaseoftradenetwork,thebarrierofreciprocationfallintobarriercategory.ThechatnetworkfallsintothelowbarrierB.MultiplereciprocationsTherecanbeseveraloverlappingforwardandbackwardarcsbetweeneachpairofplayers.Forthepurposeofmeasur-ingthereciprocationandresponsetime,we“rstpartitionthetimelineintoseveralpartitions.Weconsiderthestarttimeof“rstforwardedgeandthecorrespondingendtimeofthe“rstresponseasthe“rstpartition.Similarly,thesecondforwardarcanditsresponsemarkstheendofthesecondpartitionandsoon.Fortheremainderofthispaper,unlessspeci“ed,wealwaysrefertotheoverallreciprocationrate.InTableshowmultiplereciprocationrateforthreedifferentnetworks.C.ReciprocationinTrust,ChatandTradeNetworksInthissection,westudythereciprocationbehaviorintrust,chatandtradeasindependenthomogenousnetworks.Then,inthenextsection,weanalyzetheinteractionsofchatandtraderelationshipsinaffectingthereciprocationofthetrustnetwork.Thechatandtradenetworkshavesimpleedgetypeswithnoattributesexcepttimestamps.Forthetrustnetwork,theTrustlevelcorrespondstobeingaTrustee.Wecollapsethelowertrustlevels,(Friend,Visitor,andNone),toasingleNot-Trustlevel.WesummarizethetotalreciprocationforeachnetworktypeinTable.Inthetrustnetwork,beingahighbarrierrelationship,only14.0%oftheforwardtrust(8803one-way)linksreceiveatrustresponse(reciprocation)backandtheirresponsetimedistributionisshowninFigure.Asthechatnetworkisalowbarrieractivity,ithasthehighestamountofreciprocationwith32.6%offorwardedgesreciprocated.Thelowbarrierinthisnetworkisduetotheinstantnatureofthecommunicationandminimalriskinvolvedinmakingachatreciprocation.Oncontrary,inthemediumbarriertradeactivitiesthereciprocationrateis26.3%,lowerthanchatbutmorethanthetrustnetwork.Thereasonswhyaplayermaynot y sis and Minin g 662 100 101 102 100 101 102 103 104 105 DEGREEFREQUENCY INŠDEGREE OUTŠDEGREE 100 101 102 103 104 100 101 102 103 104 105 DEGREEFREQUENCY INŠDEGREE OUTŠDEGREE 100 101 102 103 100 101 102 103 104 105 DEGREEFREQUENCY INŠDEGREE OUTŠDEGREE Fig.2:Degreedistributionsoftrust(left),trade(right-top)andchatnetworks(right-bottom). 100 101 102 100 101 102 103 104 RESPONSE TIME IN DAYSNUMBER OF RESPONSES RESPONSE TIME m=27.304 s=42.812 #examples= 8803 (a)Trust 100 101 100 101 102 103 104 105 106 RESPONSE TIME IN DAYSNUMBER OF RESPONSES RESPONSE TIME m=0.317 s=1.397 #examples=599548 (b)Chat 100 101 102 100 101 102 103 104 105 NUMBER OF RESPONSESRESPONSE TIME IN DAYS RESPONSE TIME m=12.587 s=25.954 #examples=136809 (c)TradeFig.3:Theresponsetimedistributionforthethreenetworks.reciprocateinthisnetwork,couldbelackofeitherresourcesoraneedfordoingso.Further,wewishtoanalyzetheresponsetimedistributionsofthesenetworkstounderstandsomekeyquestions,suchas,doesallreciprocationsoccurwithinacertainnumberofdaysoraretheyspreaduniformlyoveralongerperiodoftime?showstheresponsetimedistributionforthetrustrelationship.We“ndthattheresponsetimedistributionfollowsapowerlaw,withameanresponsetimeof27days.FortheChatactivity,theresponsetimedistributionisshownin“gure.Thedistributionroughlyfollowsapowerlaw.Thereisanoutlierregionaround45days.Weinvestigatedthisregionandfoundthattheseare“rsttimeuserswhoarenotfamiliarwiththesystem.Wenotethatsuchusersareextremelyrareinthedataset(lessthan0.01%).The“gurealsoshowsthatmostoftheusersinthelowbarrier,chatnetworkreciprocatewithinthesamedayoratmostthenextday.Thisisevidentsincethemean“rstresponsetimeinthechatnetworkislessthanoneday(0.317).Inthechatnetworkthereisasharptruncation[after7days,asthesigni“canceofamessagebeyondaweekbecomescompletelyirrelevanttothecontextofthegame.Forthetradenetwork,weshowtheresponsetimedistribu-tioninFigure;thedistributionhasaheavytailandseemstofollowapowerlaw.Theslopeofthisdistributionisnotassteepasthatofthechatnetwork,implyingthetradereciprocationisnotasquickasinchat.Asthebarrierforreciprocationinthetradenetworkismorethanchat,theaverageresponsetimeintradeis43Xslowerataround13days.IV.RECIPROCATIONINETEROGENEOUSNETWORKSWewillnowoverlaythetrust,chat,andtradenetworkstoanalyzetheinteractionsamongheterogeneousedges.AsTABLEII:Thereciprocationcountsfor“rstinteraction(“rstforwardrequestand“rstreply)inaheterogeneousMMOGnetworkforaperiodofonemonth. Forward FirstForward Chat Trade Trust Type Edges Reciprocation Reciprocation Reciprocation Chat 1645623 435758 1187 105 Trade 74428 7953 11402 335 Trust 10502 907 1016 722 thechatdataisavailableonlyforamonth,werestricttheotherdatasetsalsotothisonemonthperiod.Thefocusofthissectionistoanswerquestions,suchas:Howmanytimesdoesatrustgrantingfromplayerresultinatrustreciprocation?Doesaplayerprefertoreciprocatewithtradeorsomeotherlowbarrierinteractionsuchaschatbeforegrantingahighbarrierrelationshipsuchastrusttoplayer.The“rstinteraction,asnotedearlierinFigure,capturesallthecharacteristicsoftheadditionalinteractions,henceweconsideronlythe“rstinteractionforthisexperiment.Foreachforwardedgetype,wecountthenumberof“rstreciprocationbetweenpairsofplayersintheconsolidatednetwork.Inthecaseoftiereciprocationsacrossmultipleedgetypes,weincludeallthetiededgeswhilecounting.WehavesummarizedthereciprocationsforeachforwardinteractiontypeinTableAsweseefromTable,theplayerswhoperformchatpre-dominantlymakeareciprocationusingthechatlink(26.48%).Thesameobservationismadeforthetradeactivity,whichisalsoalowbarrieractivity(buthigherthanchat).Herealsowe“ndthattraderesponsesarethepredominanttypeofresponsesforatraderelationship.However,thisisnotthesameinthe y sis and Minin g 663 caseofatrustforwardedgetypesincetrustisahighbarrierrelationshipandamoretimeconsumingactivity.Sopeopleareverycarefulbeforereciprocatingforsuchrelationshipsandreciprocationsare“rstinitializedthroughlowbarrieractivitiesbeforereciprocatingwithahighbarrierrelationship,suchastrust.Thereciprocationsfortrustforwardedgeispredominantlythroughchat(8.63%)ortrade(9.67%).Wenowanalyzethedependencybetweentrustrelationshipagainsttradeandchatinteractions.Theaimofthisexperimentistoquantifyhowchatandtrade,lowbarrierrelationships,in”uencereciprocationinahighbarrierrelationship.Foranytwonodesinthetrustnetwork,westartouranalysisfromthetimewhenaforwardTRUSTedgefromisestablished.ForsuchaforwardedgetherecaneitherbeaTRUSTreplytocompletetheTRUSTrelationshipornoTRUSTreply(incompleteTRUSTrelationship).TheTRUSTrelationshipisdeterminedasincompleteifthereisnotrustresponsefromwithintheaveragetrustresponsetimewhichis4.6daysinthiscase.Inotherwords,wetruncatetheresponsetimefortheincompletereciprocationsbythemeanresponsetime(4.6days)anduseonlytheperiodbeforethismeanresponsetimeforfurtheranalysis.Therecanalsobeseveralotherresponses(lowbarrierinteractions)fromreplieswithaTRUSTlink.Understandingtheseotherrelationships,suchaschatandtrade,beforeaTRUSTreplyisformedfromiscrucialtodecipherthenatureofsocializationrequiredforahealthymutualtrustrelationship.Asaresultofthisexperiment,we“ndthatcompleteandincompleteTRUSTdifferfromeachotherintermsofchatandtraderesponses.Weobservethattheresponsesareexactlyinanoppositeorderinthetworows.ForthecompleteTRUSTweobservethattherewere743forwardsedgesthatwererespondedbackwithTRUST.However,beforetheTRUSTiscompletedbetween,weseethatthereare408traderesponsesfrom.Thesetraderesponsesaccountfornearly63%ofthetotalresponses.Wehaveonly243chatresponses,whichiscomparativelysmallerthanthetraderesponses.Surprisingly,theamountoflowbarrierresponsesfortheincompleteTRUSTgetscompletelyreversed.Thereare9145forwardTRUSTedgeswhichdonotgetaTRUSTreplybackandremainonesided.FortheseTRUSTrequests,we“ndthatthechatresponsesfromnow6962(approximately75%ofthetotalresponses)whilethetraderesponsesaresigni“cantlylower(aslowas25%).Thisexperimentcon“rmsthataTRUSTrelationshipbe-ismorelikelytocompleteiftherearemoretraderesponsesthanchatresponsesfrom.ThisinterestingresultcanbeusedtoinferthefutureTRUSTrelationshipsbasedonsomelowbarrierrelationshipssuchaschatandtrade.V.PREDICTINGTRUSTRECIPROCATIONInthissectionweevaluatehowwellcanwepredictahighbarrierrelationship,suchasatrust,usinginformationaboutthemediumbarrieractivitiesbetweenthenodes.Theempiricalanalysisintheprevioussectionshowedthatthesuccess(completion)ofhighbarriertrustrelationshipdependsonsomemediumbarrierrelationshipliketrade.Weusethisasourmotivationtoquantifyhowwellthemediumbarrierrelationshipcanhelptopredicthighbarrierrelationshipcom-pletion.However,weusetheentire9monthsofdatainordertomakeanyconclusionsforthetrustreciprocationprediction.ThechatrelationshiphastobeexcludedfromthisexperimentTABLEIII:Tablecomparingreciprocationpredictionaccuracyusingdifferentfeaturesets. Average Average Classi“er CWA AUC Precision Recall F-measure onlytrust 0.515 0.659 0.800 0.863 0.806 trust+trade(K=0) 0.526 0.637 0.825 0.866 0.816 trust+homophily 0.519 0.604 0.788 0.849 0.808 trust+trade(K=0)+homophily 0.527 0.636 0.826 0.866 0.817 trust+trade(K=20) 0.588 0.714 0.871 0.885 0.851 becauseofitslimitedavailabilityforasinglemonth.ButweaddseveralotherfeaturestomaketheexperimentmoreForthetrustnetworkdatafor9months,thereareatotalof61006trustrequests(forwardsedges).Thetrustlinksthatarereciprocatedis8252whereas52574forwardsedgesremainedincomplete.Fortheforwardrequests,weignoredtherequeststhatstartedinthelast(9th)monthbecauseitishardtodeterminewhethertherequestswerecompleted.Thusthenumberoftrustrequestsareslightlylessthanthosementionedinearliersections.Sothecompletedtrustlink(reciprocated)formsoneclassandtheincompletetrustlinks(unreciprocated)formsthesecondclass.Thefollowingfeatureswillbeusedtobuildthepredictionmodel.Featuresfromhighbarrierrelationship(trust):Thesefeaturescharacterizethepositionofplayers(nodes)inthetrustnetwork.ForthereciprocationlinksA,Bweconsidertwostructuralfeatures.The“rststructuralfeaturedescribestheconnectivityoftoothernodesinatrustnetwork.Thesecondstructuralfeatureistheconnectivityofwithothernodesinthetrustnetwork.Forconvenience,wereferthesefeaturesastrustŽfeatures.Featuresfrommediumbarrierrelationship(trade)Thisfeaturesetconsistsoffeaturesfromthreesub-categoriesnamely,structural,past-behavioralandfuture-behavioral.ForaA,Bthestructuralfeaturecorrespondstothedegreeofanddegreeofinthetradenetwork.Thepast-behavioralfeaturesforalinkA,Bcorrespondtothecountofthetradeinteractionofthetypebeforethetrustrequestfromstarted.Thisfeaturetakesintoaccountthetradebehaviorbetweenbeforeanytrustinteractionstartedbetweenthem.Thefuture-behavioralfeaturestakesintoaccountthebehavioroftradeinteractionbetweenoncethetrustrequestissentfrom.Hereweuseatimewindow(indays)startingfromthetimewhentrustrequestwasinitiatedfrom.WecountthenumberoftradeinteractionsinthistimewindowFeaturesfromplayerdemography(homophily):Inthisfeaturesetwetakeintoaccountthetwotypesofhomophilies.The“rsttypeofhomophilyisgenderhomophily.Thegenderhomophilybetweenis1ifhasthesamegender,and0otherwise.Theseconddemographicfeatureistheexperiencehomophilybetween.ItiscomputedA,BtheexperiencelevelofAsmentionedearlier,theaimofthisexperimentistoquantitativelycomparetheimpactofusingdifferentfeatures(describedabove).Weusedthesefeaturesetstobuildbinaryclassi“er(randomforestdecisiontree)topredictwhetheratrustreciprocationwithhappenornot.Tablecomparesthetrustreciprocationaccuracyforvariousfeaturesets.Theadditionoftradefeaturesbooststheperformanceofthepredictivemodeloverthecasewhenno y sis and Minin g 664 0 5 10 15 20 25 0.5 0.52 0.54 0.56 0.58 0.6 0.62 ADDITIONAL DAYS OF TRADE INTERACTIONCWA TRUST TRUST+TRADE TRUST+TRADE+HMPLY TRUST+HMPLY (a)Classweightedaccuracy 0 5 10 15 20 25 0.8 0.81 0.82 0.83 0.84 0.85 0.86 0.87 ADDITIONAL DAYS OF TRADE INTERACTIONF1 MEASURE TRUST TRUST+TRADE TRUST+TRADE+HMPLY TRUST+HMPLY (b)F1measure 0 5 10 15 20 25 0.75 0.8 0.85 0.9 0.95 ADDITIONAL DAYS OF TRADE INTERACTIONPRECISION TRUST TRUST+TRADE TRUST+TRADE+HMPLY TRUST+HMPLY (c)Precision 0 5 10 15 20 25 0.84 0.85 0.86 0.87 0.88 0.89 0.9 0.91 ADDITIONAL DAYS OF TRADE INTERACTIONRECALL TRUST TRUST+TRADE TRUST+TRADE+HMPLY TRUST+HMPLY (d)RecallFig.4:Comparingclassweightedaccuracy,precision,andrecallfortrustreciprocationprediction.tradefeaturesareused.Weextendthisexperimenttoincludethevariationoffuturetimewindowsize(K)forallfeaturesandmonitoritsimpactintermofaccuracyofthepredictionmodel.Figureshowstheresultsofthisexperiment.Asmentionedearlier,westudytheimpactofvaryingthetimewindowssize(from0to25days)forallthefeatures.AsshowninFigures,usingtradeasanadditionalfeatureinthepredictionmodeloutperformsthemodelwhichusesonlytrustortrustandhomophilyonly.Wealso“ndthatadditionofhomophilyfeaturesdonothaveasigni“cantimpactinpredictingreciprocationintrustnetwork.Thisisaninteresting“ndingfortrustreciprocationpredictionbecauseitisageneralnotionthathomophilyissigni“cantinpredictingtrustlinks[].Apossibleexplanationofthisob-servationisthatthereciprocationphenomenonissigni“cantlydifferentfromanormaltrustformationphenomenon.Asweknow,inreciprocationthereisalreadyaonesidedrelationshipestablishedandareciprocationmightdependonentirelyotherdynamicssuchastradeinteractions.VI.CWehavestudiedvarioussocialfactorsaffectingrecip-rocationinthreedifferentinteractionnetworksfromSonyEverQuestIIMMOG.We,usingresponsetimeanalysis,showthatpeopleareslowinbuildingmutualhightrustrelationshipscomparedtolowbarrierones.Weextendouranalysisfromsingle-typenetworkstoheterogeneousnetworks,wherewecon“rmthathighdegreeofmediumbarrieractivitiesiscrucialforreciprocationinhighbarrierrelationships.CKNOWLEDGMENTSThisworkwasfundedbytheGRAPHSprogramatDARPA.SandiaNationalLaboratoriesisamulti-programlaboratorymanagedandoperatedbySandiaCorporation,awhollyownedsubsidiaryofLockheedMartinCorporation,fortheU.S.DepartmentofEnergysNationalNuclearSecurityAdministrationundercontractDE-AC04-94AL85000.[1]D.Lazer,A.Pentland,L.Adamic,S.Aral,A.-L.Barabsi,D.Brewer,N.Christakis,N.Contractor,J.Fowler,M.Gutmann,T.Jebara,G.King,M.Macy,D.Roy,andM.V.Alstyne,Computationalsocialscience,Žvol.323,no.5915,pp.721…723,2009.[2]N.Yee,Motivationsforplayinonlinegames,ŽCyberpsychologyand,vol.9,no.6,pp.772…775,2006.[3]E.Castronova,SyntheticWorlds:TheBusinessandCultureofOnline.UniversityOfChicagoPress,2005.[4]M.SzellandS.Thurner,Measuringsocialdynamicsinamassivemultiplayeronlinegame,ŽSocialNetworks,vol.32,no.4,pp.313…329,2010.[5]A.W.Gouldner,Thenormsofreciprocity:Apreliminarystatement,ŽAmericanSociologicalReview,vol.25,no.2,pp.161…178,1960.[6]S.Leider,M.M.Mbius,T.Rosenblat,andQ.-A.Do,Directedaltruismandenforcedreciprocityinsocialnetworks,ŽTheQuarterlyJournalof,vol.124,no.4,pp.1815…1851,2009.[7]K.V.Hansen,Theaskingrulesofreciprocityinnetworksofcareforchildren,ŽQualitativeSociology,vol.27,no.4,pp.421…437,2004.[8]C.A.Bliss,I.M.Kloumann,K.D.Harris,C.M.Danforth,andP.S.Dodds,Twitterreciprocalreplynetworksexhibitassortativitywithrespecttohappiness,ŽJournalofComputationalScience,vol.3,no.5,pp.388…397,2012.[9]V.ZlaticandH.Stefanc,Modelofwikipediagrowthbasedoninformationexchangeviareciprocalarcs,ŽEPL(EurophysicsLetters)vol.93,no.5,p.58005,2011.[10]N.Durak,T.G.Kolda,A.Pinar,andC.Seshadhri,Ascalablenullmodelfordirectedgraphsmatchingalldegreedistributions:In,out,andreciprocal,ŽinProc.IEEE2ndWorkshoponNetworkScience,2013.[11]M.A.Ahmad,D.A.Huffaker,J.Wang,J.W.Treem,M.S.Poole,andJ.Srivastava,GTPA:Agenerativemodelforonlinementor-apprenticenetworks,Žin,2010.[12]D.GarlaschelliandM.I.Loffredo,PatternsofLinkReciprocityinDirectedNetworks,ŽPhys.Rev.Lett.,vol.93,p.268701,2004.[13]G.Zamora-Lopez,V.Zlatic,C.Zhou,H.StefanCic,andJ.Kurths,Reciprocityofnetworkswithdegreecorrelationsandarbitrarydegreesequences,ŽPhys.Rev.E,vol.77,p.016106,2008.[14]L.Akoglu,P.V.deMelo,andC.Faloutsos,Quantifyingreciprocityinlargeweightedcommunicationnetworks,ŽinPaci“c-AsiaConferenceonKnowledgeDiscoveryandDataMining(PAKDD),2012,pp.85…96.[15]C.Wang,A.Strathman,O.Lizardo,D.Hachen,Z.Toroczkai,andN.V.Chawla,Weightedreciprocityinhumancommunicationnetworks,ŽarXivpreprintarXiv:1108.2822,2011.[16]M.Szell,R.Lambiotte,andS.Thurner,Multirelationalorganizationoflarge-scalesocialnetworksinanonlineworld,ŽProc.NationalAcademyofSciences,vol.107,no.31,pp.13636…13641,2010.[17]J.Cheng,D.M.Romero,B.Meeder,andJ.M.Kleinberg,Predictingreciprocityinsocialnetworks,ŽinIEEESocialComm/PASSAT,2011,pp.49…56.[18]V.ZlaticandH.Stefancic,In”uenceofreciprocaledgesondegreedistributionanddegreecorrelations,ŽPhysicalReviewE,vol.80,no.1,p.016117,2009.[19]M.A.Ahmad,Z.Borbora,J.Srivastava,andN.S.Contractor,Linkpredictionacrossmultiplesocialnetworks,ŽinICDMWorkshops,2010,pp.911…918.[20]L.A.N.Amaral,A.Scala,M.Barthemy,andH.E.Stanley,Classesofsmall-worldnetworks,ŽProc.NationalAcademyofSciences,vol.97,pp.11149…11152,2000.[21]M.Ahmad,I.Ahmed,J.Srivastava,andM.Poole,Trustme,imanexpert:Trust,homophilyandexpertiseinmmos,ŽinIEEESocial-Comm/PASSAT,2011,pp.882…887. 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