The K clique Densest Subgraph Problem - PDF document

TheK-cliqueDensestSubgraphProblemCharalamposE.TsourakakisHarvardSchoolofEngineeringandAppliedSciencesbabis@seas.harvard.eduABSTRACTNumerousgraphminingapplicationsrelyondetectingsub-graphswhicharelargenear-cliques.Sinceformulationsthataregearedtowardsndinglargenear-cliquesareNP-hardandfrequentlyinapproximableduetoconnectionswiththeMaximumCliqueproblem,thepoly-timesolvabledensestsubgraphproblemwhichmaximizestheaveragedegreeoverallpossiblesubgraphs\liesatthecoreoflargescaledatamining"[ 10 ].However,frequentlythedensestsubgraphprob-lemfailsindetectinglargenear-cliquesinnetworks.Inthiswork,weintroducethe-cliquedensestsubgraphproblem,2.Thisgeneralizesthewellstudieddens-estsubgraphproblemwhichisobtainedasaspecialcasefor=2.For=3weobtainanovelformulationwhichwerefertoasthetriangledensestsubgraphproblem:givenagraphG(V;E),ndasubsetofverticesSsuchthat(S)=maxSVt(S) jSj,wheret(S)isthenumberoftrianglesinducedbythesetS.Onthetheoryside,weprovethatforanyconstant,thereexistanexactpolynomialtimealgorithmforthe-cliquedensestsubgraphproblem.Furthermore,weproposeanecient1 k-approximationalgorithmwhichgeneralizesthegreedypeelingalgorithmofAsahiroandCharikar[ 8 , 18 ]for=2.Finally,weshowhowtoimplementecientlythispeelingframeworkonMapReduceforany3,generaliz-ingtheworkofBahmani,KumarandVassilvitskiiforthecase=2[ 10 ].Ontheempiricalside,ourtwomainnd-ingsarethat(i)thetriangledensestsubgraphisconsistentlyclosertobeingalargenear-cliquecomparedtothedensestsubgraphand(ii)thepeelingapproximationalgorithmsforboth=2and=3achieveonreal-worldnetworksap-proximationratioscloserto1ratherthanthepessimistic1 kguarantee.Aninterestingconsequenceofourworkisthattrianglecounting,awell-studiedcomputationalprobleminthecontextofsocialnetworkanalysiscanbeusedtode-tectlargenear-cliques.Finally,weevaluateourproposedmethodonapopulargraphminingapplication.CopyrightisheldbytheInternationalWorldWideWebConferenceCom-mittee(IW3C2).IW3C2reservestherighttoprovideahyperlinktotheauthor'ssiteiftheMaterialisusedinelectronicmedia.WWW2015,May18–22,2015,Florence,Italy.ACM978-1-4503-3469-3/15/05.http://dx.doi.org/10.1145/2736277.2741128.CategoriesandSubjectDescriptorsG.2.2[GraphTheory]:GraphAlgorithms;H.2.8[DatabaseApplications]:DataminingGeneralTermsTheory,ExperimentationKeywordsDensestsubgraphproblem;Graphalgorithms;GraphMin-ing;Near-cliqueextraction1.INTRODUCTIONAwidevarietyofdataminingapplicationsreliesonex-tractingdensesubgraphsfromlargegraphs.Inbioinformat-icsdensesubgraphsareusedfordetectingproteincomplexesinprotein-proteininteractionnetworks[ 9 ]andforndingregulatorymotifsinDNA[ 24 ].Theyarealsousedforde-tectinglinkspaminWebgraphs[ 26 ],graphcompression[ 17 ]andminingmicro-bloggingstreams[ 6 ].Amongthevariousformulationsforndingdensesub-graphs,thedensestsubgraphproblem(DS-Problem)standsoutforthefactsthatissolvableinpolynomialtime[ 28 ]and1 2-approximableinlineartime[ 18 , 40 ].TostatetheDS-Problemwerstintroducethenecessarynotation.Inthisworkwefocusonsimpleunweighted,undirectedgraphs.GivenagraphG=(V;E)andasubsetofverticesSV,letG(S)=(S;E(S))bethesubgraphinducedbyS,andlete(S)=jE(S)jbethesizeofE(S).Also,theedgeden-sityofthesetSisdenedasfe(S)=e(S)=jSj2
.TheDS-Problemmaximizesthedegreedensitye(S) jSjoverallsubgraphsSV.Noticethatthisisequivalenttomaxi-mizingtheaveragedegree 1 .TheDS-Problemisapowerfulprimitiveformanygraphapplicationsincludingsocialpig-gybacking[ 27 ],reachabilityanddistancequeryindexing[ 19 , 37 ].However,alargenumberofapplicationsaimstondsubgraphswhicharelargenear-cliquesratherthanthesub-graphthatmaximizestheaveragedegree.FrequentlytheDS-Problemfailsinndinglargenear-cliquesbecauseitfavorslargesubgraphswithlowhighedgedensityfe.For 1Ingraphtheorythetermedgedensityrefersbydefaulttofee(S)=jSj2
2[0;1].However,sincedirectmaxi-mizationoffeisnotameaningfulproblem(evenasingleedgeachievesthemaximumpossibleedgedensity),theDS-Problemmaximizestheaveragedegree.Inthefollowing,werefertotheaveragedegreeofasetasitsdegreedensity. thisreasonotherformulationshavebeenproposed,seeSec-tion 2 .Unfortunately,theseformulationsareNP-hardandalsoinapproximableduetheconnectionswiththeMaximumCliqueproblem[ 32 ].Themaincontributionofthisworkisafamilyoftractableformulationswhichattacksecientlytheproblemofex-tractinglargenear-cliquesandincontrasttowell-performingheuristicscomeswithstrongtheoreticalguarantees.Inde-tail,ourcontributionsaresummarizedasfollows.Newformulation:Weintroducethek-cliquedensestsub-graphproblem(k-Clique-DS-Problem)whichgeneralizesthewell-studiedDS-Problem[ 18 , 25 , 28 , 40 ].Thegoalistomaximizetheaveragenumberof-cliquesinducedbyasetSVoverallpossiblevertexsubsets.NoticethattheDS-Problemisobtainedasaspecialcasefor=2.Weintroducealsothespecialcaseobtainedfor=3asthetriangledensestsubgraphproblem(TDS-Problem).Exactalgorithms.WepresenttwoexactalgorithmsfortheTDS-Problem.Thealgorithmwhichachievesthebestrunningtimeisbasedonmaximum\rowcomputationsandusesO(nt)space.ItisworthoutliningthatGoldberg'snetworkconstructionfortheDS-ProblemwhichusesO(nm)space[ 25 , 28 ]doesnotgeneralizetotheTDS-Problem.ThesloweroneisbasedonsupermodularmaximizationanduseslinearspaceO(nm).Here,n;m;tarethenumberofvertices,edgesandtrianglesintheinputgraph.Ouralgo-rithmscanbemodiedtoyieldpolynomialtimealgorithmsforthek-Clique-DS-Problemwhen=(1).Approximationalgorithm.Weproposea1 3-approximationalgorithmfortheTDS-Problemwhichrunsasymptoticallyfasterthananyoftheexactalgorithms.Wealsoproposea1 3+3-approximationalgorithmforany�0whichcanbeimplementedecientlyinMapReduce.Thealgorithmre-quiresO(log(n)=)roundsandisMapReduceecient[ 39 ]duetotheexistenceofecientMapReducetrianglecount-ingalgorithms,e.g.,[ 48 ].Ouralgorithmscanbeadaptedtothek-Clique-DS-Problemforany=(1).Experimentalevaluation.Weevaluateourexactandap-proximationalgorithmsonnumerousreal-worldnetworks.Amongourndings,weobservethattheoptimaltriangledensestsubgraphisconsistentlyclosertobeingalargenear-cliquecomparedtotheoptimaldensestsubgraph.Forin-stance,intheFootballnetwork(seeTable 1 foradescriptionofthedataset)theDS-Problemreturnsthewholegraphasthedensestsubgraph,withfe=0:094whereastheTDS-Problemreturnsasubgraphon18verticeswithfe=0:48.Wealsoobservethatthepeelingapproximationalgorithmsforboth=2and=3achieveonreal-worldnetworksapproximationratioscloserto1ratherthanthepessimistic1 kguarantee.Graphminingapplication.Weproposeamodiedver-sionoftheTDS-Problem,theconstrainedtriangledensestsubgraphproblem(Constrained-TDS-Problem),whichaimstomaximizethetriangledensitysubjecttothecon-straintthattheoutputshouldcontainaspeciedsetofver-ticesQ.WeshowhowtosolveexactlytheTDS-Problem.Thisvariationisusefulinvariousdata-miningandbioinfor-maticstasks,see[ 49 ].2.RELATEDWORKSincedensesubgraphdiscoveryconstitutesamainre-searchtopicingraphanalysis,awidevarietyofrelatedmethodsexists:heuristics[ 49 , 52 , 54 ],algorithmiccontri-butionsonNP-hardformulations[ 5 , 12 , 22 , 40 , 49 ]andpoly-timesolvableformulations[ 18 , 40 , 49 ].Wefocusonthelatter.DensestSubgraph.InthedensestsubgraphproblemwearegivenagraphGandwewishtondthesetSVwhichmaximizestheaveragedegree[ 28 , 38 ].Thedensestsubgraphcanbeidentiedinpolynomialtimebysolvingamaximum\rowproblem[ 25 , 28 ].Charikar[ 18 ]provedthatthegreedyalgorithmproposedbyAsashiroetal.[ 8 ]producesa1 2-approximationofthedensestsubgraphinlin-eartime.Asashiroetal.studythecomplexityofndingdensesubgraphsbyintroducingageneralizationoftheDS-Problemandthemaximumcliqueproblem[ 7 ].A-coreisamaximalconnectedsubgraphofGinwhichallverticeshavedegreeatleast.Itisworthremarkingthatthesamealgorithmprovidesa-coredecompositionofthegraphandsolvestheproblemofndingthedegeneracy[ 11 ].Inthecaseofdirectedgraphs,thedensestsubgraphproblemissolvedinpolynomialtimeaswell[ 18 ].KhullerandSaha[ 40 ]pro-videalineartime1 2-approximationalgorithmforthecaseofdirectedgraphsamongothercontributions.TwointerestingvariationsoftheDkSproblemwereintroducedbyAnder-senandChellapilla[ 5 ].ThetwoproblemsaskforthesetSthatmaximizesthedensitysubjecttojSj(DamkS)andjSj(DalkS).WhenrestrictionsonthesizeofSareimposedtheproblembecomesNP-hard[ 40 ].Finally,thedensestsubgraphproblemhasbeenconsideredinvari-oussettings,includingMapReduce[ 10 ],thestreaming[ 10 ],thedynamicsetting[ 21 , 45 ]andtheircombinationrecently[ 13 ].TriangleCountingandListing.ThestateoftheartalgorithmforexacttrianglecountingisduetoAlon,YusterandZwick[ 4 ]andrunsinO(m2! !+1),wherecurrentlythefastmatrixmultiplicationexponent!is2.3729[ 53 ].Thus,theiralgorithmcurrentlyrunsinO(m1:4081)time.ThebestknownlistingalgorithmuntilrecentlywasduetoItaiandRodeh[ 33 ]whichrunsinO(m(G))time,where(G)isthegrapharboricity.Since(G)=O(p m),therunningtimeisalwaysO(m3=2).Recently,Bjorklund,Pagh,WilliamsandZwickgaverenedalgorithmswhichareoutputsensitivealgorithms[ 14 ].Finallyawealthofapproximatetrianglecountingmethodsexist[ 35 , 41 , 44 , 51 ].3.PROBLEMDEFINITIONWedenethenotionofaveragetriangledensity.Definition1(TriangleDensity).LetG(V;E)beanundi-rectedgraph.ForanySVwedeneitstriangledensity(S)as(S)=t(S) s,wheret(S)isthenumberoftrianglesinducedbySandsjSj.Noticethat3(S)istheaveragenumberof(induced)trian-glespervertexinS.Theoptimizationproblemwefocusonfollows.Problem1(TDS-Problem).GivenG(V;E),ndasubsetofverticesSsuchthat(S)=GwhereGmaxSV(S).ItisclearthattheDS-ProblemandtheTDS-Problemingeneralcanresultinradicallydierentsolutions.Con-siderforinstanceagraphGon2n+3verticeswhichis theunionofatriangleK3andofabipartitecliqueKn;n.TheoptimalsolutionsoftheDS-ProblemandtheTDS-Problemarethebipartitecliqueandthetrianglerespec-tively.Therefore,theinterestingquestioniswhethermaxi-mizingtheaveragedegreeandthetriangledensityresultindierentresultsinreal-worldnetworks.OurresultsinSec-tion 5 indicatethattheanswerispositivesincethetriangledensestsubgraphcomparedtodensestsubgraphissmallerwhichexhibitsastrongernear-cliquestructure.4.PROPOSEDMETHODSection 4.1 providestwoalgorithmswhichsolvetheTDS-Problemexactly.Sections 4.2 and 4.3 providetwoapprox-imationalgorithmsfortheTDS-Problem.Finally,Sec-tion 4.4 providesageneralizationoftheDS-ProblemandtheTDS-Problemtomaximizingtheaverage-cliqueden-sityandshowshowtheresultsfrompreviousSectionsadapttothisproblem.4.1ExactSolutionsLetn;m;tbethenumberofvertices,edgesandtrianglesingraphGrespectively.ThealgorithmpresentedinSec-tion 4.1.1 achievesthebestrunningtime.WepresentanalgorithmwhichreliesonthesupermodularitypropertyofourobjectiveinSection 4.1.2 .Thelatteralgorithm,evenifslower,requiresO(nm)space,whereastheformerO(nt)space.Inreal-worldnetworks,typicallymt.Finally,itisworthmentioningthatCharikar'slinearprogram,seex2in[ 18 ],canbeextendedtoalinearprogram(LP)whichsolvestheTDS-Problem,see[ 50 ]forthedetails.4.1.1AnOm3=2nt+min(n;t)3
-timeexactsolution Algorithm1triangle-densestsubgraph(G) 1:ListthesetoftrianglesT(G),tjT(G)j2:l t n;u (n1)(n2) 63:S ;4:whileul1 n(n1)do5: lu 26:H Construct-Network(G;;T(G))7:(S;T) minimumst-cutinH8:ifSfsgthen9:u 10:else11:l 12:S Snfsg
\V(G)13:endif14:ReturnS15:endwhile Ourmaintheoreticalresultisthefollowingtheorem.Itsproofisconstructive.Theorem1.ThereexistsanalgorithmwhichwhichsolvestheTDS-ProblemandrunsinOm3=2nt+min(n;t)3
time.ThersttermO(m3=2)comesfromusingtheItai-Rodehal-gorithm[ 33 ]asourtrianglelistingblackbox.IfweusethenaiveO(n3)trianglelistingalgorithmthentherunningtimeexpressionissimpliedtoO(n3nt).Ontheotherhand,ifweusethealgorithmsofBjorklundetal.[ 14 ]therstterm Algorithm2(G;;T(G)) 1:V(H) fsg[V(G)[T(G)[ftg.2:Foreachvertexv2V(G)addanarcofcapacity1toeachtriangletiitparticipatesin.3:Foreachtriangle
=(u;v;w)2T(G)addarcstou;v;wofcapacity2.4:Adddirectedarc(s;v)2A(H)ofcapacitytvforeachv2V(G).5:Addweighteddirectedarc(v;t)2A(H)ofcapacity3foreachv2V(G).6:ReturnnetworkH(V(H);A(H);w);s;t2V(H). becomesfordensegraphs~On!n3(!1)=(5!)t2(3!)=(5!)
andforsparsegraphs~Om2!=(!+1)m3!1 !+1t3! !+1
,where!isthematrixmultiplicationexponent.Currently!2:3729dueto[ 53 ].Wemaintain[ 33 ]asourblack-boxtokeeptheexpressionssimpler.However,thereadershouldkeepinmindthattheresultpresentedin[ 14 ]improvesthetotalrunningtimeoftherstterm.WeworkourwaytoprovingTheorem 1 byrstprov-ingthenextkeylemma.Then,weremovethelogarithmicfactor.Lemma1.Algorithm1solvestheTDS-ProblemandrunsinOm3=2+(nt+min(n;t)3)log(n)
time.Algorithm1usesmaximum\rowcomputationstosolvetheTDS-Problem.ItisworthoutliningthatGoldberg'smaximum\rowalgorithm[ 28 ]fortheDS-ProblemisbasedonanetworkconstructionthatdoesnotadapttothecaseoftheTDS-Problem.Algorithm1returnsanoptimalsub-graphS,i.e.,(S)=.Thealgorithmperformsabinarysearchonthetriangledensityvalue.Specically,eachbi-narysearchquerycorrespondstoqueryingdoesthereexistasetSVsuchthatt(S)=jSj�?.Foreachbinarysearch,weconstructabipartitenetworkHbyinvokingAlgorithm2.LetT(G)bethesetoftrianglesinG.ThevertexsetofHisV(H)=fsg[A[B[ftg,whereAV(G)andBT(G).Noticethatweoverloadthenotationinordertousethefre-quentlyusednotationforthesinkvertext.Itshouldalwaysbeclearfromthecontexttowhichentity(numberoftrian-glesvs.sinkvertex)wereferto.ForthepurposeofndingT(G),atrianglelistingalgorithmisrequired[ 14 , 33 ].ThearcsetofgraphHiscreatedasfollows.Foreachvertexr2Bcorrespondingtotriangle
(u;v;w)weaddthreein-comingandthreeout-comingarcs.Theincomingarcscomefromtheverticesu;v;w2Awhichformtriangle
(u;v;w).Eachofthesearcshascapacityequalto1.Theoutgoingarcsgotothesamesetofverticesu;v;w,butthecapacitiesareequalto2.Inadditiontothearcsofcapacity1fromeachvertexu2Atothetrianglesitparticipatesin,weaddanoutgoingarcofcapacity3tothesinkvertext.Fromthesourcevertexsweaddanoutgoingarctoeachu2Aofcapacitytv,wheretvisthenumberoftrianglesvertexvparticipatesinG.Aswehavealreadynoticed,HcanbeconstructedinO(m3=2)time[ 33 ].ItisworthoutliningthataftercomputingHforthersttime,subsequentnetworksneedtoupdateonlythearcsthatdependontheparameter,somethingnotshowninthepseudocodeforsimplicity.ToprovethatAlgorithm1solvestheTDS-ProblemandrunsinOm3=2+(nt+min(n;t)3)log(n)
timeweproceedinsteps. Forthesakeoftheproof,weintroducethefollowingdef-initionsandnotation.ForagivensetofverticesSletti(S)bethenumberoftrianglesthatinvolveexactlyiverticesfromS,i2f1;2;3g.Noticethatt3(S)isthenumberofinducedtrianglesbyS,forwhichwehavebeenusingthesimplernotationt(S)sofar.Weusethefollowingclaimasourcriteriontosettheinitialvaluesl;uinthebinarysearch.Claim1. t n(S)(n1)(n2) 6foranySV.Thelowerboundisobvioussince(V)=t n.Theupperboundalsofollowstriviallybyobservingthat(S)n3
=nforany;6SV.ThissuggeststhattheoptimalvalueisalwaysO(n2).Thenextclaimservesasacriteriontodecidewhentostopthebinarysearch.Claim2 Thesmallestpossibledierenceamongtwodistinctvalues(S1);(S2)isequalto1 n(n1).Toseewhy,noticethatthedierencebetweentwopossibledierenttriangledensityvaluesist(S1)jS2jt(S2)jS1j jS1jjS2j:IfjS1jjS2jthenjj1 n1 n(n1),otherwisejj1 jS1jjS2j1 n(n1).Noticethatcombiningtheabovetwoclaimsshowsthatthebinarysearchterminatesinatmostd4lognequeries.ThefollowinglemmaisastructurallemmafortheoptimalstcutthenetworkH.Lemma2.Consideranyminimumst-cut(S;T)inthenetworkH.LetA1S\A;B1S\BandA2T\A;B2T\B.Thecostofthemin-cutisequaltoXv=2A1tv+2t2(A1)+t1(A1)+3jA1j:Proof.CaseI:A1;: Inthiscasethepropositiontriv-iallyholds,asthecostisequaltoPv2Atv=3t.ItisworthnoticingthatinthiscaseB1hastobealsoempty,otherwisewecontradicttheoptimalityof(S;T).HenceSfsg;TA[B[ftg.CaseII:A1;: ConsiderthecostofthearcsfromA1[B1toA2[B2.Weconsiderthreedierentsubcases,onepereachtypeoftrianglewithrespecttosetA1.Type3:Ifthereexistthreeverticesu;v;w2A1thatformatriangle
(u;v;w),thenthevertexr2BcorrespondingtothisspecictrianglehastobeinB1.Ifnot,thenr2B2,andwecouldreducethecostofthemin-cutby3,ifwemovethetriangletoB1.Thereforethecostwepayfortrianglesoftypethreeis0.Type2:Considerthreeverticesu;v;wsuchthattheyformatriangle
(u;v;w)andu;v2A1;w2A2.Then,thevertexr2BcorrespondingtothistrianglecanbeeitherinB1orB2.Inbothcaseswealwayspay2inthecutforeachtriangleoftypetwo.Type1:Finally,inthecaseu;v;wformatriangle,u2A1;v;w2A2thevertexr2Bcorrespondingtotriangle
(u;v;w)willbeinB2.Ifnot,thenitliesinB1andwecoulddecreasethecostofthecutby3ifwemoveitinB2.Hence,wepay1inthecutforeachtriangleoftypeone.ThereforethecostduetothevarioustypesoftriangleswithrespecttoA1isequalto2t2(A1)+t1(A1).Furthermore,thecostofthearcsfromsourcestoTisequaltoPv2A2tvPv=2A1tv.ThecostofthearcsfromA1toTisequalto3jA1j.Summinguptheindividualcostterms,weobtainthatthetotalcostisequaltoPv=2A1tv2t2(A1)+t1(A1)+3jA1j. ThenextlemmaprovesthecorrectnessofthebinarysearchinAlgorithm1.Lemma3.(a)IfthereexistsasetWV(G)suchthatt3(W)�jWjthenanyminimumst-cut(S,T)inHsat-isesSnfsg6;.(b)Furthermore,iftheredoesnotexistsasetWsuchthatt3(W)�jWjthenthecut(fsg;A[B[ftg)isaminimumst-cut.Proof.(a)LetWVbesuchthatt3(W)�jWj:(1)Supposeforthesakeofcontradictionthattheminimumst-cutisachievedby(fsg;A[B[ftg).Inthiscasethecostoftheminimumst-cutisPv2Atv=3t.Now,considerthefollowing(S;T)cut.SetSconsistsofthesourcevertexs,A1WandB1bethesetoftrianglesoftype3and2inducedbyA1.LetTbetherestoftheverticesinH.Thecostofthiscutiscap(S;T)=Xv=2A1tv+2t2(A1)+t1(A1)+3jA1j:Therefore,byourassumptionthattheminimumst-cutisachievedby(fsg;A[B[ftg)weobtain3tXv=2A1tv+2t2(A1)+t1(A1)+3jA1j:(2)Now,noticethatbydoublecountingXv2A1tv=3t3(A1)+2t2(A1)+t1(A1):Furthermore,weobserveXv2A1tvXv=2A1tv=3t:Bycombiningthesetwofacts,andthefactthat3tisthecapacityoftheminimumcut,weobtainthefollowingcon-tradictionofInequality( 1 ).3tXv=2A1tv+2t2(A1)+t1(A1)+3jA1j,t3(W)jWj:(b)ByLemma 2 ,foranyminimumst-cut(S;T)thecapac-ityofthecutisequaltoPv=2A1tv+2t2(A1)+t1(A1)+3jA1j,whereA1A\S;A2A\T.Supposeforthesakeofcon-tradictionthatthecut(fsg;A[B[ftg)isnotaminimumcut.Therefore,cap(fsg;A[B[ftg)=3t�Xv=2A1tv+2t2(A1)+t1(A1)+3jA1j:Usingthesamealgebraicanalysisasin(a),theabovestatementimpliesthecontradictiont3(W)�jWj,whereWA1. NowwecancompletetheproofofLemma 1 .Proof.TheterminationofAlgorithm1followsdirectlyfromClaims1,2.ThecorrectnessfollowsfromLemmata 2 , 3 .TherunningtimefollowsfromClaims1,2whichshowthatthenumberofbinarysearchqueriesisO(log(n))andeachbinarysearchquerycanbeperformedinOnt+min(n;t)3
timeusingthealgorithmduetoAhuja,Orlin,SteinandTarjan[ 3 ] 2 orGuseld'salgorithm[ 31 ]. TheproofofTheorem 1 followsfromLemma 1 andthefactthattheparametricmaximum\rowalgorithmofAhuja,Orlin,SteinandTarjan[ 3 ],seealso[ 25 ],s(h)avestheloga-rithmicfactorfromtherunningtime.4.1.2AnO(n5m1:4081n6))log(n)
-timeexactsolu-tionInthisSectionweprovideasecondexactalgorithmfortheTDS-Problem.First,weprovidethenecessarytheoreticalbackground.Definition2(Supermodularfunction).LetVbeaniteset.Thesetfunctionf:2V!Rissupermodu-larifandonlyifforallA;BVf(A[B)f(A)+f(B)f(A\B):Afunctionfissupermodularifandonlyiffissubmodu-lar.Sub-andsupermodularfunctionsconstituteanimportantclassoffunctionswithvariousspecialproperties.Inthiswork,weareprimarilyinterestedinthefactthatmaximiz-ingasupermodularfunctionissolvableinstronglypolyno-mialtime[ 30 , 34 , 42 , 46 ].Forourpurposes,westatethefollowingresultwhichweuseasasubroutineinourproposedalgorithm.Theorem2([ 43 ]).Thereexistsanalgorithmformax-imizinganintegervaluedsupermodularfunctionfwhichrunsinOn5EOn6)
time,wherenjVjisthesizeofthegroundsetVandEOisthemaximumamountoftimetoevaluatef(S)forasubsetSV.WeshowinthefollowingthatwhenthegroundsetisthesetofverticesVandf:2V!Risdenedbyf(S)=t(S)jSjwhere2R,wecansolvetheTDS-Probleminpolynomialtime.Theorem3.Functionf:V!Rwheref(S)=t(S)jSjissupermodular.Proof.LetA;BV.Lett:2V!Rbethefunc-tionwhichforeachsetofverticesSreturnsthenumberofinducedtrianglest(S).Bycarefulcountingt(A[B)=t(A)+t(B)t(A\B)+t1(A:BnA)+t2(A:BnA);wheret1(A:BnA);t2(A:BnA)arethenumberoftrianglewithone,twoverticesinAandtwo,oneverticesinBnArespectively.Hence,foranyA;BV 2NoticethatthenetworkHhasO(nt)arcs,thereforetherunningtimeof[ 3 ]isO(min(n;t)(nt)+min(n;t)3)=O(nt+min(n;t)3).t(A[B)+t(A\B)t(A)+t(B)andthefunctiontissupermodular.Furthermore,forany�0thefunctionjSjissupermodular.Sincethesumoftwosupermodularfunctionsissupermodular,theresultfollows. Theorem 3 naturallysuggestsAlgorithm3.Thealgorithmwillruninalogarithmicnumberofrounds.IneachroundwemaximizefunctionfusingOrlin-Supermodular-OptwhichtakesasinputargumentsthegraphGandtheparameter�0[ 43 ].Weassumeforsimplicitythatwithinthepro-cedureOrlin-Supermodular-Optfunctionfisevaluatedusinganecientexacttrianglecountingalgorithm[ 4 ].Thealgo-rithmofAlon,YusterandZwick[ 4 ]runsinO(m2!=(!+1))timewhere!2:3729[ 53 ].ThissuggeststheEOO(m1:4081).TheoverallrunningtimeofAlgorithm3isO(n5m1:4081n6)log(n)
andthespaceusageO(nm)ratherthanO(nt). Algorithm3triangle-densestsubgraph(G) 1:l 0;u (n1)(n2) 6;S V2:whileul1 n(n1)do3: lu 24:(val;S) Orlin-Supermodular-Opt(G;)5:ifval0then6:u 7:else8:l 9:S S10:endif11:ReturnS12:endwhile 4.2A1 3-approximationalgorithmInthisSectionweprovideanalgorithmfortheTDS-Problemwhichprovidesa1 3-approximation.Ouralgorithmfollowsthepeelingparadigm,see[ 8 , 18 , 40 , 36 ].Specically,ineachrounditremovesthevertexwhichparticipatesinthesmallestnumberoftrianglesandreturnsthesubgraphthatachievesthelargesttriangledensity.ThepseudocodeisshowninAlgorithm4. Algorithm4(G) 1:Countthenumberoftrianglestvforeachvertexv2V2:Hn G3:fori nto2do4:LetvbethevertexofGiofminimumnumberoftri-angles5:Hi1 Hinv6:endfor7:ReturnHjthatachievesmaximumtriangledensityamongHis,i=1;:::;n. Theorem4.Algorithm4isa1 3-approximationalgorithmfortheTDS-Problem. Proof.LetSbeanoptimalset.Letv2S;jSjsandtA(v)bethenumberofinducedtrianglesbyAthatvparticipatesin.Then,Gt(S) st(Snfvg) s1,tS(v)G;sincet(Snfvg)=t(S)tS(v).ConsidertheiterationbeforethealgorithmremovestherstvertexvthatbelongsinS.CallthesetofverticesW.Clearly,SWandforeachvertexu2WthefollowinglowerboundholdstW(u)tW(v)tS(v)GduetothegreedinessofAlgorithm3.ThisprovidesalowerboundonthetotalnumberoftrianglesinducedbyWt(W)=1 3Xu2WtW(u)1 3jWjG)t(W) jWj1 3G:Tocompletetheproof,noticethatthealgorithmreturnsasubgraphSsuchthat(S)(W)1 3G. ThekeydierencecomparedtotheDS-Problempeel-ingalgorithm[ 18 ]isthatwhenweremoveavertex,thecountsofitsneighborsmaydecreasemorethan1.There-fore,whenvertexvisremoved,weupdatethecountsofitsneighborsinOdeg(v)2
time,bylookinghowmanytrian-gleseachofitsneighborshasaftervisremoved.NoticethatOPvdeg(v)2

O(mn).4.3MapReduceImplementationTheMapReduceframework[ 20 ]hasbecomethedefactostandardforprocessinglarge-scaledatasets.Inthefollow-ing,weshowhowwecanapproximateecientlytheTDS-ProbleminMapReduce.Beforewedescribethealgo-rithm,weshowthatAlgorithm5forany�0terminatesandprovidesa1 3+3-approximation.Theideabehindthisalgorithmistopeelverticesinbatches[ 10 , 29 ]ratherthanonebyone. Algorithm5(G;�0) 1:Sout;S V2:whileS;do3:A(S) fi2S:tS(i)3(1+)(S)g4:S SnA(S)5:if(S)(Sout)then6:Sout S7:endif8:endwhile9:ReturnSout. Lemma4.Forany�0,Algorithm5providesa1 (3+3)-approximationtotheTDS-Problem.Furthermore,itter-minatesinO(log1+(n))passes.Proof.LetSbeanoptimalsolutiontotheTDS-Problem.AsweprovedinTheorem 4 ,foranyv2SitistruethattS(v)G.Furthermore,ineachroundatleastonever-texisremoved.Toseewhy,assumeforthesakeofcon-tradictionthatA(S)=;forsomeSduringtheexecutionofthealgorithm.Then,weobtainthecontradictionthat3jSj(S)=Pv2StS(v)(3+3)jSj(S).Considertheroundwherethealgorithmforthersttimeremovesavertexv2S.LetWbethecorrespondingsetofvertices.Sincev2A(W)ispeeledo,weobtainanupperboundonitsin-duceddegree,namelyv2A(W))tW(v)(3+3)(W).SinceSW,weobtain(3+3)(W)tW(v)tS(v)(S);whichprovesthatAlgorithm5isa1 (3+3)-approximationtotheTDS-Problem.Toseewhythealgorithmterminatesinlogarithmicnumberofrounds,noticethat3t(S)Xv2SnA(S)tS(v)(3+3)jSjjA(S)j
t(S) jSj,jA(S)j 1+jSj,jSnA(S)j1 1+jSj:SinceSdecreasesbyafactorof1 1+ineachround,thealgo-rithmterminatesinO(log1+(n))=Olog(n) 
rounds. MapReduceImplementation:NowweareabletodescribeouralgorithminMapReduce.ItusesanyoftheecientalgorithmsofSuriandVassilvitski[ 48 ]asasubroutinetocounttrianglespervertexineachround.Theremovaloftheverticeswhichparticipateinlesstrianglesthanthethresh-old,isdoneintworounds,asin[ 10 ].Forcompleteness,wedescribetheprocedurehere.ThesetofverticesStobepeeledoineachroundaremarkedbyaddingakey-valuepairhv;$iforeachv2S.Eachedge(u;v)ismappedtohu;vi.Thereducerreceivesallendpointsoftheedgesinci-dentwithvandthesymbol$incasethevertexismarkedfordeletion.Incasethevertexismarked,thenthereducetaskreturnsnothing,otherwiseitcopiesitsinput.Inthesecondround,weperformthesameprocedurewiththeonlydier-encebeingthatwemapeachedge(u;v)tohv;ui.Therefore,theedgeswhichremainhavebothendpointsunmarked.ThealgorithmrunsinO(log(n)=),asittakesO(log(n)=)peel-ingorounds,andineachpeelinground,constantnumberofroundsisneededtocounttrianglespervertex,markver-ticesfordeletionandremovethecorrespondingvertexset.4.4k-cliqueDensestSubgraphWeoutlinethatourproposedmethodscanbeadaptedtothefollowinggeneralizationoftheDS-ProblemandtheTDS-Problem.Definition3(-clique-densestsubgraph).LetG(V;E)beanundirectedgraph.ForanySVwedeneits-cliquedensityhk(S),2ashk(S)=ck(S) s,whereck(S)isthenumberof-cliquesinducedbySandsjSj.Problem2(k-Clique-DS-Problem).GivenG(V;E),ndasubsetofverticesSsuchthathk(S)=hkwherehk=maxSVhk(S).Asinthetriangledensestsubgraphproblem,wecreateanetworkHparameterizedbythevalueonwhichweper-formourbinarysearch.TheprocedureisdescribedinAlgo-rithm6.ThesetC(G)isthesetof-cliquesinG.WetheninvokeAlgorithm1,withtheupperboundusettonk.Fol-lowingtheanalysisofTheorem 1 ,weseethatthek-Clique-DS-Problemissolvableinpolynomialtime.Forinstance,usingGuseld'salgorithm[ 31 ]or[ 3 ]ineachbinarysearch querywegetanoverallrunningtimeOnk+(njC(G)jn3)log(n)
O(nk+1log(n)).UsingtheimprovedresultduetoAhuja,Orlin,SteinandTarjanforparametricmax\rowsinunbalancedbipartitegraphs[ 3 ],wesavetheloga-rithmicfactorintherunningtime. Algorithm6(G;;C(G);k) 1:V(H) fsg[V(G)[C(G)[ftg.2:Foreachvertexv2V(G)addanarcofcapacity1toeach-cliqueciitparticipatesin.3:Foreach-clique(ui1;:::;uik)2C(G)addarcstoui1;:::;uikofcapacity1.4:Adddirectedarc(s;v)2A(H)ofcapacitycvforeachv2V(G).5:Addweighteddirectedarc(v;t)2A(H)ofcapacitykforeachv2V(G).6:ReturnnetworkH(V(H);A(H);w);s;t2V(H). Furthermore,Algorithm4canalsobemodied,byremovingineachroundthevertexwiththesmallestnumberof-cliques,toobtainCorollary 2 .AstheanalogyofTheorem 4 .Corollary1.Thealgorithmwhichpeelsoineachroundthevertexwiththeminimumnumberof-cliquesandreturnsthesubgraphthatachievesthelargest-cliquedensity,isa1 k-approximationalgorithmforthek-Clique-DS-Problem.Similarly,Algorithm5andtheMapReduceimplementa-tioncanbemodiedtosolvethek-Clique-DS-Problem.Weomitthedetails.Corollary2.Thealgorithmwhichpeelsoineachroundthesetofverticeswithlessthan(1+)h(S),whereh(S)isthe-cliquedensityinthatround,terminatesinO(log1+(n))roundsandprovidesa1 k(1+)-approximationguaranteeforthek-Clique-DS-Problem.Furthermore,using[ 23 ],weobtainanecientMapReduceimplementation.Weillustrateanexamplewherechoosingalargervalueyieldsbenets.LetGG(n;p)beanErdos-Renyigraph,wherepp(n).AssumethatweplantacliqueKofsizenforsomeconstant\r�0.Wewishtoshowanon-trivialrangeofpp(n)valuessuchthatthefollowingconditionshold:h2(C)=jE(K)j jKj(n\r2) n\rp(n2) nE[h2(V)],andfor3hk(C)=(n\rk) n\rp(k2)(nk) nE[hk(V)].BysimplealgebraicmanipulationweseethatpsatisesbothconditionsifOn(1)
pOn2 k(1)
3 .Clearly,forlargervalues,weallowourselvesalargerrangeofpvaluesforwhichwecanndthehiddencliqueinexpectation.Ourpreliminaryexperimentalresultsfor=4indicatethatthe4-clique-densestsubgraphgetsclosertoalargenear-cliquecomparedtothetriangledensestsubgraph.However,thegainofmovingfromthedensestsubgraphtothetriangledensestsubgraphwithrespecttoextractinglargenear-cliquesislargerthanthegainofmovingfromthetriangledensestsubgraphtothe4-clique-densestsubgraph. 3Noticethatforthisrangeofp,thegraphisconnectedandthecliquenumberisconstantwithhighprobability[ 15 ]5.EXPERIMENTALEVALUATIONBeforewepresentourndingsindetail,wesummarizethem:(i)theTDS-Problemandtheproposedalgorithmsconstitutenewvaluablegraphminingprimitivesforndinglargenear-cliques,(ii)the1 3-approximationalgorithm(Al-gorithm4)achievessignicantlybetterapproximationsthanthepessimistic1 3guarantee.Alsoitissignicantlyfaster.(iii)TryingasmallrangeofvaluesforAlgorithm5isingeneralasaferstrategycomparedtorunningexperimentswithxedchoice.5.1ExperimentalSetupThedatasetsweuseareshowninTable 1 .Allgraphsweremadesimpleandundirectedbyignoringtheedgedi-rection,whenthegraphisdirected,andremovingself-loopsandmultipleedges,ifany.Theexperimentswereperformedonasinglemachine,withIntel(R)Core(TM)i5CPUat2.40GHz,with3.86GBofmainmemory.Wehaveimple-mentedAlgorithm1inMatlabR2011ausingamaximum\rowimplementationduetoKolmogorovandBoykov[ 16 ]asoursubroutinewhichrunsintimeO(t(nt)3).Thisimple-mentationisprohibitivelyexpensiveevenforsmallgraphswhichhavealargenumberoftriangles.WehavecodedthepeelingalgorithminC++usingpriorityqueues.Asourtrianglelistingalgorithm,weusethesimplenodeiter-atoralgorithmwhichchecksforeachvertexthenumberofedgesamongitsneighbors.Thecodeispubliclyavailableat http://people.seas.harvard.edu/~babis/code.html .Wemeasurethequalityofeachextractedsubgraphbytwomea-sures:theedgedensityoftheextractedsubgraphfee(S)=jSj2
andtheoutputsizejSj.Noticethatwhenfeiscloseto1,theextractedsubgraphisclosetobeingaclique.5.2MainFindingsTable 2 showstheresultsobtainedonseveralpopularsmall-andmedium-sizedgraphs.Eachcolumncorrespondstoadataset.Therowscorrespondtomeasurementsforeachmethodweusetoextractasubgraph.Specically,therst(DS),second(1 2-DS),third(TDS)andfourth(1 3-TDS)rowscorrespondtothesubgraphextractedbyGold-berg'sexactalgorithm[ 28 ]fortheDS-Problem,Charikar's1 2-approximationalgorithm[ 18 ]fortheDS-Problem,Al-gorithm1andAlgorithm4fortheTDS-Problemrespec-tively.ForeachoptimalextractedsubgraphS,weshowitssizeasafractionofthetotalnumberofvertices,theedgedensityfe(S),theaveragedegree2(S)=2e(S)=jSjandtheaveragenumberoftrianglespervertex3(S)=3t(S)=jSj.Weobservethatthetriangle-densestsubgraphisclosertobeinganear-cliquecomparedtothedensestsubgraph.ApronouncedexampleistheFootballnetworkwheretheopti-maldensestsubgraphisthewholenetworkwithfe=0:0094,whereastheoptimaltriangle-densestsubgraphisasetof18verticeswithedgedensity0.48.Finally,weobservethatthequalityofAlgorithm's4outputisveryclosetotheoptimalsolutionandsometimesevenbetter.ThesameobservationholdsforthecaseofCharikar's1 2-approximationalgorithm[ 18 ].WeusetheC++implementationofAlgorithm4andaC++implementationofCharikar's1 2-approximationalgo-rithmontherestofthedatasetsofTable 1 .TheresultsareshowninTable 3 ,uptotwodecimaldigitsofaccuracy.Theca-HepThdatasetisanexception,astheoptimalsolutionscoincide.Wealsonoticethattheruntimesappearthesame Name Nodes Edges Description Adjnoun 112 425 Generatedbyprocessingtextdata AS-735 6475 12572 AutonomousSystems AS-caida 26475 53381 AutonomousSystems ca-Astro 17903 196972 Co-authorship ca-GrQC 4158 13422 Co-authorship ca-HepTh 11204 117619 Co-authorship Celegans 297 4296 NeuralnetworkofC.Elegans DBLP 53442 255936 Co-authorship Epinions 75877 405739 Socialnetwork Enron 33696 180811 Email EuAll 224832 339925 Email Football 115 613 NCAAfootballgamenetwork Karate 34 78 Socialnetwork Lesmis 77 254 Generatedbyprocessingtextdata Politicalblogs 1490 16715 Generatedbyprocessingsalesdata Politicalbooks 105 441 Blognetwork soc-Slashdot0811 77360 469180 PersontoPerson soc-Slashdot0902 82168 504230 PersontoPerson wb-cs-Stanford 8929 26320 WebGraph web-Google 855802 4291352 WebGraph web-NotreDame 325729 1090108 WebGraph Wiki-vote 7066 100736 Wikipedia\who-votes-whom" Table1:Datasetsusedinourexperiments. Method Measure AdjnounCelegansFootballKarateLesmisPolblogsPolbooks DS jSj jVj(%) 42.8645.810047.129.919.151.4 2 9.5817.1610.665.2510.7855.829.40 fe 0.200.130.0940.350.490.1960.18 3 1445.9321.125.6441.61768.8722.68 1 2-DS jSj jVj(%) 41.142.410052.929.918.757.1 2 9.5717.110.665.210.7855.89.3 fe 0.210.140.0940.310.490.200.16 3 14.1646.521.125.1641.61774.622.68 TDS jSj jVj(%) 36.610.415.717.716.98.119.1 2 9.3713.818.224.6710.6255.729.34 fe 0.230.460.480.930.890.460.50 3 1556.82288.0147.31972.3625.95 1 3-TDS jSj jVj(%) 36.69.115.717.716.98.115.2 2 9.3713.568.224.6710.6255.729.13 fe 0.230.520.480.930.890.460.61 3 1556.55288.0147.31972.3625.5 Table2:ComparisonoftheextractedsubgraphsbytheGoldberg'sexactalgorithmfortheDS-Problem(DS),Charikar's1 2-approximationalgorithm(1 2-DS),ourexactalgorithmfortheTDS-Problem(TDS)andour1 3-approximationalgorithm(1 2-TDS).Here,fe(S)=e(S)=jSj2
istheedgedensityoftheextractedsubgraph,2(S)=2e(S)=jSjistheaveragedegreeand3(S)=3t(S)=jSjistheaveragenumberoftriangles.forca-HepThduetousingtwodecimaldigitsofaccuracy.Onotherdatasets,weobservedierencesbetweenthetwosolu-tions.Forinstance,forthecollaborationnetworkca-Astrothedensestsubgraphisasubgraphwith1184verticesandfe=0:05.Thetriangle-densestsubgraphisacliquewith57vertices.Overall,weverifythefactthatthetriangle-densestsubgraphisclosertobeinganear-clique.Finally,theruntimesareshown.NoticethattheruntimesreportedforforAlgorithm4includeboththetrianglecountingandthepeelingphase.5.3ExploringparameterinAlgorithm5InthisSectionwepresenttheresultsofAlgorithm5ontheDBLPgraph.Thisisparticularlyinterestinginstanceasitindicatesthatinsteadoftryingtoselectagoodvalue,itisworthtryingoutatleastfewvalues,assumingcompu-tationalresourcesareavailable.Werangefrom0.1to1.8withastepof0.1.Figure 1 (a)plotsthenumberofroundsAlgorithm5takestoterminateasafunctionof.Weob-servethatevenforsmallvaluesthenumberofroundsis6.Thereadershouldcomparethistotheupperboundpre-dictedbyLemma 4 whichexceeds100.Figure 1 (b)plotsthe 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 2 4 6
Rounds 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.2 0.4 0.6 0.8 1
Rel.  1/(3+3
) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.2 0.4 0.6 0.8 1
Rel. fe Rel. ft (a)(b)(c)Figure1:Exploringthetrade-obetweenthenumberofroundsandaccuracyasafunctionoftheparameterforAlgorithm5.LetS;SbetheextractedsubgraphsbyAlgorithms5and1respectively.(a)Numberofrounds,(b)relativeaveragetriangledensityratio(S) (blue)andtheapproximationguarantee1=(3+3)(red),and(c)relativeratiosfe(S) fe(S);ft(S) ft(S)asfunctionsof.1 2-DS1 3-TDS jSjfeT jSjfeT AS-735 590.280.00 130.80.07 AS-caida 1430.140.02 270.520.63 ca-Astro 11840.050.06 5711.42 ca-GrQC 420.790.00 140.890.02 ca-HepTh 3210.02 3210.02 Epinions 9990.1210.08 4310.2568.75 Enron 5550.140.02 3900.192.01 EuAll 5070.130.08 2000.299.52 soc-Slashdot0811 2070.410.13 2530.496.85 soc-Slashdot0902 2190.400.16 1730.507.72 wb-cs-Stanford 840.640.48 260.800.67 web-Google 2400.232.54 1200.4479.5 web-NotreDame 13670.1150.50 4570.34516.3 Wiki-vote 8460.110.00 4640.800.19 Table3:Comparisonoftheextractedsubgraphsbythe1 2-approximationalgorithmofCharikarandthe1 3-approximationalgorithm,Algorithm4.There-spectiveruntimesareshowninseconds.relativeratioRel:(S) whereSistheoutputofAlgo-rithm5.Forconvenience,thelowerbound1 3+3isplottedwithredcolor.Besidestheratiofe(S) fe(S),gure 1 (c)plotsalsotherelativeratioft(S) ft(S)asafunctionof.Hereft(S)=t(S) (jSj3).Asweobserve,thequalityofAlgorithm5isclosetotheoptimalsolutionexceptfor=0:7and=0:8.Byinspectingwhythishappensweobservethattheoptimaltriangle-densestsubgraphisacliqueof44vertices.Itturnsoutthatfor=0:7;0:8theoptimalsubgraphwhichisfoundinthelastroundoftheexecutionofthealgorithm(thelatterhappensforallvalues)consistsof98and74verticeswhichcontainasasubgraphtheoptimalK44.Forothervaluesof,thesubgraphinthelastroundiseithertheoptimalK44orclosetoit,withfewmoreextravertices.Thisexampleshowsthepotentialdangerofusingasinglevaluefor,suggestingthattryingoutasmallnumberofvaluescanbesignicantlybenecialintermsoftheapproximationquality.6.APPLICATION:ORGANIZINGCOCKTAILPARTIESAgraphminingproblemthatcomesupinvariousappli-cationsisthefollowing:givenasetofverticesQV,ndadensesubgraphcontainingQ.Werefertothistypeofgraphminingproblemsascocktailproblems,duetothefol-lowingmotivation,c.f.[ 47 ].SupposethatasetofpeopleQwantstoorganizeacocktailparty.Howdotheyinviteotherpeopletothepartysothatthesetofalltheparticipants,includingQ,areassimilaraspossible?AvariationoftheTDS-Problemwhichaddressesthisgraphminingproblemfollows.Problem3(Constrained-TDS-Problem).GivenagraphG(V;E)andQV,ndthesubsetofverticesSthatmaximizesthetriangledensitysuchthatQS,S=argmaxQSV(S):TheConstrained-TDS-Problemcanbesolvedbymod-ifyingourproposedalgorithmsaccordingly.Ausefulcorol-laryfollows.Corollary3.TheConstrained-TDS-Problemissolv-ableinpolynomialtimebyaddingarcsfromstov2Aoflargeenoughcapacities,e.g.,capacitiesequalton3+1aresucientlylarge.Furthermore,thepeelingalgorithmwhichavoidsremovingverticesfromQisa1 3-approximationalgo-rithmfortheConstrained-TDS-Problem.Inthefollowingweevaluatethe1 3-approximationalgo-rithmontwodatasets.Thetwoexperimentsindicatetwodierenttypesofperformancesthatshouldbeexpectedinreal-worldapplications.Therstisapositivewhereasthesecondisnegativecase.Bothexperimentshereserveassan-itychecks 4 Politicalvotedata.WeobtainSenatedatafortherstsession(2006)ofthe109thcongresswhichspannedthepe-riodfromJanuary3,2005toJanuary3,2007,duringthe 4Forinstance,bypreprocessingthepoliticalvotedatafromamatrixformtoagraphusingathresholdforedgeaddi-tions,resultsininformationloss. fthandsixthyearsofGeorgeW.Bush'spresidency[ 1 ].InthisCongress,therewere55,45and1Republican,Demo-craticandindependentsenatorsrespectively.ThedatasetcanbedownloadedfromtheUSSenatewebpage http://www.senate.gov .Wepreprocessthedatasetinthefollowingway:weaddanedgebetweentwosenatorsifamongthebillsforwhichtheybothcastedavote,theyvotedatleast80%ofthetimesinthesameway.There-sultinggraphhas100verticesand2034edges.Werunthe1 3-approximationalgorithmonthisgraphusingasoursetQtherstthreerepublicansaccordingtolexicographicor-der:Alexander(R-TN),Allard(R-CO)andAllen(R-VA).Weobtainatouroutputasubgraphconsistingof47ver-tices.Byinspectingtheirparty,wendthat100%ofthemareRepublicans.Thisshowsthatouralgorithminthiscasesucceedsinndingthelargemajorityoftheclusterofrepub-licans.Itisinterestingthatthe8remainingRepublicansdonotenterthetriangle-densestsubgraph.Acarefulinspectionofthedata,c.f.[ 2 ],indicatesthat6republicansagreewiththepartyvoteonatmost79%ofthebills,and8ofthemonatmost85%ofthebills.DBLPgraph.WeinputasaquerysetQasetofscientistswhohaveestablishedthemselvesintheoryandalgorithmde-sign:RichardKarp,ChristosPapadimitriou,MihalisYan-nakakisandSantoshVempala.ThealgorithmreturnsatitsoutputthequerysetandasetSof44verticescorrespondingtoacliqueof(mostly)Italiancomputerscientists.Welistasubsetofthe44verticeshere:M.Bencivenni,M.Cana-paro,F.Capannini,L.Carota,M.Carpene,R.Veraldi,P.Veronesi,M.Vistoli,R.Zappi.TheoutputgraphinducedbyS[Qisdisconnected.Therefore,thiscanbeeasilyex-plainedbecauseofthefollowing(folklore)inequality,giventhatjQjjSjinourexample.Claim1.Leta;b;c;dbenon-negative.Then,max c;b d
b cdmin c;b d
(3)Inourexample,wegett(S);cjSj;bt(Q);djQj.Insuchascenario,wheretheoutputconsistsoftheunionofadensesubgraphandthequerysetQ,analgorithmwhichbuildsitselfupfromQ-assumingQisnotanindependentset-toVbyaddingverticeswhichcreateasmanytrianglesaspossibleandreturningthemaximumdensitysubgraph,ratherthanpeelingverticesfromVdowntoQshouldbepreferredinpractice,seealso[ 49 ].7.CONCLUSIONInthisworkweintroducetheaveragetriangledensityasanovelobjectiveforattackingtheimportantproblemofndingnear-cliques.Weproposeexactandapproximationalgorithms.Furthermore,ourtechniquescansolvethemoregeneralproblemofmaximizingthe-cliquedensity.Exper-imentallyweverifythevalueoftheTDS-Problemasanoveladditiontothegraphminingtoolbox.Ourworkleavesnumerousproblemsopen,includingthefollowing:(a)Canweobtainafasterexactalgorithmbyimprovingthespaceusageofthenetworkconstruction?(b)Canweusesparsicationtoobtainfasterapproximatesolu-tions[ 44 ]?8.REFERENCES[1] http://tinyurl.com/bwgpka . 6 [2] http://tinyurl.com/zgdam . 6 [3]R.K.Ahuja,J.B.Orlin,C.Stein,andR.E.Tarjan.Improvedalgorithmsforbipartitenetwork\row.SIAMJournalonComputing,23(5):906{933,1994. 4.1.1 , 2 , 4.4 [4]N.Alon,R.Yuster,andU.Zwick.Findingandcountinggivenlengthcycles.Algorithmica,17(3):209{223,1997. 2 , 4.1.2 [5]R.AndersenandK.Chellapilla.Findingdensesubgraphswithsizebounds.InWAW,2009. 2 [6]A.Angel,N.Sarkas,N.Koudas,andD.Srivastava.Densesubgraphmaintenanceunderstreamingedgeweightupdatesforreal-timestoryidentication.Proc.VLDBEndow.,5(6):574{585,Feb.2012. 1 [7]Y.Asahiro,R.Hassin,andK.Iwama.Complexityofndingdensesubgraphs.Discr.Ap.Math.,121(1-3),2002. 2 [8]Y.Asahiro,K.Iwama,H.Tamaki,andT.Tokuyama.Greedilyndingadensesubgraph.J.Algorithms,34(2),2000. 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