Mining Attributestructure Correlated Patterns in Large - PDF document

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473 471 477 470 469 466 468 476 472 467 474 475 MiningAttribute­structureCorrelatedPatternsinLargeAttributedGraphsArleiSilvaUniversidadeFederaldeMinasGeraisBeloHorizonte,Brasilarlei@dcc.ufmg.brWagnerMeiraJr.UniversidadeFederaldeMinasGeraisBeloHorizonte,Brasilmeira@dcc.ufmg.brMohammedJ.ZakiRensselaerPolytechnicInstituteTroy,NYzaki@cs.rpi.eduABSTRACTInthiswork,westudythecorrelationbetweenattributesetsandtheoccurrenceofdensesubgraphsinlargeattributedgraphs,ataskwecallstructuralcorrelationpatternmin-ing.Astructuralcorrelationpatternisadensesubgraphinducedbyaparticularattributeset.Existingmethodsarenotabletoextractrelevantknowledgeregardinghowvertexattributesinteractwithdensesubgraphs.Structuralcorre-lationpatternminingcombinesaspectsoffrequentitemsetandquasi-cliqueminingproblems.Weproposestatisticalsignicancemeasuresthatcomparethestructuralcorrela-tionofattributesetsagainsttheirexpectedvaluesusingnullmodels.Moreover,weevaluatetheinterestingnessofstruc-turalcorrelationpatternsintermsofsizeanddensity.Anecientalgorithmthatcombinessearchandpruningstrate-giesintheidenticationofthemostrelevantstructuralcor-relationpatternsispresented.Weapplyourmethodfortheanalysisofthreereal-worldattributedgraphs:acollab-oration,amusic,andacitationnetwork,verifyingthatitprovidesvaluableknowledgeinafeasibletime.1.INTRODUCTIONInseveralreal-lifegraphs,attributescanbeassociatedwithverticesinordertorepresentvertexproperties.Inso-cialnetworks,forexample,vertexattributesareusefultomodelpersonalcharacteristics.Moreover,vertexattributescanbeassociatedwithcontent(e.g.,keywords,tags)inthewebgraph.Suchanextendedgraphrepresentation,whichiscalledanattributedgraph,maysupportgraphpatternsthatproviderelevantknowledgeinvariousapplicationscenarios.Aninterestingquestionrelatedtoattributedgraphsishowparticularattributesareassociatedwiththetopologyofrealgraphs.Inotherwords,dothereexistpatternsthatexplainhowvertexattributesinteractwiththegraphstruc-ture?Howcanweextractandevaluatesuchpatterns?Inthispaper,westudytheproblemofcorrelatingattributesetswithanimportanttopologicalpropertyofgraphs,whichistheorganizationofverticesintodensesubgraphs.Forin-stance,weaimtoaddressquestionssuchas:Howdoesaparticularsetofinterestsinducecommunitiesinasocialnetwork?Whatarethecommunitiesthatemergearoundsuchinterests?Suchquestionsarerelatedtoimportantso-cialphenomenasuchashomophily[11]andin uence[2].Althoughseveraldenitionsofdensesubgraphshavebeenproposedintheliterature,mostofthemdonottakevertexattributesintoconsideration.Furthermore,suchdenitionsdonotprovideanyknowledgeregardinghowdierentsetsofattributesinducedensesubgraphs.Thisworkstudiesthecorrelationbetweenvertexattributesanddensesubgraphs,ataskwecallstructuralcorrelationpatternmining.Thestructuralcorrelationofanattributesetistheprobabilityofavertextobememberofadensesubgraphinitsinducedgraph.Moreover,astructuralcor-relationpatternisadensesubgraphinducedbyaparticularattributeset.Figure1illustratesadatasetforstructuralcorrelationpatternmining.ThevertexattributesaregiveninFigure1(a)andthegraphisshowninFigure1(b).Ex-ampledensesubgraphsareshowninFigures1(c)and1(d).ThestructuralcorrelationoftheattributeAis0.82,since9outof11verticesarecoveredbydensesubgraphsinitsinducedgraph.Ontheotherhand,thestructuralcorrela-tionofCis0,becausethereisnodensesubgraphinsidethegraphinducedbyC.ThestructuralcorrelationoffA,Bgis1,duetothefactthateveryvertexisamemberofadensesubgraphinthegraphinducedbyfA,Bg.Thepair(fA,Bg,f6,7,8,9,10,11g)isanexampleofastructuralcor-relationpattern,forwhichthesubgraphisshowninFigure1(d).Anotherexampleisthepattern(fAg,f3,4,5,6g),forwhichtheinducedsubgraphisshowninFigure1(d).Thestructuralcorrelationofattributesetsandthestruc-turalcorrelationpatternsarecomplementaryinformation,whiletherstisameasureofthecorrelationbetweenagivenattributesetandtheoccurrenceofdensesubgraphs,thesec-ondprovidesrepresentativesforsuchacorrelationthroughspecicsubgraphs.Weformulatethestructuralcorrelationpatternminingintermsoftwoexistingdataminingprob-lems:frequentitemsetandquasi-cliquemining.Frequentitemsetmining[1,19]isappliedtohandlethepossiblelargenumberofattributesetsfromthegraphandquasi-cliques[14,10]areusedasadenitionfordensesubgraphs.Westudystructuralcorrelationpatternminingfocusingontwoimportantaspects.Therstaspectisthesignicanceofthepatterns.Morespecically,itisrelevanttoprovidesignicancemeasuresforthestructuralcorrelationofat-tributesetsandthestructuralcorrelationpatterns.The vertexattributes 1A,C 2A 3A,C,D 4A,D 5A,E 6A,B,C 7A,B,E 8A,B 9A,B 10A,B,D 11A,B (a)Vertexattributes (b)Graph (c)Densesubgraph (d)DensesubgraphFigure1:Structuralcorrelationpatternmining(illustrativeexample)secondaspectisrelatedtothecomputationalcostoftheproposedtask.Ourobjectiveistoenabletheanalysisoflargerealgraphsinafeasibletime.Althoughsignicanceandhigh-performancearenotnecessarilyconcordantgoals,weproposesignicancemetricsthatmayleadtoecientpruningstrategiesforstructuralcorrelationpatternmining.Regardingthesignicanceofpatterns,weformulatenor-malizationapproachesforstructuralcorrelationpatternmin-inginordertomeasurethestatisticalsignicanceofthestructuralcorrelationofagivenattributeset.Theideaistocomparethestructuralcorrelationagainstitsexpectedvalue,whichisprovidedbyanullmodel.Moreover,weevaluatethestructuralcorrelationpatternsintermsofsize(i.e.,numberofvertices)anddensity(i.e.,cohesion).Suchevaluationisusefultorankthemostinterestingpatterns.Wecombinethestatisticalsignicanceofthestructuralcorrelationofattributesetsandthesizeanddensityofstruc-turalcorrelationpatternswitheectiveconstraintstoprunedownthesearchspace.Moreover,weproposetwostrategiesforcomputingthestructuralcorrelationofattributesetse-ciently.ThesepruningandsearchtechniquesareintegratedintotheSCPM(StructuralCorrelationPatternMining)al-gorithm,whichisdescribedandevaluatedinthispaper.Inparticular,weapplySCPMtotheanalysisofthreerealat-tributedgraphs:collaboration,musicandcitationnetworks.TheresultsshowthatSCPMisabletoextractrelevantknowledgeregardinghowvertexattributesarecorrelatedwithdensesubgraphsinlargeattributedgraphs.2.STRUCTURALCORRELATIONPATTERNMINING2.1Denitions2.1.1StructuralCorrelationWedeneanattributedgraphasa4-tupleG=(V;E;A;F)whereVisthesetofvertices,Eisthesetofedges,A=fa1;a2;:::angisthesetofattributes,andF:V!P(A)isafunctionthatreturnsthesetofattributesofavertex.Pisthepowersetfunction.EachvertexviinVhasasetofattributesF(vi)=fai1;ai2;:::aipg,wherep=jF(vi)jandF(vi)A.Figure1(b)showsanexampleofanattributedgraphwherethevertexattributesaregiveninFigure1(a).GiventhesetofattributesA,wedeneanattributesetSasasubsetofA(SA).Moreover,wedenotebyV(S)VthevertexsetinducedbyS(i.e.,V(S)=fvi2VjSF(vi)g)andbyE(S)EtheedgesetinducedbyS(i.e.,E(S)=f(vi;vj)2Ejvi;vj2V(S)g).ThegraphG(S),in-ducedbyS,isthepair(V(S);E(S)).Wealsodeneasup-portfunction,whichgivesthenumberofoccurrencesofanattributesetintheinputgraph((S)=jV(S)j),i.e.,thenumberofverticesthatcontainS.Thestructuralcorrelationfunctionmeasuresthecorre-lationbetweenagivenattributesetandtheoccurrenceofdensesubgraphsinanattributedgraph.Weapplyquasi-cliquesasadenitionfordensesubgraphs.Quasi-cliquesareanaturalextensionofthetraditionalcliquedenition.DEFINITION1.(Quasi-clique)Givenaminimumden-sitythreshold min(0 min1)andaminimumsizethresholdmin size,aquasi-cliqueisamaximalvertexsetQsuchthatforeachv2Q,thedegreeofvinQisatleastd min:(jQj�1)eandjQjmin size.Figures1(c)and1(d)areexamplesofan1-quasi-cliqueofsize4anda0.6-quasi-cliqueofsize6,respectively,fromthegraphshowninFigure1(b).Thequasi-cliqueminingprob-lemconsistsofidentifyingthequasi-cliquesfromagraphconsideringminimumsizeanddensityparameters,aprob-lemknowntobe#P-hard[14,17].WedenethestructuralcorrelationofanattributesetSastheprobabilityofavertexvwithattributeStobepartofaquasi-cliqueinG(S).DEFINITION2.(Structuralcorrelationfunction)GivenanattributesetS,thestructuralcorrelationofS,(S),isgivenas:(S)=jKSj jV(S)j(1)whereKSisthesetofverticesinquasi-cliquesinG(S).InthegraphfromFigure1,KfAg=f3;4;5;6;7;8;9;10;11g,KfCg=fgandKfA;Bg=f6;7;8;9;10;11g,andthusthecorrespondingvaluesof(fAg),(fCg),and(fA;Bg)are0.82,0,and1,respectively.StructuralcorrelationmeasuresthedependencebetweenattributesetSandthedensityoftheassociatedvertices.ItindicateshowlikelySistobepartofdensesubgraphs.Ourformulationenablestheiden-ticationofattributesthatinduceverticesthatarewellcon-nectedinthegraph.Inasocialnetwork,forinstance,suchattributesareofgreatinterestsincetheymayberelatedtohomophilyorin uence.Nevertheless,itisalsorelevantto understandthedensesubgraphsinducedbyattributesets.Wecallstructuralcorrelationpatternaquasi-cliquethatishomogeneousw.r.t.anattributeset.DEFINITION3.(Structuralcorrelationpattern).Astructuralcorrelationpatternisapair(S;Q),whereSisanattributeset(SA),andQisaquasi-cliquefromthegraphinducedbyS(QV(S)),giventhequasi-cliqueparameters minandmin size.Thepair(fAg,f3;4;5;6g)isanexampleofasize4struc-turalcorrelationpatternwithdensity1inducedbytheat-tributeAinthegraphfromFigure1.Anotherexampleofastructuralcorrelationpatternis(fA;Bg,f6;7;8;9;10;11g),whichisasize6structuralcorrelationpatternwithdensity0.6inducedbytheattributesetfA;Bg.2.1.2StructuralCorrelationPatternMiningProblemBasedonthedenitionofstructuralcorrelationpatternsandstructuralcorrelationfunction,weformulatethestruc-turalcorrelationpatternminingproblem.Itcomprisestheidenticationoftheattributesetscorrelatedwithdensesub-graphsandthedensesubgraphsinducedbysuchattributesets.Weapplyaminimumsupportthresholdminforat-tributesetsinordertoprunedownthenumberofpatterns.DEFINITION4.(Structuralcorrelationpatternmin-ingproblem).GivenanattributedgraphG(V;E;A;F),aminimumsupportthresholdmin,aminimumquasi-cliquedensity minandsizemin size,andaminimumstructuralcorrelationmin,thestructuralcorrelationpatternminingconsistsofidentifyingthesetofstructuralcorrelationpat-terns(S,Q)fromG,suchthatSisanattributesetforwhich(S)min,(S)min,andQisa min-quasi-cliqueforwhichQV(S)andjQjmin size.Asanexample,weconsidertheattributedgraphshowninFigure1andtheparametersmin, min,min sizeandminsetto3,0.6,4,and0.5,respectively.ThesetofstructuralcorrelationpatternsareshowninTable1.Foreachpattern,wegivethepair(attributeset,densesubgraph),therespec-tivequasi-cliquesizeanddensity( ),andtheattributesetsupport()andstructuralcorrelation(). patternsize  (fAg,f6;7;8;9;10;11g)60.60110.82 (fAg,f3;4;5;6g)41110.82 (fAg,f3;4;6;7g)40.67110.82 (fAg,f3;5;6;7g)40.67110.82 (fAg,f3;6;7;8g)40.67110.82 (fBg,f6;7;8;9;10;11g)60.6061.0 (fA;Bg,f6;7;8;9;10;11g)60.6061.0 Table1:PatternsfromthegraphshowninFigure1Similartothequasi-cliquemining,thestructuralcorre-lationpatternminingis#P-hard[17].Thisisbecausethequasi-cliqueminingproblemcanbereducedtothestructuralcorrelationpatternminingbyassigningthesameattributetoeachvertexfromthegraphandsettingminto1.Structuralcorrelationpatternminingisbasedonthestruc-turalcorrelationfunction,whichmeasureshowagivenat-tributesetisassociatedwiththeoccurrenceofdensesub-graphsinanattributedgraph.However,itisimportanttoassessthesignicance/interestingnessofagivenstructuralcorrelation,whichisthesubjectofthenextsection.2.1.3StatisticalSignicanceoftheStructuralCor­relationGiventhestructuralcorrelationofanattributeset,howcanweevaluateit?Inotherwords,whatcanbeconsideredahighorlowstructuralcorrelation?Inthissection,wead-dresssuchquestionsbyproposingnullmodelsforstructuralcorrelation.Thesemodelsspecifytheexpectedstructuralcorrelationofanattributesetassumingthatthecorrela-tionbetweenvertexattributesanddensesubgraphsisran-dom.Normalizedstructuralcorrelationmeasureshowthestructuralcorrelationofanattributesetdeviatesfromitsexpectedvalue,andallowsustoassessthestatisticalsignif-icanceofagivenstructuralcorrelationvalue.DEFINITION5.(Normalizedstructuralcorrelation).GivenanattributesetSwithsupport(S)andafunc-tionexp,whichgivestheexpectedstructuralcorrelationofanattributesetbasedonitssupportandtheattributedgraphG,thenormalizedstructuralcorrelationofSisgivenby:(S;G)=(S) exp((S);G)(2)AccordingtoDenition5,thenormalizedstructuralcor-relationfunctiongiveshowmuchthestructuralcorrelationofanattributesetSishigherthanexpected.Therefore,itrequiresthedenitionofthefunctionexp,whichreceivesthesupportofS((S))andtheattributedgraphGasargu-ments.Bynormalizingthestructuralcorrelation,weexpecttoobtainameasureofthecorrelationofanattributesetSthatisindependentofitssupportandtheinputgraph.WeassumethattheinputgraphGcomprisestheobjectofinterest,i.e.,itisthe\population"graph.Assumethatwearegiventheattributesetsupportvalue(S)(independentoftheactualattributesetS).Tocomputetheexpectedstructuralcorrelation,oursamplespaceisthesetofallver-texsubsetsofsize(S)drawnrandomlyfromG.Thestatis-ticofinterestisthemeanstructuralcorrelationvalue,exp.Thatis,theexpectedprobabilitythatarandomvertexinagivensampleinducesdensesubgraphs(quasi-cliques)inthatsampleofsize(S).Thequasi-cliqueparameters, minandmin size,areassumedtobexedaswell.Anintuitiveapproachforcomputingexpisthroughsim-ulation.Giventhesupport(S)oftheattributeset,aran-domsampleof(S)verticesfromGisselected.Eachvertexfromthesampleischeckedtobeinaquasi-clique,accordingtothequasi-cliqueparameters.Thestructuralcorrelationofthesampleisthefractionofverticesfromitthatareinatleastonequasi-clique.Thesimulation-basedexpectedstruc-turalcorrelationsim-expisgivenbytheaveragestructuralcorrelationofrrandomsamples.Thesimulation-basedstructuralcorrelationisverysimpleconceptuallybutmayrequireahighrtoachieveaccurateestimates,whichisprohibitiveinrealsettings.Thuswealsoproposeananalyticalformulationforanupperboundontheexpectedstructuralcorrelationofanattributeset.Theideaisthatavertexmusthaveaminimumdegreeofd min:(min size�1)einordertobememberofa min-quasi-cliqueofminimumsizemin size.Consequently,theprobabilityofavertextohaveadegreeofd min:(min size�1)einarandomsubgraphofsize(S)fromGgivesanupperboundontheexpectedstructuralcorrelationofS.Givenarandomsize(S)subgraphG(S)fromG,thedegreeofvinGandG(S)arerelatedasfollows. THEOREM1.(ProbabilityofavertexthathasadegreeinGtohaveadegreeinG(S)).IfarandomvertexvfromGwithdegreeisselectedtobepartofG(S),theprobabilityofsuchvertextohaveadegreeinG(S)isgivenbythefollowingbinomialfunction:F(;;)= !::(1�)�(3)whereistheprobabilityofaspecicvertexufromGtobeinG(S),ifvisalreadychosen,whichisgivenas:=(S)�1 jVj�1(4)Proofsketch.ThereareverticesadjacenttovinG,thus,theprobabilityofvtohaveadegreeofinG(S)istheprobabilityofselectingoutofverticestobepartofG(S).Sincevisalreadyselected,theprobabilityofselectinganyremainingvertexfromGisgivenbyequation4.BasedonTheorem1,wedeneanupperboundontheexpectedstructuralcorrelationastheprobabilityofavertextohaveadegreeofatleastd min:(min size�1)einG(S).THEOREM2.(Upperboundontheexpectedstruc-turalcorrelation).Giventhequasi-cliqueparameters minandmin size,thestructuralcorrelationofanattributesetwithsupport(S)isupperboundedby:max-exp((S))=mX=zp():X=zF(;;)(5)wherez=d min:(min size�1)e,misthemaximumdegreeofavertexfromG,andpisthedegreedistributionofG.Proofsketch.GivenavertexwithdegreeinG,theprob-abilityofsuchvertextohaveadegreeofatleastd min:(min size�1)einG(S)isthesumofexpression3overthedegreeintervalfromd min:(min size�1)eto.IfwemultiplythissumbytheprobabilityofavertexofdegreefromGtobeinG(S),i.e.,p(),itgivestheprobabil-ityofanyvertexwithdegreefromGtohaveadegreeofatleastd min:(min size�1)einG(S).Equation5isthesumofsuchproductsoverthevertexdegreeshigherthand min:(min size�1)e.TheproposedupperboundontheexpectedstructuralcorrelationofanattributeSisbasedontheexpectedde-greedistributionofarandomgraphofsize(S)fromG.However,thedegreeisnottheonlycriteriaforavertextobepartofaquasi-clique.Verticesthatsatisfytheminimumdegreethresholdmaynotbepartofaquasi-cliqueiftheyareconnectedtolowdegreevertices.Nevertheless,sinceweapplytheproposedformulationinordertonormalizethestructuralcorrelationofattributesetswithdierentsup-ports,ourobjectiveistoprovideafunctionthatpresentsaslopethatissimilartoexpectedstructuralcorrelation.InSection4.1,wecomparetheexpectedstructuralcorrelationcomputedusingsimulationwiththeproposedupperbound.Wecallsimandlbthenormalizedstructuralcorrela-tionfunctionsthatapplytheexpectedstructuralcorrela-tionbasedonsimulationsim-expandthetheoreticalupperboundmax-exp,respectively.Sincemax-expsim-exp,lb=(S) max�exp(S) sim�exp=sim,thus,lbisalowerboundonsim.Itisimportanttonoticethatmax-expismonotonicallynon-decreasing,i.e.,max-exp(1)�max-exp(2)ifandonlyif12.Itfollowsdirectlyfromthefactthattheanalyticalupperbound(Equation5)isbasedonacumula-tivebinomialfunction,whichisknowntobemonotonicallynon-decreasingw.r.t..Wealsoassumethatsim-expismonotonicallynon-decreasingforsucientlyhighvaluesofr,sinceanincreaseinthesizeoftherandomgraphsselectedfromGisnotexpectedtodecreasetheprobabilityofndingavertexinaquasi-clique.Suchpropertieswillbeexploitedbyourpruningtechniques,whichwillbeproposedfurtherinthispaper(seeSection3.2.1).Weapplythenormalizedstructuralcorrelationintheiden-ticationofstatisticallysignicantstructuralcorrelationval-ues.Therefore,weextendthestructuralcorrelationpat-ternminingproblem(Denition4)byaddingaminimumnormalizedstructuralcorrelationthresholdmin.Suchathresholdmayalsobeusefultoimprovetheperformanceofstructuralcorrelationpatternminingalgorithms,aswillbediscussedinSection3.2.Sinceausermaybeinterestedinpatternsthathavehighstructuralcorrelation()aswellasbeingstatisticallysignicant(),wepresentresultsusingbothregularandnormalizedstructuralcorrelation.2.2RelatedWorkFindingcommunities[6,3]anddensesubgraphs[5,10,8,20]hasbeenanactiveresearchtopic.Acommunityisusuallydenedassetofverticessignicantlymorecon-nectedamongthemselvesthanwithverticesoutsideit[3].Ontheotherhand,densesubgraphs,suchascliques[18],arestronglybasedoninternalcohesionandmaximality.Thisworkappliesadensesubgraphdenitioncalledquasi-clique,whichisasetofverticeswhereeachvertexiscon-nectedatleasttoafractionoftheothers.[14]introducestheproblemofminingcross-graphquasi-cliques.Theyfurtherstudiedtheproblemofminingfrequentcross-graphquasi-cliques[8].In[20]and[21]theauthorsstudytheproblemofminingfrequentcoherentclosedquasi-cliques.[10]stud-iestheproblemofndingquasi-cliquesfromasinglegraph,proposingpruningtechniquesforquasi-cliquemining.Graphclusteringanddensesubgraphdiscoverymethodsthatconsidervertexattributesascomplementaryinforma-tionhaveattractedtheinterestoftheresearchcommunityintherecentyears[12,4,22,13].Ageneralassumptionofthesemethodsisthatclustersbasedonboththetopologyofthegraphandtheattributesofverticesaremoremeaningfulthanthosebasedonlyonthetopologyortheattributes.[4]proposestwoecientalgorithmsfortheconnectedk-centerproblem,whichhasasobjectivetopartitionagraphconsid-eringboththeattributesandthetopology.[22]proposesarandomwalk-baseddistancemetricinanaugmentedgraphwhereverticesfromtheoriginalgraphareconnectedtonewverticesthatrepresentvertexattributes.In[12],theauthorsintroducetheproblemofminingcohesivepatterns,whicharedenseconnectedsubgraphswhereverticeshavehomo-geneousattributes(orfeatures).[13]considerstheproblemofcomputingmaximalhomogeneouscliquesinattributedgraphs.Dierentfromthesemethods,structuralcorrela-tionpatternminingdoesnotassumethatvertexattributesarecomplementaryinformation.Infact,weareinterestedinndingattributesetsthatexplaintheformationofdensesubgraphsthroughcorrelation.Assessinghowvertexattributesarerelatedtothegraph topologyhasledtothedenitionofnewpatterns.[15]proposedtheproblemofndingitemset-sharingsubgraphs,whichconsistsofextractingsubgraphswithcommonitem-sets.Itisimportanttonoticethatsuchmethoddonotcon-siderthedensityofsubgraphs.[9]denestheproximitypat-ternmining,whichevaluateshowclosevertexattributesareinthegraph.Aproximitypatternisasetoflabelsthatco-occurinneighborhoods.Therefore,proximitypatternsarenotnecessarilydensesubgraphsorcohesive,dierentlyfromstructuralcorrelationpatterns.In[7],theauthorsproposeadierentdenitionforthestructuralcorrelation,whichcom-parestheclosenessamongverticesinducedbyagivensingleattributeagainstasubgraphwhereattributesarerandomlydistributed.Ourworkdiersfrom[7]bycombiningmultipleattributesandconsideringaparticulartopologicalpropertywhichistheorganizationintodensesubgraphs.Moreover,besidestheevaluationofstructuralcorrelationofattributesets,weareinterestedinthediscoveryofrelevantdensesub-graphstoberepresentativesofthestructuralcorrelation.In[16],weintroducethestructuralcorrelationpatternminingandpresentanalgorithmforthisproblemcalledSCORP.Inthispaper,westudytheproblemofidentifyingstatisticallysignicantstructuralcorrelationpatternsbasedonanormalizationofthestructuralcorrelation.WealsopresenttheSCPMalgorithm,whichextendsSCORPwithnewpruningandsearchstrategiesforstructuralcorrelationpatternmining.DierentfromSCORP,SCPMenumeratesthetopstructuralcorrelationpatternsintermsofsizeanddensityeciently,insteadofthecompletesetofpatterns.3.ALGORITHMS3.1NaiveAlgorithmSincestructuralcorrelationpatternminingcombinesas-pectsofthefrequentitemsetminingandthequasi-cliqueminingproblems,wemaycombineafrequentitemsetmin-ingalgorithmandaquasi-cliqueminingalgorithmintoanaivealgorithmforstructuralcorrelationpatternmining.Thenaivealgorithmsolvesthestructuralcorrelationpat-ternminingproblem(seeDenition4)byrstenumeratingthesetoffrequentattributesetsFfromGandtheniden-tifyingthesetofquasi-cliquesQfromthegraphinducedbyeachfrequentattributesetSfromF.ThestructuralcorrelationofeachfrequentattributesetSiscomputedbycheckingwhethereachvertexv2V(S)ispartofaquasi-cliqueinQ.Frequentattributesetscanbeidentiedusingafrequentitemsetminingalgorithm[1,19].Inthiswork,weapplytheEclatalgorithm[19].Moreover,anyalgorithmforquasi-cliqueminingcanbeappliedbysuchnaivealgorithm.WeapplytheQuickalgorithm[10].Themaindrawbackofthenaivealgorithmisthatitenu-meratesthecompletesetoffrequentattributesetsfromGandthecompletesetofquasi-cliquesfromeachinducedgraphG(S),whereSisafrequentattributeset.Sincethefrequentitemsetminingandthequasi-cliqueminingprob-lemsareknowntobe#P-hard,thenaivealgorithmisex-pectedtonotbeabletoprocesslargeattributedgraphs.Inordertoachievesuchgoal,intheupcomingsections,wedescribeseveralstrategiesforecientstructuralcorrelationpatternmining.Wecombinesuchstrategiesintoanewal-gorithm,whichisdescribedinSection3.2.Furtherinthispaper,wecomparetheperformanceoftheproposedalgo-rithmagainstthisnaivemethod.3.2SCPMAlgorithmThissectionpresentstheSCPM(StructuralCorrelationPatternMining)algorithm,whichappliesseveralstrategiesinordertoenablethestructuralcorrelationpatternmin-inginlargeattributedgraphs.Unlikethenaivealgorithm,SCPMdoesnotenumerateeveryfrequentattributesetbutprunesthoseattributesetsthatcannotsatisfyaminimumstructuralcorrelationthreshold.Moreover,insteadofidenti-fyingeachquasi-cliquefromaninducedgraph,SCPMcheckswhetherverticesareinquasi-cliquesbyverifyingareducednumberofquasi-cliquecandidates.Finally,SCPMreturnsthesetofthetop-kmostrelevantstructuralcorrelationpat-ternsfromtheattributedgraph.3.2.1PruningStrategiesforSCPMiningThissectionpresentspruningtechniquesforstructuralcorrelationpatternmining.Theobjectiveofthesepruningtechniquesistoreducetheexecutiontimeofthestructuralcorrelationpatternminingalgorithmswithoutcompromis-ingitscorrectness.Theorem3allowsthepruningofverticesduringthelevel-wiseenumerationofattributesets.THEOREM3.(Vertexpruningforattributesets).LetKSbethesetofverticesindensesubgraphsinthegraphinducedbyanattributesetS.IfSiSj,thenKSjKSi.Proofsketch.Letssupposethatthereexistsavertexvsuchthatv2KSjandv=2KSi.Sincev2KSj,thereexistsadensesubgraphVV(Sj),suchthatv2V.Moreover,ifv=2KSi,theredoesnotexistanydensesubgraphUV(Si)suchthatv2U.Nevertheless,ifSiSj,thenV(Sj)V(Si),whichimpliesthatVV(Si)(contradiction).BasedonTheorem3,wecanpruneverticesthatarenotindensesubgraphsinthegraphinducedbyagivenattributesetbeforeextendingittogeneratelargerattributesets.At-tributesetscanalsobeprunedbasedonanupperboundonthestructuralcorrelationfunction,asstatedbyTheorem4.THEOREM4.(Attributesetpruningbasedontheupperboundonthestructuralcorrelation).Fortwoat-tributesetsSiandSj,ifSiSjand(Sj)min,then(Sj)(Si):jV(Si)j=minProofsketch.AccordingtoTheorem3,(Si):jV(Si)j(Sj):jV(Sj)j,sinceeveryvertexcoveredbyadensesub-graphinV(Sj)isalsocoveredbyadensesubgraphinV(Si).Moreover,since(Sj)min,(Sj)isupperboundedby(Si):jV(Si)j=minbasedonthedenitionofthestructuralcorrelationfunction(seeDenition2).GivenanattributesetSi,ofsizei,if(Si):jV(Si)j=minmin,thenSiisnotincludedinthesetofattributesetstobecombinedforthegenerationofsizei+1attributesets.Theorem4guaranteesthattheredoesnotexistanattributesetSj,suchthatSiSjand(Sj)min.Asimilarpruningrulecanbeformulatedbasedonthenormalizedstructuralcorrelationfunctiondenition.THEOREM5.(Attributesetpruningbasedontheupperboundonthenormalizedstructuralcorrelation).FortwoattributesetsSiandSj,ifSiSj,expisamono-tonicallynon-decreasing,and(Sj)min,then(Sj)(Si):jV(Si)j=(exp(min):min)Proofsketch.AccordingtoTheorem4,(Sj)(Si):jV(Si)j=min.Since(Sj)minandexpis Figure2:Setenumerationtree Algorithm1GeneralStructuralCorrelationAlgorithm Require:G(S), min,min sizeEnsure:Q1:Q ;2:X ;3:candExts(X) V(S)4:ApplyvertexpruningincandExts(X)5:qcCands f(X;candExts(X))g6:whileqcCands6=;do7:q qcCands:get()8:Applycandidatequasi-cliquepruninginq9:ifq:X[q:candExts(X)isaquasi-cliquethen10:Q Q[fq:X[q:candExts(X)g11:else12:ifq:Xisaquasi-cliquethen13:Q Q[fq:Xg14:endif15:insertextensionsofqintoqcCands16:endif17:endwhile monotonicallynon-decreasing,thenexp((Sj))exp(min).Therefore,(Sj)(Si):jV(Si)j=(exp(min):min).If(Si):jV(Si)j=(exp(min):min)min,theattributesetSi,ofsizei,isnotincludedinthesetofattributesetstobecombinedforthegenerationofsizei+1attributesets.Sincelbgivesalowerboundonthenormalizedstructuralcorrelation,thewholepruningpotentialofTheorem5maynotbeexplored.Nevertheless,theresultsshowthatuseoflbenablessignicantperformancegains(seeSection4.2).ThepruningstrategystatedbyTheorem3reducesthenumberofverticestobecheckedtobeinquasi-cliquesinthecomputationofstructuralcorrelation.Theorems4and5enablethereductionoftheattributesetsforwhichthestructuralcorrelationiscomputedtoasetthatisexpectedtobesmallerthanthesetoffrequentattributesets.3.2.2ComputingtheStructuralCorrelationAsdiscussedinSection3.1,thenaivealgorithmcomputesthestructuralcorrelationofanattributesetSthroughtheenumerationofthequasi-cliquesfromG(S).Inthissection,wedescribehowthestructuralcorrelationcanbecomputedbyidentifyingareducednumberofquasi-cliquecandidates.Quasi-cliquescanbeenumeratedbasedonavertexsetX,initiallysetas;,andasetofcandidateextensionsofX,candExts(X),initiallysetasV.VerticesaremovedfromcandExts(X)toX,oneatatime,untilthecompletesetofquasi-cliquecandidatesaregenerated.Figure2showsasetenumerationtreethatrepresentsthesearchspaceofquasi-cliquesconsideringasetof4vertices(1-4).Inordertoprunedownsuchsearchspace,quasi-cliqueminingalgo-rithmsapplyseveralpruningtechniques.Wedividethesetechniquesintotwogroups:1.Vertexpruning:Removalofverticesthatcannotbepartofanyquasi-cliqueinGaccordingtothequasi-cliquedenitionandthequasi-cliqueparameters.Ver-texpruningisperformediterativelyoverthegraphinordertominimizethesearchspaceofquasi-cliques.2.Candidatequasi-cliquepruning:Removalofcan-didatequasi-cliques(i.e.,pairs(X,candExts(X)))fromthesearchspaceofquasi-cliques.SuchremovalisbasedonthepropertiesofthesubgraphcomposedbyverticesfromXandcandExts(X).Algorithm1givesageneraldescriptionofhowquasi-cliquesareidentiedinthecomputationofstructuralcorre-lation.Thisalgorithmisalsousedasthebasisfortheenu-merationofthetop-kstructuralcorrelationpatterns.ThealgorithmreceivesaninducedgraphG(S),andthemini-mumdensity( min)andsize(min size)forquasi-cliques.Itgivesasoutputasetofquasi-cliquesQfromG.Vertexandquasi-cliquecandidatepruningsareappliedinlines4and8,respectively.Candidatequasi-cliquesaremanagedbythedatastructureqcCands,whichwillbediscussedlater.Eachcandidatepatternischeckedtobealookaheadquasi-clique(i.e.,q:X[q:candExts(X)isaquasi-clique)rst,duetothefactthatquasi-cliquesaremaximal.Incasesuchacondi-tiondoesnothold,q:Xischeckedtobeaquasi-cliqueandtheextensionsofqareinsertedintoqcCands(line15).ThealgorithmnisheswhenqcCandsbecomesempty.ThesetKS,whichiscomposedofverticescoveredbyquasi-cliquesinG(S),canbeobtaineddirectlyfromQ.Sincethequasi-cliqueminingproblemisknowntobe#P-hard,theidenticationofquasi-cliquesmayrequireprocess-ingalargenumberofquasi-cliquecandidates,whichwouldconstituteanimportantlimitationtothecomputationofthestructuralcorrelationoflargeinducedgraphs.Neverthe-less,computingthestructuralcorrelationdoesnotrequiretheenumerationofthecompletesetofquasi-cliques.Thenecessaryinformationiswhethereachvertexfromthein-ducedgraphiscoveredbyaquasi-cliqueornot.Therefore,candidatequasi-cliquescomposedofverticesalreadyknowntobecoveredbyquasi-cliquescanbeprunedfromthenewquasi-cliquecandidatesgeneratedinline15ofAlgorithm1.Besidespruningcandidatequasi-cliquesthatarealreadyknowntobecoveredbydensesubgraphs,wealsoproposesearchstrategiesforcomputingthestructuralcorrelation.Thesesearchstrategiesdeterminetheorderinwhichcan-didatequasi-cliquesareenumerated.Abreadth-rstsearch(BFS)strategyforcomputingthestructuralcorrelationtra-versesthesearchspaceofquasi-cliquesinabreadth-rstorder,startingfromtherootandvisitingthesmallervertexsetsbeforethelargerones.Ontheotherhand,adepth-rstsearch(DFS)strategyextendsvertexsetsasmuchaspossible.TheBFSstrategyisexpectedtoperformbetterincasecoveringverticeswithsmallerquasi-cliquesismoreecientthanwithlargerquasi-cliques.Consideringasetof4vertices,forwhichthesearchspaceofquasi-cliquesisshowninFigure2,theBFSandtheDFSstrategyvisitthequasi-cliquecandidatesasfollows: Algorithm2SCPMAlgorithm Require:G,min, min,min size,min,min,kEnsure:P1:P ;2:T ;3:I frequentattributesfromG4:forallS2Ido5: structuralcorrelationofS6:ifminAND=exp(S)minthen7:Q top-kpatternsfromG(S)8:forallq2Qdo9:P P[(S;q)10:endfor11:endif12:if:(S)min:minAND:(S)min:exp(min):minthen13:T T[S14:endif15:endfor16:P P[enumerate-patterns(T;G;min; min;min size;min;min;k) Algorithm3enumerate-patterns Require:T;G;min; min;min size;min;min;kEnsure:P1:P ;2:forallSi2Tdo3:R ;4:forallSj2Tdo5:ifi�jthen6:S Si[Sj7:if(S)minthen8: structuralcorrelationofS9:ifminAND=exp(S)minthen10:Q top-kpatternsfromG(S)11:forallq2Qdo12:P P[(S;q)13:endfor14:endif15:if:(S)min:minAND:(S)min:exp(min):minthen16:R R[S17:endif18:endif19:endif20:endfor21:P P[enumerate-patterns(R;G;min; min;min size;min;min;k)22:endfor BFS:f1g,f2g,f3g,f4g,f1;2g,:::f1;2;3;4g.DFS:f1g,f1;2g,f1;2;3g,f1;2;3;4g,f1;3g,:::f4g.Quasi-cliquescanbeenumeratedinBFSorderbyusingaqueueasadatastructuretomanagequasi-cliquecandidatesinAlgorithm1.Similarly,aDFSstrategyforenumeratingquasi-cliquescanapplyastackinordertomanipulatecandi-datepatterns.Furtherinthispaper,weevaluatethesearchstrategiespresentedinthissection.3.2.3EnumeratingTop­kPatternsAsdiscussedinSection2.1.2,enumeratingstructuralcor-relationpatternsisacomputationallyexpensivetask.Inthissection,westudyhowtoreducethecostofenumerat-ingstructuralcorrelationpatternsbyrestrictingtheoutputsettoonlythetop-kmostrelevantpatternsintermsofsize(primarycriteria)anddensity(secondarycriteria).Theenumerationofthetop-kstructuralcorrelationpat-ternsfollowsthesameproceduredescribedinAlgorithm1.WeuseaDFSstrategyinthediscoveryofthetop-kpat-ternsbecausestructuralcorrelationpatternsaremaximal(seeDenition3).However,sincethenumberofpatternstobediscoveredisknown,acurrentsetofpatternscanbeap-pliedtoprunethesearchspaceofnewcandidates.Newcan-didatequasi-cliquesaregeneratedinline15.Incasethecur-rentsetoftoppatternscontainskpatternsandacandidatepatternpcannotproduceapatternlargerthanthesmallestcurrenttop-kpatternt(i.e.,jp:X[p:candExts(X)jtj),pcanbepruned.Byupdatingthesetoftop-kpatterns,theminimumsizethresholdisincreasediteratively.Asaconse-quence,thetop-kpatternsareenumeratedmoreecientlythanthecompletesetofpatternsfromaninducedgraph.Algorithm2isahigh-leveldescriptionoftheSCPMal-gorithm,whichappliesthestrategiesforecientstructuralcorrelationpatternminingpresentedinthissection.TheinitialsetofattributesIiscomposedbythosewithasup-portofatleastmin(line3).ThestructuralcorrelationofeachsizeoneattributesetS2IiscomputedasdescribedinSection3.2.2.IncasethestructuralcorrelationofSsat-isesminimumstructuralcorrelation(min)andnormalizedstructuralcorrelation(min)thresholds,thetop-kpatternsinducedbySareidentiedusingthealgorithmdescribedinthissection(line7).ThesepatternsareincludedintoasetofpatternsPthatwillbegivenasoutput.Thepruningrulesforattributesetsbasedonand(seeSection3.2.1)areappliedinline12.PrunedattributesarenotincludedintothesetofattributesTtobeextended.Theseattributesareextendedbythefunctionenumerate-patterns(line16).Algorithm3describesthefunctionenumerate-patterns.ItreceivesthesameinputparametersofSCPM,andalsothesetofpatternstobeextendedT.Itreturnsthesetoftop-kpatterns(S;V)thathaveattributesetsextendedfromthoseinTregardingtheinputparameters.Newat-tributesetsareextendedthroughtheunionofexistingones(line6).AttributesetsaretraversedinaDFSorder(e.g.,fAg;fA;Bg;fA;B;Cg:::fEg).Theenumerate-patternsfunctionissimilartoAlgorithm2,exceptthateachnewattributesetSischeckedtosatisfytheminimumsupportthresholdmin(line7).Allvalidattributesetsaregener-atedthroughrecursivecallstoenumerate-patterns(line21).4.EXPERIMENTALRESULTSThissectionpresentscasestudiesonthestructuralcor-relationpatternminingusingrealdatasets.Moreover,weevaluatetheperformanceandstudythesensitivityofimpor-tantinputparametersofSCPM.Experimentswereexecutedona16-coreIntelXeon2.4Ghzwith50GBofRAM.Theimplementationsareavailableasopen-source1.4.1CaseStudies4.1.1DBLPIntheattributedgraphextractedfromtheDBLP2digitallibrary,eachvertexrepresentsanauthorandtwoauthorsareconnectediftheyhaveco-authoredapaper.Theattributesofauthorsaretermsthatappearinthetitlesofpapersau-thoredbythem3.IntheDBLPdatasetanattributesetde-nesatopic(i.e.,setoftermsthatcarryaspecicmeaningintheliterature)andadensesubgraphisacommunity.TheDBLPdatasethas108,030vertices,276,658edgesand23,285attributes.Table2showsthetop10attribute 1http://code.google.com/p/scpm/2http://www.informatik.uni-trier.de/~ley/db3Stemmingandremovalofstopwordswereapplied. (a)Graphinducedbyfsearch;rankg (b)Patterninducedbyfperform;systemgFigure3:ExamplesofresultsfromtheDBLPdataset   lb Slb Slb Slb basesystem5492.0414.0 gridapplic840.2641577 searchrank420.19635349 baseus5421.0413.5 gridservic599.23154703 performle404.14555067 basemodel4852.0313.3 environgrid525.21256793 structurindex404.14555067 modelus4168.0321.0 querixml615.21123533 searchmine413.14490932 systemus3989.0536.8 searchweb1031.2013738 usxml400.11442638 basenetwork3774.0541.8 searchrank420.19635349 searchwebdata424.14431589 modelsystem3460.0221.7 dynamsimul469.19383169 basesearchanalysi414.12416385 basedata3452.0771.6 queridata1540.192758 modelinternet401.10406059 baseimag3424.0217.6 chipsystem702.1963351 processdatadatabas416.12405363 imagus3345.0219.6 datastream1073.1810653 performdistributparallel416.11388818 Table2:DBLP-Topsupport(),str.correlation(),andnormalizedstr.correlation(lb)attributesets. Figure4:DBLP-Expectedcomputedbythesim-ulation(sim-exp)andanalytical(max-exp)models.setsw.r.tsupport(),structuralcorrelation(),andnor-malizedstructuralcorrelation(lb).Theminimumsize(min size)anddensity( min)parametersweresetto10and0.5,respectively.Theminimumsupportthreshold(min)wassetto400andweconsideredonlyattributesetsofsizeatleast2.Theparametersusedinourcasestudieswereselectedempirically.Top-attributesetspresentalowcorrelationwiththeformationofdensesubgraphsintheDBLPdataset.Suchtermsarepopularinpapertitles,butdonotcarrymuchknowledgeregardingtheformationofresearchcommuni-ties.Ontheotherhand,top-structuralcorrelationmaybemoreeasilyassociatedtoknowntopicsincomputersci-ence.Theattributesetfgrid;applicghasthehigheststruc-turalcorrelation(0.26),i.e.,26%oftheauthorsthathavethekeywords\grid"and\applic"areinsideacommunityofresearchersofsizeatleast10whereeachofthemhavecol-laboratedwithhalfoftheothermembers.Itisinterestingtopointoutthatthegraphinducedbyfgrid;applicghasmoreverticesindensesubgraphsthanthegraphinducedbyfbase;systemg,thoughfbase;systemgismorethan6timesmorefrequentthanfgrid;applicg.Ingeneral,highsupportattributesetsdonotpresenthighstructuralcorrelation.Figure4,showstheexpectedstructuralcorrelationfordierentsupportvaluesintheDBLPdataset.Theinputparametersarethesameasthoseusedtogeneratethere-sultsshowninTable2.Forthesimulationmodel,weex-ecuted1000simulationsforeachsupportvalueandshowalsothestandarddeviationoftheexpectedstructuralcor-relationestimated.Theanalyticalupperboundisnottightw.r.t.thesimulationresults,butpresentsasimilargrowth,whichshowsthatitenablesaccuratecomparisonsbetweenthestructuralcorrelationofattributesets.Basedontheproposedanalyticalmodel,thethirdcolumnofTable2showsthetopattributesetsintermsofanalyti-calnormalizedstructuralcorrelation(lb).Theattributeset (a)GraphinducedbyfSStevens,Wilcog (b)PatterninducedbyfVanMorrisongFigure5:ExamplesofresultsfromtheLastFmdataset   lb Slb Slb Slb Radiohead121892.11.37 Radiohead121892.11.37 SStevens,Wilco28798.041.14 Coldplay118053.09.33 Coldplay118053.09.33 SStevens,OfMontreal28621.041.13 Beatles109037.09.36 Beatles109037.09.36 Beirut27605.041.11 RPeppers105984.09.35 RPeppers105984.09.35 SStevens,Decemberists,Beatles27415.041.11 Nirvana100604.07.31 Metallica83587.08.41 NHotel,SStevens29260.041.10 TKillers96305.07.32 DCforCutie82025.07.41 SStevens,FLips,Beatles27571.041.09 Muse94382.07.33 Beck83360.07.40 ACollective33555.051.09 Oasis87875.06.30 Muse94382.07.33 BSScene,NMHotel27308.041.09 FFighters87001.06.33 Nirvana100604.07.31 Radiohead,Spoon,SStevens27113.041.06 PFloyd86807.07.34 TheShins68480.07.50 NHotel,Radiohead,Beatles28776.041.04 Table3:LastFm-Topsupport(),str.correlation(),andnormalizedstr.correlation(lb)attributesets.fsearch;rankghasthehighestnormalizedstructuralcorre-lation(635,349),i.e.,thestructuralcorrelationis635,349timestheupperboundonitsexpectedstructuralcorrela-tiongivenbytheanalyticalmodel.Figure3(a)presentsthegraphinducedbyfsearch;rankg.Verticescontainedinadensesubgraphareindicated.Densesubgraphscoverthedensestcomponentsoftheinducedgraph.Ingeneral,top-attributesetshavelowlbwhencomparedtothetop-lbattributesets.Moreover,highvaluesofdonotnecessar-ilyleadtohighvaluesoflb.Figure3(b)showsthelargeststructuralcorrelationpatternintermsofnumberofverticesfromDBLP,whichrepresentstwoimportantinterconnectedresearchgroupsonhighperformancesystems.4.1.2LastFmLastFm4isanonlinesocialmusicnetwork.Weuseasam-pleoftheLastFmuserscrawledthroughanAPIprovidedbyLastFm.IntheLastFmnetwork,verticesrepresentusersandedgesrepresentfriendships.Theattributesofavertexaretheartiststherespectiveuserhaslistenedto.Anat-tributesetintheLastFmdatasetrepresents,inamoregen-eralinterpretation,amusicaltaste(i.e.,setofartists)andadensesubgraphisacommunity.TheLastFmdatasetcontains272,412vertices,350,239edges,and3,929,101attributes.Table3showsthetop10attributesetsintermsofsupport(),structuralcorrelation()andnormalizedstructuralcorrelation(lb)discoveredfromLastFm.Theminimumsize(min size)anddensity 4http://www.last.fm Figure7:LastFm-Expectedcomputedbythesim-ulation(sim-exp)andanalytical(max-exp)models.( min)parametersweresetto5and0.5,respectively.Theminimumsupportthreshold(min)wassetto27,000.Ingeneral,thetop-attributesetsarethemostfrequentones.However,suchattributesetspresentlownormalizedstructuralcorrelation.Inotherwords,althoughtheseat-tributesarefrequentandhaveseveralverticescoveredbycommunities,thiscoverageisnotmuchhigherthanex-pected.Consideringthenormalizedstructuralcorrelation,whichtakesintoaccounttheexpectedstructuralcorrela-tionofanattributeset,thetoppatternschangesigni-cantly.Figure7showstheexpectedstructuralcorrelationforsupportvaluesvaryingfrom20,000to100,000.Eachsimulation-basedexpectedstructuralcorrelationvaluecor-respondstoanaverageof100simulations.Thetoplbat-tributesetfSStevens,WilcogincludestheAmericansingerandsongwriterSufjanStevensandtheAmericanbandWilco. (a)Graphinducedbyfnode,wirelessg (b)Patterninducedbyfperform,systemgFigure6:ExamplesofresultsfromtheCiteSeerdataset   lb Slb Slb Slb systempaper57906.16.77 networksensor3276.47108.7 nodewireless2086.35164.4 basepaper56566.10.47 networkhoc2744.47141.2 protocolrout2134.35157.6 paperresult47516.08.45 adnetworkhoc2725.44134.6 memoricach2150.32143.8 papermodel43929.09.59 networkrout5084.4148.0 networkhoc2744.47141.2 uspaper43573.05.32 networkwireless5242.4044.7 protocolwireless2048.29138.7 systembase42079.09.63 nodewireless2086.35164.4 adnetworkhoc2725.44134.7 approachpaper38690.05.40 protocolrout2134.35157.6 networknoderout2075.25118.3 performpaper37349.131.04 adnetwork3563.3469.3 optimqueri2094.26118.2 paperpropos37243.06.46 programlogic5895.3331.2 performinstruct2111.25115.95 paperalgorithm37027.12.95 memoricach2150.32143.8 paperadnetwork2081.23108.86 Table4:CiteSeer-Topsupport(),str.correlation(),andnormalizedstr.correlation(lb)attributesets.Figure5(a)showsthegraphinducedbytheattributesetfSStevens,Wilcog.Forclarity,weremovedverticeswithdegreelowerthan2.Byvisualizingverticesinsideandout-sidestructuralcorrelationpatterns,wecanunderstandhowthestructuralcorrelationcapturestherelationshipbetweenattributesanddensesubgraphs.Thelargeststructuralcor-relationpatternfoundispresentedinFigure5(b).Itrep-resentsacommunityof34userswhohavelistenedtotheNorthernIrishsingerandsongwriterVanMorrison.Vertexidentiersarenotshownduetoprivacyissues.4.1.3CiteSeerCiteSeerX5isascienticliteraturedigitallibraryandsearchengine.WebuiltacitationgraphfromCiteSeerXasofMarchof2010.IntheCiteSeergraph,papersarerep-resentedbyverticesandcitationsbyundirectededges.Eachpaperhasasattributestermsextractedfromitsabstract6.AttributesetsrepresenttopicsanddensesubgraphsdenegroupsofrelatedworkintheCiteSeergraph.TheCiteSeerdatasethas294,104vertices,782,147edges,and206,430attributes.Theparameterssettingappliedinthiscasestudyismin=2000,min size=5,and min=0:5.Table4showsthetopstructuralcorrelationattribute 5http://citeseerx.ist.psu.edu6Stemmingandstopwordsremovalwereapplied. Figure9:CiteSeer-Expectedforthesimulation(sim-exp)andanalytical(max-exp)models.setsw.r.t.,,andlbdiscovered.Top-attributesetspresentlowstructuralcorrelationandnormalizedstructuralcorrelationwhencomparedtothetop-andtop-lbattributesets,respectively.Moreover,similartotheDBLPdataset,whilethetop-attributesetsfromtheCiteSeerdatasetaregenericterms,thetop-andtop-lbattributesetsmaybeeasilyassociatedtoknownresearchtopics(e.g,computernetworks,queryoptimization).Figure9showstheexpectedstructuralcorrelationfordif-ferentsupportvaluesinCiteSeer.Theattributesetfnode;wirelessghasthehighestnormalizedstructuralcorrelation(lb=164.40).Figure6(a)showsthegraphinducedby (a)Runtimex min (b)Runtimexmin size (c)Runtimexmin (d)Runtimexmin (e)Runtimexmin (f)RuntimexkFigure8:Performanceevaluationtheattributesetfnode;wirelessginCiteSeer.Figure6(b)presentsthelargeststructuralcorrelationpatterndiscoveredintheCiteSeerdataset.Vertexlabelsaretheinitialsofpa-pertitles.Thepapersincludedinthepatterncovertopicssuchascaching,memorymanagement,computernetworks,processordesign,andinstructionleveloptimization(e.g.,AttributeCaches,SystemsforLateCodeModication,Lim-itsofInstructionLevelParallelism,Link-timeOptimizationofAddressCalculationona64-bitArchitecture).Wedonotshowthefulllistofpapertitlesduetospacelimitations.4.2PerformanceEvaluationThissectionevaluatestheperformanceofthestructuralcorrelationpatternminingalgorithms.ThedatasetusedisasmallerversionoftheDBLPdataset(SmallDBLP),whichhas32,908vertices,82,376edges,and11,192attributes.TheSCPM-BFSandSCPM-DFSareversionsoftheSCPMalgorithmusingtheBFSandDFSstrategy,respec-tively.TheNaivealgorithmenumeratesthecompletesetofquasi-cliquesfromtheinducedgraphs,asdescribedinSec-tion3.1.Wevaryeachparameterofthealgorithmskeepingtheothersconstant.Defaultvaluesfor min,min size,andminare0.5,11,and100.Moreovermin,min,andkaresetto0.1,1,and5,respectively,unlessstatedotherwise.Figures8(a),8(b),and8(c)showtheruntimeoftheal-gorithmsvaryingthevaluesof min,min size,andmin,respectively.Ingeneral,SCPM-DFSachievesthebestre-sults,beingupto3ordersofmagnitudefasterthantheNaivealgorithm.Moreover,SCPM-BFSperformsbetterthantheNaivealgorithminalltheexperiments.Intermsofthemin(Figure8(d))andmin(Figure8(e))parameters,boththeSCPM-BFSandSCPM-DFSapplythepruningtechniquesdescribedinSection3.2.1.BasedontheresultsshowninFigures8(d)and8(e),wecannoticethatsuchtechniquesleadtosignicantperformancegainswhenthevaluesofminandminareincreased.InFigure8(f),weshowtheruntimeofSCPM-DFSandtheNaivealgorithmfordierentvaluesofk.TheresultsofSCPM-BFSareomittedbecausebothSCPM-BFSandSCPM-DFSalgorithmsapplythesamestrategyforiden-tifyingthetop-kstructuralcorrelationpatterns(seeSection3.2.3).TheinsetalsoshowstheexecutiontimeofSCPM-DFSusingalinearscaleforthey-axis,tomoreclearlyseetheeectofkontheruntime.Theresultsshowthatforlowvaluesoftopk,SCPM-DFSisabletoachievelowrunningtimes,outperformingtheNaivealgorithmsignicantly.4.3ParameterSensitivityandSettingWenowassesshowdierentinputparametersaecttheoutputofstructuralcorrelationpatternmining.Ourobjec-tiveistoprovideguidelinesforsettingtheparametersofSCPM.Figure10showstheaveragestructuralcorrelationandnormalizedstructuralcorrelationofthecompleteoutput(global)andthetop-10%attributesetsfromtheSmallDBLPdatasetvaryingthe min,min size,andminparameters.Defaultvaluesfor min,min size,andminare0.5,10and100.Theresultsshowthatmorerestrictivequasi-cliquepa-rameters(i.e.,highvaluesof minandmin size)reducetheaveragebutmayincrease,sincedensesubgraphsbecomelessexpected.Moreover,highvaluesofminarerelatedtohighvaluesofstructuralcorrelation.However,suchat-tributesetsalsopresenthighvaluesofexp,leadingtolowvaluesofnormalizedstructuralcorrelation.SCPMisanexploratorypatternminingmethod,andthusreasonablevaluesforthedierentparameterscanbeob-tainedbysearchingtheparameterspace.Theminimumdensityparameter, min,andtheminimumquasi-cliquesize,min size,willdependontheapplication.Formin,ause-fulguidelineistoselectvaluesthatproduceasignicantexpectedstructuralcorrelation.Infrequentattributesetsmaynotbeexpectedtoinduceanydensesubgraph.Theotherparameters(min,min,andk)haveasobjectivestospeedupthealgorithmandmustbesetaccordingtotheavailablecomputationalresourcesandtime.5.CONCLUSIONSInthispaper,westudiedtheproblemofcorrelatingvertexattributesanddensesubgraphsinlargeattributedgraphs.Theconceptofstructuralcorrelation,whichmeasureshowanattributesetinducesdensesubgraphsinanattributedgraphwasproposed.Wealsopresentednormalizationap-proachesthatcomparethestructuralcorrelationofagivenattributesetagainstitsexpectedvalue,whichprovidesa (a)x min (b)xmin size (c)xmin (d)x min (e)xmin size (f)xminFigure10:Parametersensitivitymeasureofthestatisticalsignicanceforthestructuralcor-relation.Inordertoenabletheanalysisoflargedatabases,weintroducedsearchandpruningstrategiesforstructuralcorrelationpatternmining.Wealsoproposedanalgorithmfortheidenticationofthetopstructuralcorrelationpat-terns,whicharethelargestanddensestsubgraphsinducedbyagivensetofattributes.Thepatternsandalgorithmsproposedwereappliedtothreerealdatasets.Theattributesetsandpatternsfoundrepresentrelevantknowledgeintermsofthecorrelationbetweenattributesanddensesubgraphs.Acknowledgements:ThisworkwassupportedbyCNPQ,CAPES,Fapemig,FINEP,InWeb,NSFawardEMT-0829835,andNIHaward1R01EB0080161.WewouldliketothankAlbertoLaenderandLocCerffortheircomments.6.REFERENCES[1]R.Agrawal,T.Imielinski,andA.Swami.Miningassociationrulesbetweensetsofitemsinlargedatabases.InSIGMOD,pages207{216,1993.[2]A.Anagnostopoulos,R.Kumar,andM.Mahdian.In 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