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subgraphofGinducedbytherstnverticesf1;2;:::;ng.Thenthesequence!(Gn)appearstobealmostcompletelydeterminedfora.e.G2G(N;p). Foranaturall,letusdenotebykl(Gn)thenumberofcliquesspanninglverticesofGn.Then,obviously,!(Gn)=maxfl:kl(Gn)>0g: Usingthisobservationandthesecondmomentmethod,BollobasandErdos[ 32 ]provedthatifp=p(n)satisesn
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shownthatfora.e.G2G(N;p)ifnislargeenoughthenbl0(n)2loglogn=lognc!(Gn)bl0(n)+2loglogn=lognc 2: 56 ]andJansonetal.[ 73 ]extendedtheseresultsbyshowingthatfor>0thereexistsaconstantc,suchthatforc b2log1=pn2log1=plog1=pn+2log1=p(e=2)+1+=pc: 61 ]werethersttostudytheproblemofcoloringrandomgraphs.Manyotherresearcherscontributedtosolvingthisproblem[ 12 31 ].Wewillmentionsomefactsemergedfromthesestudies. Luczak[ 85 ]improvedtheresultsabouttheconcentrationof(G(n;p))previouslyprovedbyShamirandSpencer[ 110 ],provingthatforeverysequencep=p(n)suchthatpn6=7thereisafunctionch(n)suchthata:a:s:ch(n)(G(n;p))ch(n)+1: 12 ]provedthatforanypositiveconstantthechromaticnumberofauniformrandomgraphG(n;p),wherep=n1 2,isa.a.s.concentratedintwoconsecutivevalues.Moreover,theyprovedthataproperchoiceofp(n)mayresultinaonepointdistribution.Thefunctionch(n)isdiculttond,butinsomecasesitcanbecharacterized.Forexample,Jansonetal.[ 73 ]provedthatthereexistsaconstantc0suchthatforanyp=p(n)satisfyingc0
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30 ]yieldsthefollowingestimate:(G(n;p))=n 2.1.1 ).Ontheotherhand,probabilisticmethodssimilartothosediscussedinSubsection 2.1.4 couldbeutilizedinordertondtheasymptoticaldistributionofthecliquenumberinthesamenetwork'srandomgraphmodel,andthereforeverifythismodel.
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Oneofthemostimportantproblemsinthemodernnanceisndingecientwaysofsummarizingandvisualizingthestockmarketdatathatwouldallowonetoobtainusefulinformationaboutthebehaviorofthemarket.Nowadays,agreatnumberofstocksaretradedintheUSstockmarket;moreover,thisnumbersteadilyincreases.Theamountofdatageneratedbythestockmarketeverydayisenormous.Thisdataisusuallyvisualizedbythousandsofplotsreectingthepriceofeachstockoveracertainperiodoftime.Theanalysisoftheseplotsbecomesmoreandmorecomplicatedasthenumberofstocksgrows. Itturnsoutthatthestockmarketdatacanbeeectivelyrepresentedasanetwork,althoughthisrepresentationisnotsoobviousasinthecaseoftelephonetracorinternetdata.Wehavedevelopedthenetworkbasedmodelofthemarketreferredtoasthemarketgraph.Thischapterisbasedontheresultsdescribedin[ 26 27 28 ]. Anaturalgraphrepresentationofthestockmarketisbasedonthecrosscorrelationsofpriceuctuations.Amarketgraphcanbeconstructedasfollows:eachnancialinstrumentisrepresentedbyavertex,andtwoverticesareconnectedbyanedgeifthecorrelationcoecientofthecorrespondingpairofinstruments(calculatedforacertainperiodoftime)exceedsaspeciedthreshold;11. Nowadays,agreatnumberofdierentinstrumentsaretradedintheUSstockmarket,sothemarketgraphrepresentingthemisverylarge.Themarketgraphthatweconstructhas6546verticesandseveralmillionedges. 33
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Inthischapter,wepresentadetailedstudyofthepropertiesofthisgraph.Itturnsoutthatthemarketgraphcanberatheraccuratelydescribedbythepowerlawmodel.Weanalyzethedistributionofthedegreesoftheverticesinthisgraph,theedgedensityofthisgraphwithrespecttothecorrelationthreshold,aswellasitsconnectivityandthesizeofitsconnectedcomponents. Furthermore,welookformaximumcliquesandmaximumindependentsetsinthisgraphfordierentvaluesofthecorrelationthreshold.Analyzingcliquesandindependentsetsinthemarketgraphgivesusaveryvaluableknowledgeabouttheinternalstructureofthestockmarket.Forinstance,acliqueinthisgraphrepresentsasetofnancialinstrumentswhosepriceschangesimilarlyovertime(achangeofthepriceofanyinstrumentinacliqueislikelytoaectallotherinstrumentsinthisclique),andanindependentsetconsistsofinstrumentsthatarenegativelycorrelatedwithrespecttoeachother;therefore,itcanbetreatedasadiversiedportfolio.Basedontheinformationobtainedfromthisanalysis,wewillbeabletoclassifynancialinstrumentsintocertaingroups,whichwillgiveusadeeperinsightintothestockmarketstructure. 3.1.1ConstructingtheMarketGraph 92 ]:
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Figure3{1. Distributionofcorrelationcoecientsinthestockmarket whereRi(t)=lnPi(t) ThecorrelationcoecientsCijcanvaryfrom1to1.Figure 3{1 showsthedistributionofthecorrelationcoecientsbasedonthepricesdatafortheyears20002002.Itcanbeseenthatthisplothasashapesimilartothenormaldistributionwiththemean0.05. Themainideaofconstructingamarketgraphisasfollows.Letthesetofnancialinstrumentsrepresentthesetofverticesofthegraph.Also,wespecifyacertainthresholdvalue;11andaddanundirectededgeconnectingtheverticesiandjifthecorrespondingcorrelationcoecientCijisgreaterthanorequalto.Obviously,dierentvaluesofdenethemarketgraphswiththesamesetofvertices,butdierentsetsofedges. Itiseasytoseethatthenumberofedgesinthemarketgraphdecreasesasthethresholdvalueincreases.Infact,ourexperimentsshowthattheedgedensity
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Figure3{2. Edgedensityofthemarketgraphfordierentvaluesofthecorrelationthreshold. ofthemarketgraphdecreasesexponentiallyw.r.t..ThecorrespondinggraphispresentedonFigure 3{7 2.1.3 wementionedtheconnectivitythresholdsinrandomgraphs.Themainideaofthisconceptisndingathresholdvalueoftheparameterofthemodelthatwilldeneifthegraphisconnectedornot. Asimilarquestionarisesforthemarketgraph:whatisitsconnectivitythreshold?Sincethenumberofedgesinthemarketgraphdependsonthechosencorrelationthreshold,weshouldndavalue0thatdeterminestheconnectivityofthegraph.Asitwasmentionedabove,thesmallervalueofwechoose,themoreedgesthemarketgraphwillhave.So,ifwedecrease,afteracertainpoint,thegraphwillbecomeconnected.Wehaveconductedaseriesofcomputational
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Figure3{3. Plotofthesizeofthelargestconnectedcomponentinthemarketgraphasafunctionofcorrelationthreshold. experimentsforcheckingtheconnectivityofthemarketgraphusingthebreadthrstsearchtechnique,andweobtainedarelativelyaccurateapproximationoftheconnectivitythreshold:0'0:14382.Moreover,weinvestigatedthedependencyofthesizeofthelargestconnectedcomponentinthemarketgraphw.r.t..ThecorrespondingplotisshowninFigure 3{3 Itturnsoutthatifasmall(inabsolutevalue)correlationthresholdisspecied,thedistributionofthedegreesoftheverticesdoesnothaveanywelldenedstructure.Notethatforthesevaluesofthemarketgraphhasarelativelyhighedgedensity(i:e.theratioofthenumberofedgestothemaximumpossiblenumberofedges).However,asthecorrelationthresholdisincreased,thedegree
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Table3{1. Leastsquaresestimatesoftheparameterinthemarketgraphfordierentvaluesofcorrelationthreshold(complementarygraph) 0.2 0.15 0.2 0.4931 0.25 0.5820 0.3 0.6793 0.35 0.7679 0.4 0.8269 0.45 0.8753 0.5 0.9054 0.55 0.9331 0.6 0.9743 distributionmoreandmoreresemblesapowerlaw.Infact,for0:2thisdistributionisapproximatelyastraightlineinthelogarithmicscale,whichrepresentsthepowerlawdistribution,asitwasmentionedabove.Figure 3{4 demonstratesthedegreedistributionsofthemarketgraphforsomepositivevaluesofthecorrelationthreshold,alongwiththecorrespondinglinearapproximations.Theslopesoftheapproximatinglineswereestimatedusingtheleastsquaresmethod.Table 3{1 summarizestheestimatesoftheparameterofthepowerlawdistribution(i.e.,theslopeoftheline)fordierentvaluesof. Fromthistable,itcanbeseenthattheslopeofthelinescorrespondingtopositivevaluesofisrathersmall.Accordingtothepowerlawmodel,inthiscaseagraphwouldhavemanyverticeswithhighdegrees,therefore,onecanintuitivelyexpecttondlargecliquesinapowerlawgraphwithasmallvalueoftheparameter. Wealsoanalyzethedegreedistributionofthecomplementofthemarketgraph,whichisdenedasfollows:anedgeconnectsinstrumentsiandjifthecorrelationcoecientbetweenthemCij.Studyingthiscomplementarygraphisimportantforthenextsubjectofourconsiderationndingmaximumindependent
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Figure3{4. Degreedistributionofthemarketgraphfor=0:4(left);=0:5(right)(logarithmicscale) setsinthemarketgraphwithnegativevaluesofthecorrelationthreshold.Obviously,amaximumindependentsetintheinitialgraphisamaximumcliqueinthecomplement,sothemaximumindependentsetproblemcanbereducedtothemaximumcliqueprobleminthecomplementarygraph.Therefore,itisusefultoinvestigatethedegreedistributionsofthecomplementarygraphsfordierentvaluesof.AsitcanbeseenfromFigure 3{1 ,thedistributionofthecorrelationcoecientsisnearlysymmetricaround=0:05,soforthevaluesofcloseto0theedgedensityofboththeinitialandthecomplementarygraphishighenough.Forthesevaluesofthedegreedistributionofacomplementarygraphalsodoesnotseemtohaveanywelldenedstructure,asinthecaseofthecorrespondinginitialgraph.Asdecreases(i.e.,increasesintheabsolutevalue),thedegreedistributionofacomplementarygraphstartstofollowthepowerlaw.Figure 3{5 showsthedegreedistributionsofthecomplementarygraph,alongwiththeleastsquareslinearregressionlines.However,asonecanseefromTable 3{1 ,theslopesoftheselinesarehigherthaninthecaseofthegraphswithpositivevaluesof,whichimpliesthattherearefewerverticeswithahighdegreeinthesegraphs,sointuitively,thesizeofacliquesinacomplementarygraph(i.e.,thesize
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Figure3{5. Degreedistributionofthecomplementarymarketgraphfor=0:15(left);=0:2(right)(logarithmicscale) ofindependentsetsintheoriginalgraph)shouldbesignicantlysmallerthaninthecaseofthemarketgraphwithpositivevaluesofthecorrelationthreshold(seeSection 3.2 ). Forthispurpose,wechosethemarketgraphwithahighcorrelationthreshold(=0:6),calculatedthedegreesofeachvertexinthisgraphandsortedtheverticesinthedecreasingorderoftheirdegrees.
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Interestingly,eventhoughtheedgedensityoftheconsideredgraphisonly0.04%(onlyhighlycorrelatedinstrumentsareconnectedbyanedge),therearemanyverticeswithdegreesgreaterthan100. Accordingtoourcalculations,thevertexwiththehighestdegreeinthismarketgraphcorrespondstotheNASDAQ100IndexTrackingStock.Thedegreeofthisvertexis216,whichmeansthatthereare216instrumentsthatarehighlycorrelatedwithit.AninterestingobservationisthatthedegreeofthisvertexistwicehigherthanthenumberofcompanieswhosestockpricestheNASDAQindexreects,whichmeansthatthese100companiesgreatlyinuencethemarket. InTable 3{2 wepresentthe\top25"instrumentsintheU.S.stockmarket,accordingtotheirdegreesintheconsideredmarketgraph.Thecorrespondingsymbolsdenitionscanbefoundonseveralwebsites,forexamplehttp://www.nasdaq.com.Notethatmostofthemareindicesthatincorporateanumberofdierentstocksofthecompaniesindierentindustries.Althoughthisresultisnotsurprisingfromthenancialpointofview,itisimportantasapracticaljusticationofthemarketgraphmodel. 3{3 .Forinstance,asonecanseefromthistable,themarketgraphwith=0:6hasalmostthesameedgedensityasthecomplementarymarketgraphwith=0:15,however,theirclusteringcoecientsdierdramatically.Thisfactalsointuitivelyexplainsthe
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Table3{2. Top25instrumentswithhighestdegreesinthemarketgraph(=0:6) symbol vertexdegree QQQ 216IWF 193IWO 193IYW 193XLK 181IVV 175MDY 171SPY 162IJH 159IWV 158IVW 156IAH 155IYY 154IWB 153IYV 150BDH 144MKH 143IWM 142IJR 134SMH 130STM 118IIH 116IVE 113DIA 106IWD 106 resultspresentedinthenextsection,whichdealswithcliquesandindependentsetsinthemarketgraph. Themaximumcliqueproblem(aswellasthemaximumindependentsetproblem)isknowntobeNPhard[ 59 ].Moreover,itturnsoutthatthemaximumcliqueisdiculttoapproximate[ 18 62 ].Thismakestheseproblemsespeciallychallenginginlargegraphs.However,aswewillseeinthenextsubsection,even
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Table3{3. Clusteringcoecientsofthemarketgraph(complementarygraph) clusteringcoef. 0.15 2:64105 0.0012 0.3 0.0178 0.4885 0.4 0.0047 0.4458 0.5 0.0013 0.4522 0.6 0.0004 0.4872 0.7 0.0001 0.4886 thoughthemaximumcliqueproblemisgenerallyveryhardtosolveinlargegraphs,thespecialstructureofthemarketgraphallowsustondtheexactsolutionrelativelyeasily. Astandardintegerprogrammingformulation[ 33 ]wasusedtocomputetheexactmaximumcliqueinthemarketgraph,however,beforesolvingthisproblem,weappliedagreedyheuristicforndingalowerboundofthecliquenumber,andaspecialpreprocessingtechniquewhichreducestheproblemsize.Tondalargeclique,weapplythe\bestin"greedyalgorithmbasedondegreesofvertices.LetCdenotetheclique.StartingwithC=;,werecursivelyaddtothecliqueavertexvmaxoflargestdegreeandremoveallverticesthatarenotadjacenttovmaxfromthegraph.Afterrunningthisalgorithm,weappliedthefollowingpreprocessing
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procedure[ 2 ].WerecursivelyremovefromthegraphalloftheverticeswhicharenotinCandwhosedegreeislessthanjCj,whereCisthecliquefoundbythegreedyalgorithm. DenotebyG0=(V0;E0)thegraphinducedbyremainingvertices.ThenthemaximumcliqueproblemcanbeformulatedandsolvedforG0.Thefollowingintegerprogrammingformulationwasused[ 33 ]: maximizejV0jXi=1xis:t:xi+xj1;(i;j)=2E0xi2f0;1g 26 ].Thiscanbeintuitivelyexplainedbythefactthattheseinstancesofthemarketgraphareclustered(i.e.twoverticesinagrapharemorelikelytobeconnectediftheyhaveacommonneighbor),sotheclusteringcoecient,whichisdenedastheprobabilitythatforagivenvertexitstwoneighborsareconnectedbyanedge,ismuchhigherthantheedgedensityinthesegraphs(seeTable 3{8 ).Thischaracteristicisalsotypicalforotherpowerlawgraphsarisingindierentapplications. Afterreducingthesizeoftheoriginalgraph,theresultingintegerprogrammingproblemforndingamaximumcliquecanberelativelyeasilysolvedusingtheCPLEXintegerprogrammingsolver[ 71 ]. Table 3{4 summarizestheexactsizesofthemaximumcliquesfoundinthemarketgraphfordierentvaluesof.Itturnsoutthatthesecliquesare
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ratherlarge,whichagreeswiththeanalysisofdegreedistributionsandclusteringcoecientsinthemarketgraphswithpositivevaluesof. Table3{4. Sizesofthemaximumcliquesinthemarketgraphwithpositivevaluesofthecorrelationthreshold(exactsolutions) cliquesize 0.35 0.0090 193 0.4 0.0047 144 0.45 0.0024 109 0.5 0.0013 85 0.55 0.0007 63 0.6 0.0004 45 0.65 0.0002 27 0.7 0.0001 22 Theseresultsshowthatinthemodernstockmarkettherearelargegroupsofinstrumentswhosepriceuctuationsbehavesimilarlyovertime,whichisnotsurprising,sincenowadaysdierentbranchesofeconomyhighlyaecteachother. 3{5 presentsthesizesoftheindependentsetsfoundusingthegreedyheuristicthatwasdescribedintheprevioussection.
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Table3{5. Sizesofindependentsetsinthecomplementarymarketgraphfoundusingthegreedyalgorithm(lowerbounds) indep.setsize 0.05 0.4794 45 0.0 0.2001 12 0.05 0.0431 5 0.1 0.005 3 0.15 0.0005 2 Thistabledemonstratesthatthesizesofcomputedindependentsetsarerathersmall,whichisinagreementwiththeresultsoftheprevioussection,wherewementionedthatinthecomplementarygraphthevaluesoftheparameterofthepowerlawdistributionareratherhigh,andtheclusteringcoecientsareverysmall. Thesmallsizeofthecomputedindependentsetsmeansthatndingalarge\completelydiversied"portfolio(whereallinstrumentsarenegativelycorrelatedtoeachother)isnotaneasytaskinthemodernstockmarket. Moreover,itturnsoutthatonecanmakeatheoreticalestimationofthemaximumsizeofadiversiedportfolio,whereallstocksarestrictlynegativelycorrelatedwitheachother.Intuitively,thelower(higherbytheabsolutevalue)thresholdweset,thesmallerdiversiedportfolioonewouldexpecttond.Theseconsiderationsareconrmedbythefollowingtheorem. Proof.
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maximumcorrelationis=maxi;jCij<0.Considerthevarianceofthesumofthesevariables: Notethatif<0,m2max(1+(m1))<0form>1+1 Therefore,thenumberofstockswithpairwisecorrelationsCij<0cannotbegreaterthanm=1+1 Anothernaturalquestionnowarises:howmanycompletelydiversiedportfolioscanbefoundinthemarket?Inordertondananswer,wehavecalculatedmaximalindependentsetsstartingfromeachvertex,byrunning6546iterationsofthegreedyalgorithmmentionedabove.Thatis,foreachoftheconsidered6546nancialinstruments,wehavefoundacompletelydiversiedportfoliothatwouldcontainthisinstrument.Interestinglyenough,foreveryvertexinthemarketgraph,wewereabletodetectanindependentsetthatcontainsthisvertex,andthesizesoftheseindependentsetswereratherclose.Moreover,alltheseindependentsetsweredistinct.Figure 3{6 showsthefrequencyofthesizesoftheindependentsetsfoundinthemarketgraphscorrespondingtodierentcorrelationthresholds. Theseresultsdemonstratethatitisalwayspossibleforaninvestortondagroupofstocksthatwouldformacompletelydiversiedportfoliowithanygivenstock,andthiscanbeecientlydoneusingthetechniqueofndingindependentsetsinthemarketgraph.
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Figure3{6. Frequencyofthesizesofindependentsetsfoundinthemarketgraphwith=0:00(left),and=0:05(right) Nontrivialinformationabouttheglobalpropertiesofthestockmarketisobtainedfromtheanalysisofthedegreedistributionofthemarketgraph.Highlyspecicstructureofthisdistributionsuggeststhatthestockmarketcanbeanalyzedusingthepowerlawmodel,whichcantheoreticallypredictsomecharacteristicsofthegraphrepresentingthemarket. Ontheotherhand,theanalysisofcliquesandindependentsetsinthemarketgraphisalsousefulfromthedataminingpointofview.Asitwaspointedoutabove,cliquesandindependentsetsinthemarketgraphrepresentgroupsof\similar"and\dierent"nancialinstruments,respectively.Therefore,informationaboutthesizeofthemaximumcliquesandindependentsetsisalsoratherimportant,sinceitgivesonetheideaaboutthetrendsthattakeplaceinthestock
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market.Besidesanalyzingthemaximumcliquesandindependentsetsinthemarketgraph,onecanalsodividethemarketgraphintothesmallestpossiblesetofdistinctcliques(orindependentsets).Partitioningadatasetintosets(clusters)ofelementsgroupedaccordingtoacertaincriterionisreferredtoasclustering,whichisoneofthewellknowndataminingproblems[ 34 ]. Asdiscussedabove,themaindicultyoneencountersinsolvingtheclusteringproblemonacertaindatasetisthefactthatthenumberofdesiredclustersofsimilarobjectsisusuallynotknownapriori,moreover,anappropriatesimilaritycriterionshouldbechosenbeforepartitioningadatasetintoclusters. Clearly,themethodologyofndingcliquesinthemarketgraphprovidesanecienttoolofperformingclusteringbasedonthestockmarketdata.Thechoiceofthegroupingcriterionisclearandnatural:\similar"nancialinstrumentsaredeterminedaccordingtothecorrelationbetweentheirpriceuctuations.Moreover,theminimumnumberofclustersinthepartitionofthesetofnancialinstrumentsisequaltotheminimumnumberofdistinctcliquesthatthemarketgraphcanbedividedinto(theminimumcliquepartitionproblem).Similarpartitioncanbedoneusingindependentsetsinsteadofcliques,whichwouldrepresentthepartitionofthemarketintoasetofdistinctdiversiedportfolios.Inthiscasetheminimumpossiblenumberofclustersisequaltoapartitionofverticesintoaminimumnumberofdistinctindependentsets.Thisproblemiscalledthegraphcoloringproblem,andthenumberofsetsintheoptimalpartitionisreferredtoasthechromaticnumberofthegraph. Weshouldalsomentionanothermajortypeofdataminingproblemswithmanyapplicationsinnance.Theyarereferredtoasclassicationproblems.Althoughthesetupofthistypeofproblemsissimilartoclustering,oneshouldclearlyunderstandthedierencebetweenthesetwotypesofproblems.
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Inclassication,onedealswithapredenednumberofclassesthatthedataelementsmustbeassignedto.Also,thereisasocalledtrainingdataset,i.e.,thesetofdataelementsforwhichitisknownaprioriwhichclasstheybelongto.Itmeansthatinthissetuponeusessomeinitialinformationabouttheclassicationofexistingdataelements.Acertainclassicationmodelisconstructedbasedonthisinformation,andtheparametersofthismodelare\tuned"toclassifynewdataelements.Thisprocedureisknownas\trainingtheclassier".Anexampleoftheapplicationofthisapproachtoclassifyingnancialinstrumentscanbefoundin[ 40 ]. Themaindierencebetweenclassicationandclusteringisthefactthatunlikeclassication,inthecaseofclustering,onedoesnotuseanyinitialinformationabouttheclassattributesoftheexistingdataelements,buttriestodetermineaclassicationusingappropriatecriteria.Therefore,themethodologyofclassifyingnancialinstrumentsusingthemarketgraphmodelisessentiallydierentfromtheapproachescommonlyconsideredintheliteratureinthesensethatitdoesnotrequireanyaprioriinformationabouttheclassesthatcertainstocksbelongto,butclassiesthemonlybasedonthebehavioroftheirpricesovertime. Inordertoinvestigatethedynamicsofthemarketgraphstructure,wechosetheperiodof1000tradingdaysin1998{2002andconsideredeleven500dayshiftswithinthisperiod.Thestartingpointsofeverytwoconsecutiveshiftsareseparated
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Table3{6. Datesandmeancorrelationscorrespondingtoeachconsidered500dayshift Period#StartingdateEndingdateMeancorrelation 109/24/199809/15/20000.0403212/04/199811/27/20000.0373302/18/199902/08/20010.0381404/30/199904/23/20010.0426507/13/199907/03/20010.0444609/22/199909/19/20010.0465712/02/199911/29/20010.0545802/14/200002/12/20020.0561904/26/200004/25/20020.05281007/07/200007/08/20020.05701109/18/200009/17/20020.0672 bytheintervalof50days.Therefore,everypairofconsecutiveshiftshad450daysincommonand50daysdierent.DatescorrespondingtoeachshiftandthecorrespondingmeancorrelationsaresummarizedinTable 3{6 Thisprocedureallowsustoaccuratelyreectthestructuralchangesofthemarketgraphusingrelativelysmallintervalsbetweenshifts,butatthesametimeonecanmaintainsucientlylargesamplesizesofthestockpricesdataforcalculatingcrosscorrelationsforeachshift.Weshouldnotethatinouranalysisweconsideredonlystockswhichwereamongthosetradedasofthelastofthe1000tradingdays,i.e.forpracticalreasonswedidnottakeintoaccountstockswhichhadbeenwithdrawnfromthemarket.
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Therstsubjectofourconsiderationisthedistributionofcorrelationcoecientsbetweenallpairsofstocksinthemarket.Asitwasmentionedabove,thisdistributionon[1;1]hadashapesimilartoapartofnormaldistributionwithmeancloseto0.05forthesampledataconsideredin[ 26 27 ].Oneoftheinterpretationsofthisfactisthatthecorrelationofmostpairsofstocksisclosetozero,therefore,thestructureofthestockmarketissubstantiallyrandom,andonecanmakeareasonableassumptionthatthepricesofmoststockschangeindependently.Asweconsidertheevolutionofthecorrelationdistributionovertime,itturnsoutthattheshapeofthisdistributionremainsstable,whichisillustratedbyFigure 3{7 Figure3{7. DistributionofcorrelationcoecientsintheUSstockmarketforseveraloverlapping500dayperiodsduring20002002(period1istheearliest,period11isthelatest). Thestabilityofthecorrelationcoecientsdistributionofthemarketgraphintuitivelymotivatesthehypothesisthatthedegreedistributionshouldalsoremainstablefordierentvaluesofthecorrelationthreshold.Toverifythisassumption,
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wehavecalculatedthedegreedistributionofthegraphsconstructedforallconsideredtimeperiods.Thecorrelationthreshold=0:5waschosentodescribethestructureofconnectionscorrespondingtosignicantlyhighcorrelations.Ourexperimentsshowthatthedegreedistributionissimilarforalltimeintervals,andinallcasesitiswelldescribedbyapowerlaw.Figure 3{8 showsthedegreedistributions(inthelogarithmicscale)forsomeinstancesofthemarketgraph(with=0:5)correspondingtodierentintervals. (a)period1 (b)period4 (c)period7 (d)period11 Figure3{8. Degreedistributionofthemarketgraphfordierent500dayperiodsin20002002with=0:5:(a)period1,(b)period4,(c)period7,(d)period11. Thecrosscorrelationdistributionandthedegreedistributionofthemarketgraphrepresentthegeneralcharacteristicsofthemarket,andtheaforementioned
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resultsleadustotheconclusionthattheglobalstructureofthemarketisstableovertime.However,aswewillseenow,someglobalchangesinthestockmarketstructuredotakeplace.Inordertodemonstrateit,welookatanothercharacteristicofthemarketgraph{itsedgedensity. Inouranalysisofthemarketgraphdynamics,wechosearelativelyhighcorrelationthreshold=0:5thatwouldensurethatweconsideronlytheedgescorrespondingtothepairsofstocks,whicharesignicantlycorrelatedwitheachother.Inthiscase,theedgedensityofthemarketgraphwouldrepresenttheproportionofthosepairsofstocksinthemarket,whosepriceuctuationsaresimilarandinuenceeachother.Thesubjectofourinterestistostudyhowthisproportionchangesduringtheconsideredperiodoftime.Table 3{7 summarizestheobtainedresults.Asitcanbeseenfromthistable,boththenumberofverticesandthenumberofedgesinthemarketgraphincreaseastimegoes.Obviously,thenumberofverticesgrowssincenewstocksappearinthemarket,andwedonotconsiderthosestockswhichceasedtoexistbythelastof1000tradingdaysusedinouranalysis,sothemaximumpossiblenumberofedgesinthegraphincreasesaswell.However,itturnsoutthatthenumberofedgesgrowsfaster;therefore,theedgedensityofthemarketgraphincreasesfromperiodtoperiod.AsonecanseefromFigure 3{9(a) ,thegreatestincreaseoftheedgedensitycorrespondstothelasttwoperiods.Infact,theedgedensityforthelatestintervalisapproximately8.5timeshigherthanfortherstinterval!Thisdramaticjumpsuggeststhatthereisatrendtothe\globalization"ofthemodernstockmarket,whichmeansthatnowadaysmoreandmorestockssignicantlyaectthebehavioroftheothers. Itshouldbenotedthattheincreaseoftheedgedensitycouldbepredictedfromtheanalysisofthedistributionofthecrosscorrelationsbetweenallpairsofstocks.FromFigure 3{7 ,onecanobservethateventhoughthedistributionscorrespondingtodierentperiodshaveasimilarshapeandthesamemean,
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Table3{7. Numberofverticesandnumberofedgesinthemarketgraphfordierentperiods(=0:5) PeriodNumberofVerticesNumberofEdgesEdgedensity 1543022580.015%2550726140.017%3559337720.024%4566652760.033%5576868410.041%6586677700.045%76013104280.058%86104124570.067%96262129110.066%106399197070.096%116556278850.130% the\tail"ofthedistributioncorrespondingtothelatestperiod(period11)issomewhat\heavier"thanfortheearlierperiods,whichmeansthattherearemorepairsofstockswithhighervaluesofthecorrelationcoecient. (b) Dynamicsofedgedensityandmaximumcliquesizeinthemarketgraph:Evolutionoftheedgedensity(a)andmaximumcliquesize(b)inthemarketgraph(=0:5)
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Table 3{8 presentsthesizesofthemaximumcliquesfoundinthemarketgraphfordierenttimeperiods.Asintheprevioussubsection,weusedarelativelyhighcorrelationthreshold=0:5toconsideronlysignicantlycorrelatedstocks.Asonecansee,thereisacleartrendoftheincreaseofthemaximumcliquesizeovertime,whichisconsistentwiththebehavioroftheedgedensityofthemarketgraphdiscussedabove(seeFigure 3{9(b) ).Thisresultprovidesanotherconrmationoftheglobalizationhypothesisdiscussedabove. Anotherrelatedissuetoconsiderishowmuchthestructureofmaximumcliquesisdierentforthevarioustimeperiods.Table 3{9 presentsthestocksincludedintothemaximumcliquesfordierenttimeperiods.Itturnsoutthatinmostcasesstocksthatappearinacliqueinanearlierperiodalsoappearinthecliquesinlaterperiods. Therearesomeotherinterestingobservationsaboutthestructureofthemaximumcliquesfoundfordierenttimeperiods.Itcanbeseenthatallthecliquesincludeasignicantnumberofstocksofthecompaniesrepresentingthe\hightech"industrysector.Astheexamples,onecanmentionwellknowncompaniessuchasSunMicrosystems,Inc.,CiscoSystems,Inc.,IntelCorporation,etc.Moreover,eachcliquecontainsstocksofthecompaniesrelatedtothesemiconductorindustry(e.g.,CypressSemiconductorCorporation,Cree,Inc.,LatticeSemiconductorCorporation,etc.),andthenumberofthesestocksinthecliquesincreaseswiththetime.Thesefactssuggestthatthecorrespondingbranchesofindustryexpandedduringtheconsideredperiodoftimetoformamajorclusterofthemarket. Inaddition,weobservedthatinthelaterperiods(especiallyinthelasttwoperiods)themaximumcliquescontainaratherlargenumberofexchangetradedfunds,i.e.,stocksthatreectthebehaviorofcertainindicesrepresentingvariousgroupsofcompanies.Itshouldbementionedthatallmaximumcliquescontain
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Table3{8. Greedycliquesizeandthecliquenumberfordierenttimeperiods(=0:5) PeriodjVjEdgeDens.ClusteringjCjjV0jEdgeDens.CliqueinGCoecientinG0Number 154300:000150.50515760.28618255070:000170.50418430.73119355930:000240.49926490.81727456660:000330.51734700.77434557680:000410.55042820.78742658660:000450.55845860.80445760130:000580.553511100.76951861040:000670.566601140.81960962620:000660.553621070.869621063990:000960.486771340.841771165560:001300.452841460.84485 Nasdaq100trackingstock(QQQ),whichwasalsofoundtobethevertexwiththehighestdegree(i.e.,correlatedwiththemoststocks)inthemarketgraph[ 26 ]. Anothernaturalquestionthatonecanposeishowthesizeofindependentsets(i.e.,diversiedportfoliosinthemarket)changesovertime.Asitwaspointedoutin[ 26 27 ],ndingamaximumindependentsetinthemarketgraphturnsouttobeamuchmorecomplicatedtaskthanndingamaximumclique.Inparticular,inthecaseofsolvingthemaximumindependentsetproblem(or,equivalently,themaximumcliqueprobleminthecomplementarygraph),thepreprocessingproceduredescribedabovedoesnotreducethesizeoftheoriginalgraph.Thiscanbeexplainedbythefactthattheclusteringcoecientinthecomplementarymarketgraphwith=0ismuchsmallerthanintheoriginalgraphcorrespondingto=0:5(seeTable 3{10 ). SimilarlytoSection 3.2 ,wecalculatemaximalindependentsets(amaximalindependentsetisanindependentsetthatisnotasubsetofanotherindependentset)inthemarketgraphusingtheabovegreedyalgorithm.AsonecanseefromTable 3{10 ,thesizesofindependentsetsfoundinthemarketgraphfor=0arerathersmall,whichisconsistentwiththeresultsofSection 3.2
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Table3{9. Structureofmaximumcliquesinthemarketgraphfordierenttimeperiods(=0:5) Stocksincludedintomaximumclique BK,EMC,FBF,HAL,HP,INTC,NCC,NOI,NOK,PDS,PMCS,QQQ,RF,SII,SLB,SPY,TER,WM 2 ADI,ALTR,AMAT,AMCC,ATML,CSCO,KLAC,LLTC,LSCC,MDY,MXIM,NVLS,PMCS,QQQ,SPY,SUNW,TXN,VTSS,XLNX 3 AMAT,AMCC,CREE,CSCO,EMC,JDSU,KLAC,LLTC,LSCC,MDY,MXIM,NVLS,PHG,PMCS,QLGC,QQQ,SEBL,SPY,STM,SUNW,TQNT,TXCC,TXN,VRTS,VTSS,XLK,XLNX 4 AMAT,AMCC,ASML,ATML,BRCM,CHKP,CIEN,CREE,CSCO,EMC,FLEX,JDSU,KLAC,LSCC,MDY,MXIM,NTAP,NVLS,PMCS,QLGC,QQQ,RFMD,SEBL,SPY,STM,SUNW,TQNT,TXCC,TXN,VRSN,VRTS,VTSS,XLK,XLNX 5 ALTR,AMAT,AMCC,ASML,ATML,BRCM,CIEN,CREE,CSCO,EMC,FLEX,IDTI,IRF,JDSU,JNPR,KLAC,LLTC,LRCX,LSCC,LSI,MDY,MXIM,NTAP,NVLS,PHG,PMCS,QLGC,QQQ,RFMD,SEBL,SPY,STM,SUNW,SWKS,TQNT,TXCC,TXN,VRSN,VRTS,VTSS,XLK,XLNX 6 ADI,ALTR,AMAT,AMCC,ASML,ATML,BEAS,BRCM,CIEN,CREE,CSCO,CY,ELX,EMC,FLEX,IDTI,ITWO,JDSU,JNPR,KLAC,LLTC,LRCX,LSCC,LSI,MDY,MXIM,NTAP,NVLS,PHG,PMCS,QLGC,QQQ,RFMD,SEBL,SPY,STM,SUNW,TQNT,TXCC,TXN,VRSN,VRTS,VTSS,XLK,XLNX 7 ALTR,AMAT,AMCC,ATML,BEAS,BRCD,BRCM,CHKP,CIEN,CNXT,CREE,CSCO,CY,DIGL,EMC,FLEX,HHH,ITWO,JDSU,JNPR,KLAC,LLTC,LRCX,LSCC,MDY,MERQ,MXIM,NEWP,NTAP,NVLS,ORCL,PMCS,QLGC,QQQ,RBAK,RFMD,SCMR,SEBL,SPY,SSTI,STM,SUNW,SWKS,TQNT,TXCC,TXN,VRSN,VRTS,VTSS,XLK,XLNX 8 ALTR,AMAT,AMCC,AMKR,ARMHY,ASML,ATML,AVNX,BEAS,BRCD,BRCM,CHKP,CIEN,CMRC,CNXT,CREE,CSCO,CY,DIGL,ELX,EMC,EXTR,FLEX,HHH,IDTI,ITWO,JDSU,JNPR,KLAC,LLTC,LRCX,LSCC,MDY,MERQ,MRVC,MXIM,NEWP,NTAP,NVLS,ORCL,PMCS,QLGC,QQQ,RFMD,SCMR,SEBL,SNDK,SPY,SSTI,STM,SUNW,SWKS,TQNT,TXCC,TXN,VRSN,VRTS,VTSS,XLK,XLNX 9 ADI,ALTR,AMAT,AMCC,ARMHY,ASML,ATML,AVNX,BDH,BEAS,BHH,BRCM,CHKP,CIEN,CLS,CREE,CSCO,CY,DELL,ELX,EMC,EXTR,FLEX,HHH,IAH,IDTI,IIH,INTC,IRF,JDSU,JNPR,KLAC,LLTC,LRCX,LSCC,LSI,MDY,MXIM,NEWP,NTAP,NVLS,PHG,PMCS,QLGC,QQQ,RFMD,SCMR,SEBL,SNDK,SPY,SSTI,STM,SUNW,SWKS,TQNT,TXCC,TXN,VRSN,VRTS,VTSS,XLK,XLNX 10 ADI,ALTR,AMAT,AMCC,AMD,ASML,ATML,BDH,BHH,BRCM,CIEN,CLS,CREE,CSCO,CY,CYMI,DELL,EMC,FCS,FLEX,HHH,IAH,IDTI,IFX,IIH,IJH,IJR,INTC,IRF,IVV,IVW,IWB,IWF,IWM,IWV,IYV,IYW,IYY,JBL,JDSU,KLAC,KOPN,LLTC,LRCX,LSCC,LSI,LTXX,MCHP,MDY,MXIM,NEWP,NTAP,NVDA,NVLS,PHG,PMCS,QLGC,QQQ,RFMD,SANM,SEBL,SMH,SMTC,SNDK,SPY,SSTI,STM,SUNW,TER,TQNT,TXCC,TXN,VRTS,VSH,VTSS,XLK,XLNX 11 ADI,ALA,ALTR,AMAT,AMCC,AMD,ASML,ATML,BDH,BEAS,BHH,BRCM,CIEN,CLS,CNXT,CREE,CSCO,CY,CYMI,DELL,EMC,EXTR,FCS,FLEX,HHH,IAH,IDTI,IIH,IJH,IJR,INTC,IRF,IVV,IVW,IWB,IWF,IWM,IWO,IWV,IWZ,IYV,IYW,IYY,JBL,JDSU,JNPR,KLAC,KOPN,LLTC,LRCX,LSCC,LSI,LTXX,MCRL,MDY,MKH,MRVC,MXIM,NEWP,NTAP,NVDA,NVLS,PHG,PMCS,QLGC,QQQ,RFMD,SANM,SEBL,SMH,SMTC,SNDK,SPY,SSTI,STM,SUNW,TER,TQNT,TXN,VRTS,VSH,VTSS,XLK,XLNX
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Table3{10. Sizeofindependentsetsinthemarketgraphfoundusingthegreedyheuristic(=0:0).Edgedensityandclusteringcoecientaregivenforthecomplementarygraph. PeriodNumberofEdgeClusteringIndependentverticesdensitycoecientsetsize 154300.2580.29311255070.2750.30711355930.2810.30710456660.2650.29711557680.2600.29211658660.2540.28811760130.2280.26911861040.2270.26810962620.2380.277121063990.2280.269121165560.2010.24511 Forndingacliquepartition,wechoosetheinstanceofthemarketgraphwithalowcorrelationthreshold=0:05(themeanofthecorrelationcoecientsdistributionshowninFigure 3{7 ),whichwouldensurethattheedgedensityoftheconsideredgraphishighenoughandthenumberofisolatedvertices(whichwouldobviouslyformdistinctcliques)issmall. Weusethestandardgreedyheuristictocomputeacliquepartitioninthemarketgraph:recursivelyndamaximalcliqueandremoveitfromthegraph,untilnovertexremain.Cliquesarecomputedusingthepreviouslydescribedgreedyalgorithm.Thecorrespondingresultsforthemarketgraphwiththreshold=0:05arepresentedinTable 3{11 .Notethatthesizeofthelargestcliqueinthepartitionisincreasingfromoneperiodtoanother,withthelargestcliqueinthelastperiod
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Table3{11. Thelargestcliquesizeandthenumberofcliquesincomputedcliquepartitions(=0:05) PeriodNumberofEdgeLargestclique#ofcliquesinverticesdensityinthepartitionthepartition 154300.400469494255070.377552517355930.379636513456660.405743503557680.413789501658660.425824496760130.469929471861040.475983470962620.4569975091063990.47411595011165560.5211372479 containingaboutthreetimesasmanyverticesasthecorrespondingcliqueintherstpartition.Atthesametime,thenumberofcliquesinthepartitioniscomparablefordierentperiods,withaslightoveralltrendtowardsdecrease,whereasthenumberofverticesisincreasingastimegoes. Anotherimportantresultisthefactthattheedgedensityofthemarketgraph,aswellasthemaximumcliquesize,steadilyincreaseduringthelastseveralyears,whichsupportsthewellknownideaabouttheglobalizationofeconomywhichhasbeenwidelydiscussedrecently.
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Wehavealsoindicatedthenaturalwayofdividingthesetofnancialinstrumentsintogroupsofsimilarobjects(clustering)bycomputingacliquepartitionofthemarketgraph.Thismethodologycanbeextendedbyconsideringquasicliquesinthepartition,whichmayreducethenumberofobtainedclusters.Moreover,ndingindependentsetsinthemarketgraphprovidesanewapproachtochoosingdiversiedportfolioswhereallstocksarepairwiseuncorrelated,whichispotentiallyusefulinpractice.
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Humanbrainisoneofthemostcomplexsystemseverstudiedbyscientists.Enormousnumberofneuronsandthedynamicnatureofconnectionsbetweenthemmakestheanalysisofbrainfunctionespeciallychallenging.Oneofthemostimportantdirectionsinstudyingthebrainistreatingdisordersofthecentralnervoussystem.Forinstance,epilepsyisacommonformofsuchdisorders,whichaectsapproximately1%ofthehumanpopulation.Essentially,epilepticseizuresrepresentexcessiveandhypersynchronousactivityoftheneuronsinthecerebralcortex. Duringthelastseveralyears,signicantprogressintheeldofepilepticseizurespredictionhasbeenmade.Theadvancesareassociatedwiththeextensiveuseofelectroencephalograms(EEG)whichcanbetreatedasaquantitativerepresentationofthebrainfunction.RapiddevelopmentofcomputationalequipmenthasmadepossibletostoreandprocesshugeamountsofEEGdataobtainedfromrecordingdevices.TheavailabilityofthesemassivedatasetsgivesarisetoanotherproblemutilizingmathematicaltoolsanddataminingtechniquesforextractingusefulinformationfromEEGdata.Isitpossibletoconstructa\simple"mathematicalmodelbasedonEEGdatathatwouldreectthebehavioroftheepilepticbrain? Inthischapter,wemakeanattempttocreatesuchamodelusinganetworkbasedapproach. InthecaseofthehumanbrainandEEGdata,weapplyarelativelysimplenetworkbasedapproach.WerepresenttheelectrodesusedforobtainingtheEEG 62
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readings,whicharelocatedindierentpartsofthebrain,astheverticesoftheconstructedgraph.ThedatareceivedfromeverysingleelectrodeisessentiallyatimeseriesreectingthechangeoftheEEGsignalovertime.LaterinthechapterwewilldiscussthequantitativemeasurecharacterizingstatisticalrelationshipsbetweentherecordingsofeverypairofelectrodessocalledTindex.ThevaluesoftheTindexTijmeasuredforallpairsofelectrodesiandjenableustoestablishcertainrulesofplacingedgesconnectingdierentpairsofverticesiandjdependingonthecorrespondingvaluesofTij.Usingthistechnique,wedevelopseveralgraphbasedmathematicalmodelsandstudythedynamicsofthestructuralpropertiesofthesegraphs.Aswewillsee,thesemodelscanprovideusefulinformationaboutthebehaviorofthebrainpriorto,during,andafteranepilepticseizure. 4.1.1Datasets. 4{1 67 69 101 ]). Sincethebrainisanonstationarysystem,algorithmsusedtoestimatemeasuresofthebraindynamicsshouldbecapableofautomaticallyidentifyingandappropriatelyweighingexistingtransientsinthedata.Inachaoticsystem,orbitsoriginatingfromsimilarinitialconditions(nearbypointsinthestatespace)divergeexponentially(expansionprocess).TherateofdivergenceisanimportantaspectofthesystemdynamicsandisreectedinthevalueofLyapunovexponents.The
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Electrodeplacementinthebrain:(A)Inferiortransverseand(B)lateralviewsofthebrain,illustratingapproximatedepthandsubduralelectrodeplacementforEEGrecordingsaredepicted.Subduralelectrodestripsareplacedovertheleftorbitofrontal(AL),rightorbitofrontal(AR),leftsubtemporal(BL),andrightsubtemporal(BR)cortex.Depthelectrodesareplacedinthelefttemporaldepth(CL)andrighttemporaldepth(CR)torecordhippocampalactivity.
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methodusedforestimationoftheshorttimelargestLyapunovexponentSTLmax,anestimateofLmaxfornonstationarydata,isexplainedindetailin[ 66 68 118 ]. BysplittingtheEEGtimeseriesrecordedfromeachelectrodeintoasequenceofnonoverlappingsegments,each10.24secinduration,andestimatingSTLmaxforeachofthesesegments,prolesofSTLmaxovertimearegenerated. HavingestimatedtheSTLmaxtemporalprolesatanindividualcorticalsite,andasthebrainproceedstowardstheictalstate,thetemporalevolutionofthestabilityofeachcorticalsiteisquantied.ThespatialdynamicsofthistransitionarecapturedbyconsiderationoftherelationsoftheSTLmaxbetweendierentcorticalsites.Forexample,ifasimilartransitionoccursatdierentcorticalsites,theSTLmaxoftheinvolvedsitesareexpectedtoconvergetosimilarvaluespriortothetransition.Suchparticipatingsitesarecalled\criticalsites",andsuchaconvergence\dynamicalentrainment".Morespecically,inorderforthedynamicalentrainmenttohaveastatisticalcontent,weallowaperiodoverwhichthedierenceofthemeansoftheSTLmaxvaluesattwositesisestimated.Weuseperiodsof10minutes(i.e.movingwindowsincludingapproximately60STLmaxvaluesovertimeateachelectrodesite)totestthedynamicalentrainmentatthe0.01statisticalsignicancelevel.WeemploytheTindex(fromthewellknownpairedTstatisticsforcomparisonsofmeans)asameasureofdistancebetweenthemeanvaluesofpairsofSTLmaxprolesovertime.TheTindexattimetbetweenelectrodesitesiandjisdenedas: whereEfgisthesampleaveragedierencefortheSTLmax;iSTLmax;jestimatedoveramovingwindowwt()denedas:wt()=8><>:1if2[tN1;t]0if62[tN1;t];
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whereNisthelengthofthemovingwindow.Then,i;j(t)isthesamplestandarddeviationoftheSTLmaxdierencesbetweenelectrodesitesiandjwithinthemovingwindowwt().TheTindexfollowsatdistributionwithN1degreesoffreedom.FortheestimationoftheTi;j(t)indicesinourdataweusedN=60(i.e.,averageof60dierencesofSTLmaxexponentsbetweensitesiandjpermovingwindowofapproximately10minuteduration).Therefore,atwosidedttestwithN1(=59)degreesoffreedom,atastatisticalsignicancelevelshouldbeusedtotestthenullhypothesis,Ho:\brainsitesiandjacquireidenticalSTLmaxvaluesattimet".Inthisexperiment,wesettheprobabilityofatypeIerror=0:01(i.e.,theprobabilityoffalselyrejectingHoifHoistrue,is1%).FortheTindextopassthistest,theTi;j(t)valueshouldbewithintheinterval[0,2.662].WewillrefertotheupperboundofthisintervalasTcritical. 4.2.1KeyIdeaoftheModel
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70 108 111 ],whichisessentiallythedivergenceoftheprolesoftheSTLmaxtimeseries.Asitwasindicatedabove,thisdivergenceischaracterizedbythevaluesofTindexgreaterthanTcritical. 4{2
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Figure4{2. NumberofedgesinGRAPHII
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ThesizeofthelargestconnectedcomponentoftheGRAPHIIispresentedinFigure 4{3 .OnecanseethatGRAPHIIisconnectedduringtheinterictalperiod(i.e.,thebrainisaconnectedsystem),however,itbecomesdisconnectedaftertheseizure(duringtheposticalstate):thesizeofthelargestconnectedcomponentsignicantlydecreases.Thisfactisnotsurprisingandcanbeintuitivelyexplained,sinceaftertheseizurethebrainneedssometimeto\reset"[ 70 108 111 ]andrestoretheconnectionsbetweenthefunctionalunits.
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Figure4{3. ThesizeofthelargestconnectedcomponentinGRAPHII.Numberofnodesinthegraphis30. hypothesisispartiallysupportedbythebehavioroftheaverageTindexoftheedgescorrespondingtotheMinimumSpanningTreeofGRAPHI,whichisshowninFigure 4{4 However,thishypothesiscannotbeveriedusingtheconsidereddata,sincethevaluesofaverageTindicesarecalculatedovera10minuteinterval,whereasthetheseizuresignalpropagatesinafractionofasecond.Therefore,inordertocheckiftheseizuresignalactuallyspreadsalongtheminimumspanningtree,oneneedstointroduceothernonlinearmeasurestoreectthebehaviorofthebrainovershorttimeintervals.
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Figure4{4. AveragevalueofTindexoftheedgesinMinimumSpanningTreeofGRAPHI. Also,notethattheaveragevalueoftheTindexintheMinimumSpanningTreeislessthanTcritical,whichalsosupportstheabovestatementabouttheconnectivityofthesystem. WelookatthebehavioroftheaveragedegreeoftheverticesinGRAPHIIovertime.Clearly,thisplotisverysimilartothebehavioroftheedgedensityofGRAPHII(seeFigure 4{5 ).
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Figure4{5. AveragedegreeoftheverticesinGRAPHII. Wearealsoparticularlyinterestedinhighdegreevertices,i.e.,thefunctionalunitsofthebrainthatareatacertaintimemomentconnected(entrained)withmanyotherbrainsites.Interestinglyenough,thevertexwithamaximumdegreeinGRAPHIIusuallycorrespondstotheelectrodewhichislocatedinRTD(righttemporaldepth)orRST(rightsubtemporalcortex),inotherwords,thevertexwiththemaximumdegreeislocatedneartheepileptogenicfocus. 69 ].Infact,thisapproachutilizesthesamepreprocessingtechnique(i.e.,calculatingthevaluesofTindicesforallpairsofelectrodesites)asweapplyinthischapter.Inthis
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subsection,wewillbrieydescribethisquadraticprogrammingtechniqueandrelateittothegraphmodelsintroducedabove. Themainideaoftheconsideredquadraticprogrammingapproachistoconstructamodelthatwouldselectacertainnumberofsocalled\critical"electrodesites,i.e.,thosethatarethemostentrainedduringtheseizure.AccordingtoSection3,suchgroupofelectrodesitesshouldproduceaminimalsumofTindicescalculatedforallpairsofelectrodeswithinthisgroup.Ifthenumberofcriticalsitesissetequaltok,andthetotalnumberofelectrodesitesisn,thentheproblemofselectingtheoptimalgroupofsitescanbeformulatedasthefollowingquadratic01problem[ 69 ]: minxTAx s.t.Pni=1xi=k: Inthissetup,thevectorx=(x1;x2;:::;xn)consistsofthecomponentsequaltoeither1(ifthecorrespondingsiteisincludedintothegroupofcriticalsites)or0(otherwise),andtheelementsofthematrixA=[aij]i;j=1;:::;narethevaluesofTij'sattheseizurepoint. However,asitwasshowninthepreviousstudies,onecanobservethe\resetting"ofthebrainafterseizures'onset[ 111 70 108 ],thatis,thedivergenceofSTLmaxprolesafteraseizure.Therefore,toensurethattheoptimalgroupofcriticalsitesshowsthisdivergence,onecanreformulatethisoptimizationproblembyaddingonemorequadraticconstraint:
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wherethematrixB=[bij]i;j=1;:::;nistheTindexmatrixofbrainsitesiandjwithin10minutewindowsaftertheonsetofaseizure. Thisproblemisthensolvedusingstandardtechniques,andthegroupofkcriticalsitesisfound.Itshouldbepointedoutthatthenumberofcriticalsiteskispredetermined,i.e.,itisdenedempirically,basedonpracticalobservations.Also,notethatintermsofGRAPHImodelthisproblemrepresentsndingasubgraphofGRAPHIofaxedsize,satisfyingthepropertiesspeciedabove. Now,recallthatweintroducedGRAPHIIIusingthesameprinciplesasintheformulationoftheaboveoptimizationproblem,thatis,weconsideredtheconnectionsonlybetweenthepairsofsitesi;jsatisfyingbothofthetwoconditions:Tij
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level.ThemainideaofthismodelistousethepropertiesofGRAPHI,GRAPHII,andGRAPHIIIasacharacterizationofthebehaviorofthebrainpriorto,during,andafterepilepticseizures.Accordingtothisgraphmodel,thegraphsreectingthebehavioroftheepilepticbraindemonstratethefollowingproperties: Moreover,oneoftheadvantagesoftheconsideredgraphmodelisthepossibilitytodetectspecialformationsinthesegraphs,suchascliquesandminimumspanningtrees,whichcanbeusedforfurtherstudyingofvariouspropertiesoftheepilepticbrain. Amongthedirectionsoffutureresearchinthiseld,onecanmentionthepossibilityofdevelopingdirectedgraphmodelsbasedontheanalysisofEEGdata.Suchmodelswouldtakeintoaccountthenatural\asymmetry"ofthebrain,wherecertainfunctionalunitscontroltheotherones.Also,onecouldapplyasimilarapproachtostudyingthepatternsunderlyingthebrainfunctionofthepatientswithothertypesofdisorders,suchasParkinson'sdisease,orsleepdisorder.
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Therefore,themethodologyintroducedinthischaptercanbegeneralizedandappliedinpractice.
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Inthischapter,wewilldiscussoneofthemostinterestingreallifegraphapplications{socalled\socialnetworks"wheretheverticesarerealpeople[ 63 116 ].Themainideaofthisapproachistoconsiderthe\acquaintanceshipgraph"connectingtheentirehumanpopulation.Inthisgraph,anedgeconnectstwogivenverticesifthecorrespondingtwopersonsknoweachother. Socialnetworksareassociatedwithafamous\smallworld"hypothesis,whichclaimsthatdespitethelargenumberofvertices,thedistancebetweenanytwovertices(or,thediameterofthegraph)issmall.Morespecically,theideaof\sixdegreesofseparation"hasbeenintroduced.Itstatesthatanytwopersonsintheworldarelinkedwitheachotherthroughasequenceofatmostsixpeople[ 63 116 117 ]. Clearly,onecannotverifythishypothesisforthegraphincorporatingmorethan6billionpeoplelivingontheEarth,however,smallersubgraphsoftheacquaintanceshipgraphconnectingcertaingroupsofpeoplecanbeinvestigatedindetail.Oneofthemostwellknowngraphsofthistypeisthescienticcollaborationgraphreectingtheinformationaboutthejointworksbetweenallscientists.Twoverticesareconnectedbyanedgeifthecorrespondingtwoscientistshaveajointresearchpaper.Anothergraphofthistypeisknownasthe\Hollywoodgraph":itlinksallthemovieactors,andanedgeconnectstwoactorsiftheyeverappearedinthesamemovie.Wellknownconceptsassociatedwiththesegraphsaresocalled\Erdosnumber"(inthescienticcollaborationgraph)and\Baconnumber"(intheHollywoodgraph),whichareassignedtoeveryvertexandcharacterizethedistancefromthisvertextothevertexdenotingthe\center"ofthegraph. 77
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Inthecollaborationgraph,thecentralvertexcorrespondstothefamousgraphtheoreticianPaulErdos,whereasintheHollywoodgraphthesamepositionisassignedtoKevinBacon. Inthischapter,wediscussgraphsofasimilartypearisinginsports,thatrepresenttheplayers'\collaboration".Inthesegraphs,theplayersarethevertices,andanedgeisaddedtothegraphifthecorrespondingtwoplayerseverplayedtogetherinthesameteam.Oneoftheexamplesofthistypeofgraphsisthegraphrepresentingbaseballplayers.ForanytwobaseballplayerswhoeverplayedintheMajorLeagueBaseball(MLB),apathconnectingthemcanbefoundinthisgraph. Asanotherinstanceofsocialnetworksinsports,westudythe\NBAgraph"wheretheverticesrepresentallthebasketballplayerswhoarecurrentlyplayingintheNBA.Weapplystandardgraphtheoreticalalgorithmsforinvestigatingthepropertiesofthisgraph,suchasitsconnectivityanddiameter(i.e.,themaximumdistancebetweenallpairsofverticesinthegraph).Aswewillseelaterinthechapter,thisstudyalsoconrmsthe\smallworldhypothesis".Moreover,weintroduceadistancemeasureintheNBAgraphsimilartotheErdosnumberandtheBaconnumber.ThecentralroleinthisgraphisgiventoMichaelJordan,thegreatestbasketballplayerofalltimes,andwerefertothismeasureastheJordannumber.
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distancesinthisgraph,the\centralvertex"isintroduced.ThisvertexcorrespondstoPaulErdos,thefatherofthetheoryofrandomgraphs.ThisvertexisassignedErdosnumberequalto0.Forallotherverticesinthegraph,theErdosnumberisdenedasthedistance(i.e.,theshortestpathlength)fromthecentralvertex.Forexample,thosescientistswhohadajointpaperwithErdoshaveErdosnumber1,thosewhodidnotcollaboratewithErdos,butcollaboratedwithErdos'collaboratorshaveErdosnumber2,etc. Followingthislogic,onecanconstructtheconnectedcomponentofthecollaborationgraphwith\concentriccircles",whichwouldincorporatealmostallscientistsintheworld,exceptthosewhonevercollaboratewithanybody.Thisconnectedcomponentisexpectedtohavearelativelysmalldiameter. Theideaofconstructingcollaborationgraphsencompassingpeopleindierentareasgavearisetoseveralotherapplications.Next,wediscusstheHollywoodgraphandthebaseballgraph,wherethenumberofverticesissignicantlysmallerthaninthescienticcollaborationgraph,whichallowsonetostudytheirstructureinmoredetail.
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Figure5{1. NumberofverticesintheHollywoodgraphwithdierentvaluesofBaconnumber.AverageBaconnumber=2.946. actor.ItturnsoutthatmostoftheactorshaveBaconnumbersequalto2or3,andthemaximumpossibleBaconnumberisequalto8,whichisthecaseonlyfor3vertices. ThedistributionofBaconnumbersintheHollywoodgraphisshowninFigure 5{1 .TheaverageBaconnumber(i.e.,theaveragepathlengthfromagivenactortoBacon)isequalto2.946.Asonecansee,boththeaverageandthemaximumBaconnumbersoftheHollywoodgraphareverysmall,whichprovidesanargumentinfavorofthe\smallworldhypothesis"mentionedabove.
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Figure5{2. NumberofverticesinthebaseballgraphwithdierentvauesofWynnnumber.AverageWynnnumber=2.901 has15817vertices.Linksbetweenanypairofbaseballplayerscanbefoundatthe\OracleofBaseball"website. 5{2 showsthedistributionofWynnnumbersinthebaseballgraph.ThemaximumWynnnumberis6,whichissmallerthanthemaximumBaconnumbersincetotalnumberofbaseballplayersislessthanthenumberofHollywoodactors.
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Hollywoodgraph,andEarlyWynnasthecenterofthebaseballgraphisthefactthatitisreasonabletoexpectthemtobeconnectedtomanyvertices:Baconappearedinmanymovies,andWynnplayedinseveralbaseballteamshadalotofteammatesduringhislongcareer.However,onecanchooseless\connected"centersofthesegraphs,andinthiscasethemaximumdistancefromthenewcenterofthegraphmaysignicantlyincrease.Forexample,ifonechoosesBarryBondsasthecenterofthebaseballgraph,themaximumBondsnumberwillbe9insteadof6.Moreover,intheHollywoodgraph,itispossibletochoosethecentersothatthemaximumdistancefromitisequalto14,andtheaveragedistanceisgreaterthan6(insteadof2.946).Therefore,inordertohaveamorecompleteinformationaboutthestructureofthesegraphs,oneshouldcalculatethemaximumpossibledistanceamongallpairsofverticesinthegraph.Recallthatthisquantityisreferredtoasthediameterofthegraph.Clearly,thediametercanbefoundbyconsideringeachvertexasthecenterofthegraph,calculatingcorrespondingmaximaldistances,andthenchoosingthemaximumamongthem. Inthenextsection,westudythepropertiesoftheNBAgraphincorporatingbasketballplayersplayingintheworld'sbestbasketballleague.Inasimilarfashion,weintroducetheJordannumber,investigateitsvaluescorrespondingtodierentvertices,andcalculatethediameterofthisgraph.
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Asonecaneasilysee,thisgraphhasahighlyspecicstructure:theplayersofeveryteamformacliqueinthegraph(i.e.,thesetofcompletelyinterconnectedvertices),becausealltheverticescorrespondingtotheplayersofthesameteammustbeinterconnected.Sincemanyplayerschangeteamsduringorbetweentheseasons,thereareedgesconnectingtheverticesfromdierentcliques(teams).Notethatthistypeofstructureiscommonforall\collaborationnetworks"(seeFigure 5{3 ). Itshouldbepointedoutthatthenumberofplayersinabasketballteamisrelativelysmall,andtheplayers'transfersbetweendierentteamsoccurratheroften,therefore,itwouldbelogicaltoexpectthattheNBAgraphshouldbeconnected,i.e.,thereisapathfromeveryvertextoeveryvertex,moreover,thelengthofthispathmustbesmallenough.Aswewillseebelow,calculationsconrmtheseassumptions.
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Figure5{3. GeneralstructureoftheNBAgraphandothercollaborationnetworks First,weusedastandardbreadthrstsearchtechniqueforcheckingtheconnectivityoftheconsideredgraph.Startingfromanarbitraryvertex,wewereabletolocateallotherverticesinthegraph,whichmeansthateveryvertexisreachablefromanother,therefore,thegraphisconnected.Inthenextsubsection,wewillalsoseethateverypairofverticesinthisgraphareconnectedbyashortpath,whichisinagreementwiththe\smallworldhypothesis".
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Figure5{4. NumberofverticesintheNBAgraphwithdierentvaluesofJordannumber.AverageJordannumber=2.270 Similarlytothesocialgraphsmentionedabove,wedenethe\centralvertex"intheNBAgraphcorrespondingtoMichaelJordan,whoplayedforWashingtonWizardsduringhisnalNBAseason.Obviously,allotherplayersintheWizards'rosterfor20022003,aswellasalltheplayerswhohaveplayedwithJordanduringatleastoneseasoninthepast,haveJordannumber1.ItshouldbenotedthatMichaelJordanplayedonlyfortwoteams(ChicagoBullsandWashingtonWizards)throughhisentirecareer,therefore,onecanexpectthatthenumberofplayerswithJordannumber1israthersmall.Infact,only24playerscurrentlyplayingintheNBAhaveJordannumber1.
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Table5{1. JordannumbersofsomeNBAstars(endofthe20022003season). PlayerTeamJordanNumber KobeBryantLosAngelesLakers2VinceCarterTorontoRaptors2VladeDivacSacramentoKings2TimDuncanSanAntonioSpurs2MichaelFinleyDallasMavericks2SteveFrancisHoustonRockets3KevinGarnettMinnesotaTimberwolves3PauGasolMemphisGrizzlies3RichardHamiltonDetroitPistons1AllenIversonPhiladelphia76ers2JasonKiddNewJerseyNets2ToniKukocMilwaukeeBucks1KarlMaloneUtahJazz2StephonMarburyPhoenixSuns2ShawnMarionPhoenixSuns2KenyonMartinNewJerseyNets3JamalMashburnNewOrleansHornets2TracyMcGradyOrlandoMagic2ReggieMillerIndianaPacers3YaoMingHoustonRockets3DikembeMutomboNewJerseyNets2SteveNashDallasMavericks2DirkNowitzkiDallasMavericks2JermaineO'NealIndianaPacers2ShaquilleO'NealLosAngelesLakers2GaryPaytonMilwaukeeBucks2PaulPierceBostonCeltics2ScottiePippenPortlandTrailBlazers1DavidRobinsonSanAntonioSpurs2ArvydasSabonisPortlandTrailBlazers2JerryStackhouseWashingtonWizards1PredragStojakovicSacramentoKings2AntoineWalkerBostonCeltics2BenWallaceDetroitPistons2ChrisWebberSacramentoKings2
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Followingsimilarlogic,theplayerswhohaveplayedwithJordan's\collaborators"haveJordannumber2,andsoon.However,itturnsoutthatthemaximumJordannumberinthisinstanceoftheNBAgraphisonly3,i.e.,alltheplayersarelinkedwithJordanthroughatmosttwovertices,whichiscertainlynotsurprising:with29teamsandonlyaround15playersineachteam,NBAisreallya\smallworld".Figure 5{4 showsthedistributionofJordannumbersintheNBAgraph.TheaverageJordannumberisequalto2.27,whichissmallerthantheaverageBaconnumberintheHollywoodgraph,andtheaverageWynnnumberinthebaseballgraph,duetosmallernumberofvertices. Table 5{1 presentsJordannumberscorrespondingtosomewellknownNBAplayers.Notsurprisingly,mostofthemhaveJordannumber2,exceptforseveralplayerswithJordannumber3:thosewhojoinedthisleaguerecently,andthereforedidnothavemanyteammatesthroughtheircareer,aswellasReggieMillerwhospent16seasonsinthesameteam(IndianaPacers),andKevinGarnettwhoplayedinMinnesotafor8years.ScottiePippen,ToniKukoc,andJerryStackhousewereJordan'steammatesatdierenttimes,therefore,theyhaveJordannumber1. Furthermore,wecalculatedthediameteroftheNBAgraph,i.e.,themaximumpossibledistancebetweenanytwoverticesinthegraph.SincethemaximumJordannumberintheNBAgraphisequalto3,onewouldexpectthatthevalueofthediametertobeofthesameorderofmagnitude.Asitwasmentionedintheprevioussection,thediameteroftheNBAgraphcanbefoundasfollows:foreverygivenvertex,wecalculatethedistancesbetweenthisvertexandallothers.Inthisapproach,weneedtorepeatthisprocedure404times,andeverytimeadierentvertexisconsideredtobethe\center"ofthegraph.OurcalculationsshowthatthediameteroftheNBAgraph(themaximumdistancebetweenallpairsofvertices)isequalto4.Therefore,onecanclaimthattheNBAgraphactuallyfollowsthesmallworldhypothesis,sinceitsdiameterissmallenough.
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Table5{2. DegreesoftheVerticesintheNBAgraph degreeintervalnumberofvertices 1120134213011631401034150425160861+2 5{2 presentsthenumberofverticesintheNBAgraphcorrespondingtodierentintervalsofthedegreevalues. ItwouldbereasonabletoassumethatifonepicksavertexwithahighdegreeasthecenteroftheNBAgraph,theaveragedistanceinthegraphcorrespondingtothisvertexwouldbesmallerthantheaverageJordannumber.Wehavefoundthemost\connected"playersintheNBAgraphwiththesmallestcorrespondingaveragedistances.Table 5{3 presentsveplayerswhocouldbethemost\connected"centersoftheNBAgraph.Asonecannotice,allofthemare\benchplayers"whohavechangedmanyteamsduringtheircareer,therefore,theyhavehighdegreesintheNBAgraph.Also,aninterestingobservationisthatalthoughCorieBlount'svertexisdegreesmallerthanJimJackson's,theaverageconnectivityishigherforCorieBlount,whichcouldbeexplainedbythefactthathisteammateswerehighly\connected"themselves.
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Table5{3. Themost\connected"playersintheNBAgraph PlayerTeamDegreeAv.Distance CorieBlountChicagoBulls631.906JimJacksonSacramentoKings681.923RobertPackNewOrleansHornets571.936GrantLongBostonCeltics501.946BimboColesBostonCeltics541.958 AlthoughtheinstanceoftheNBAgraphconsideredinthischaptercontainsonlycurrentlyactivebasketballplayers,itcanbeeasilyextendedtoreectallplayersinthehistoryoftheNBA.Moreover,sincealotofforeignplayersfromdierentcountriesandcontinentshavecometotheNBAinrecentyears,onewouldexpectthatthegraphcoveringallbasketballplayersplayinginmajorforeignchampionshipsisalsoconnectedandhasasmalldiameter.
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Inthisdissertation,wehaveaddressedseveralissuesregardingtheuseofnetworkbasedtechniquesforsolvingvariousproblemsarisinginthebroadareaoftheanalysisofcomplexsystems.Wehavedemonstratedthatapplyingtheseapproachesiseectiveinmanyapplications,includingnance,biomedicine,telecommunications,sociology,etc.Ifarealworldmassivedatasetcanbeappropriatelyrepresentedasanetworkstructure,itsanalysisusinggraphtheoreticaltechniquesoftenyieldsimportantpracticalresults. Clearly,theresearchinthisareaisfarfromcomplete.Astechnologicalprogresscontinues,newtypesofdatasetsemergeindierentpracticalelds,whichleadstofurtherresearchintheeldofmodelingandinformationretrievalfromthesedatasets.Moreover,theapproachesdiscussedinthisdissertationcanbepotentiallyextendedtoobtainamoredetailedpictureofthestructureoftheconsidereddatasets.Inthefuturework,thenetworkmodelsdescribedabovecanbegeneralizedtotakeintoaccountthedirectionoflinksbetweenvertices(directedgraphs),whichcanhelptounderstandthemechanismsofinuencebetweendierentelementsofthesystems(e.g.,stocks,brainunits,etc.)Inaddition,someparameterscanbeassignedtoverticesrepresentingelementsofthesystem(e.g.,stockscanbecharacterizedbytheirexpectedreturnsandliquidities).Thisleadstosolvingoptimizationproblemsonweightedgraphs(e.g.,maximumweightedclique/independentset),whichmaybemorechallengingtosolveinpracticeforlargegraphs;however,thisanalysismayprovidevaluableinformationabouttheconsideredsystems. 90
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VladimirBoginskiwasbornonSeptember23,1980,inBryansk,Russia.Hereceivedhisbachelor'sdegreeinAppliedMathematicsfromMoscowInstituteofPhysicsandTechnology(StateUniversity)in2000.In2001,heenteredthegraduateprograminIndustrialandSystemsEngineeringattheUniversityofFlorida.HereceivedhisM.S.andPh.D.degreesinIndustrialandSystemsEngineeringfromtheUniversityofFloridainMay2003andAugust2005,respectively. 100
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OPTIMIZATION AND INFORMATION RETRIEVAL TECHNIQUES FOR COMPLEX NETWORKS By VLADIMIR L. BOGINSKI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005 Copyright 2005 by Vladimir L. Boginski I dedicate this to my parents. ACKNOWLEDGMENTS I would like to thank my advisor Prof. Panos Pardalos for his support and guidance that made my studies in the University of Florida enjoi, 1l and produc tive. His energy and enthusiasm inspired me during these four years, and I believe that this was crucial for my success. I also want to thank my committee members Prof. Stan Uryasev, Prof. Joseph Geunes, and Prof. William Hager for their concern and encouragement. I am grateful to all my collaborators, especially Sergiy Butenko and Oleg Prokopyev, who were ahvb a great pleasure to work with. Finally, I would like to express my greatest appreciation to my family and friends, who ahvb believed in me and supported me in all circumstances. TABLE OF CONTENTS page ACKNOWLEDGMENTS ................... ...... iv LIST OF TABLES ................... .......... viii LIST OF FIGURES ................... ......... ix ABSTRACT ...................... ............. xi CHAPTER 1 INTRODUCTION .................... ....... 1 1.1 Basic Concepts from Graph Theory and Data Mining Interpretation 3 1.1.1 Connectivity and Degree Distribution ............. 3 1.1.2 Cliques and Independent Sets ........ ........ 5 1.1.3 Clustering via Clique Partitioning ...... . . 6 2 REVIEW OF NETWORKBASED MODELING AND OPTIMIZATION TECHNIQUES IN MASSIVE DATA SETS .... . .. 9 2.1 Modeling and Optimization in Massive Graphs . . ... 9 2.1.1 Examples of Massive Graphs ................. .. 10 2.1.1.1 Call Graph .................. ..... 10 2.1.1.2 Internet and Web Graphs . . ...... 13 2.1.2 External Memory Algorithms ................ .. 17 2.1.3 Modeling Massive Graphs .................. .. 18 2.1.3.1 Uniform Random Graphs . . ..... 19 2.1.3.2 Potential Drawbacks of the Uniform Random Graph Model ....... . . .... 21 2.1.3.3 Random Graphs with a Given Degree Sequence .23 2.1.3.4 PowerLaw Random Graphs . . 24 2.1.4 Optimization in Random Massive Graphs . .... 29 2.1.4.1 Clique Number ............ .. .. .. 29 2.1.4.2 C('.!i. ii ,,1 Number ................. .. 31 2.1.5 Remarks .................. .......... .. 32 3 NETWORKBASED APPROACHES TO MINING STOCK MARKET DATA ................. .................. ..33 3.1 Structure of the Market Graph ............... .. .. 34 3.1.1 Constructing the Market Graph ............... .. 34 3.1.2 Connectivity of the Market Graph . . 36 3.1.3 Degree Distribution of the Market Graph . .... 37 3.1.4 Instruments Corresponding to HighDegree Vertices . 40 3.1.5 Clustering Coefficients in the Market Graph . ... 41 3.2 Analysis of Cliques and Independent Sets in the Market Graph 42 3.2.1 Cliques in the Market Graph ............. .. 43 3.2.2 Independent Sets in the Market Graph ......... .45 3.3 Data Mining Interpretation of the Market Graph Model ...... ..48 3.4 Evolution of the Market Graph . . . ... 50 3.4.1 Dynamics of Global 'C!i 'lteristics of the Market Graph 51 3.4.2 Dynamics of the Size of Cliques and Independent Sets in the Market Graph .................. ...... 55 3.4.3 Minimum Clique Partition of the Market Graph ...... ..59 3.5 Concluding Remarks ............... ...... .. 60 4 NETWORKBASED TECHNIQUES IN ELECTROENCEPHALOGRAPHIC (EEG) DATA ANALYSIS AND EPILEPTIC BRAIN MODELING ... 62 4.1 Statistical Preprocessing of EEG Data ... . . 63 4.1.1 Datasets. .................. .. ..... 63 4.1.2 Tstatistics and STLmax ................ 63 4.2 Graph Structure of the Epileptic Brain ............... ..66 4.2.1 Key Idea of the Model . . . .. ..66 4.2.1.1 Interpretation of the Considered Graph Models .67 4.2.2 Properties of the Graphs ................ 67 4.2.2.1 Edge Density ................ .. .. 67 4.2.2.2 Connectivity ................ .. .. 69 4.2.2.3 Minimum Spanning Tree .............. ..69 4.2.2.4 Degrees of the Vertices ............... 71 4.2.2.5 Maximum Cliques . . ...... 72 4.3 Graph as a Macroscopic Model of the Epileptic Brain . ... 74 4.4 Concluding Remarks and Directions of Future Research ...... ..75 5 COLLABORATION NETWORKS IN SPORTS ........... .77 5.1 Examples of Social Networks . . . .... ...... 78 5.1.1 Scientific Collaboration Graph and Erdos Number . 78 5.1.2 Hollywood Graph and Bacon Number . ..... 79 5.1.3 Baseball Graph and Wynn Number ............. ..80 5.1.4 Diameter of Collaboration Networks ............. .81 5.2 NBA Graph ........ .... ... ............ 82 5.2.1 General Properties of the NBA Graph ............ ..83 5.2.2 Diameter of the NBA Graph and Jordan Number . 85 5.2.3 Degrees and "Connectedness" of the Vertices in the NBA Graph ...... ......... ......... .... 88 5.3 Concluding Remarks .................. ....... .. 89 6 CONCLUSIONS AND DIRECTIONS FOR FUTURE RESEARCH ... 90 REFERENCES .............. .......... ... .... 91 BIOGRAPHICAL SKETCH ......... ....... ......... 100 LIST OF TABLES Table page 31 Leastsquares estimates of the parameter 7 in the market graph . 38 32 Top 25 instruments with highest degrees in the market graph ...... 42 33 C('!1i. i ig coefficients of the market graph ................ ..43 34 Sizes of the maximum cliques in the market graph ............ ..45 35 Sizes of independent sets in the complementary market graph . 46 36 Dates and mean correlations corresponding to each considered 500d,v shift .... ................ ....... ...... .. 51 37 Number of vertices and number of edges in the market graph for differ ent periods .................. ............... .. 55 38 Greedy clique size and the clique number for different time periods 57 39 Structure of maximum cliques in the market graph for different time pe riods ...... ............. ................. .. 58 310 Size of independent sets in the market graph found using the greedy heuris tic . . . . . . .... ... ..... 59 311 The largest clique size and the number of cliques in computed clique par titions . . . . . . . . ... .. 60 51 Jordan numbers of some NBA stars (end of the 20022003 season). . 86 52 Degrees of the Vertices in the NBA graph ................ 88 53 The most "connected" ph ,l rs in the NBA graph ............ ..89 LIST OF FIGURES Figure page 21 Frequencies of clique sizes in the call graph ..... . . 11 22 Pattern of connections in the call graph .................. 12 23 Number of Internet hosts for the period 01/199101/2002. . ... 13 24 Pattern of connections in the Web graph ............. .. 14 25 Connectivity of the Web (BowTie model) ............ .. 16 31 Distribution of correlation coefficients in the stock market . ... 35 32 Edge density of the market graph for different values of the correlation threshold ............... ............ .. .. 36 33 Plot of the size of the largest connected component in the market graph as a function of correlation threshold 0. ................. 37 34 Degree distribution of the market graph ................. 39 35 Degree distribution of the complementary market graph . ... 40 36 Frequency of the sizes of independent sets found in the market graph 48 37 Distribution of correlation coefficients in the US stock market for several overlapping 500day periods during 20002002 . . ..... 52 38 Degree distribution of the market graph for different 500div periods in 20002002 .. ....... ............... 53 39 Dynamics of edge density and maximum clique size in the market graph 55 41 Electrode placement in the brain .................. ..... 64 42 Number of edges in GRAPHII .................. .. 68 43 The size of the largest connected component in GRAPHII . ... 70 44 Average value of Tindex of the edges in Minimum Spanning Tree of GRAPH I........................ ......... ...... 71 45 Average degree of the vertices in GRAPHII. . . ..... 72 51 Number of vertices in the Hollywood graph with different values of Ba con number .................. ............... .. 80 52 Number of vertices in the baseball graph with different vaues of Wynn number ..... .............. ............... .. 81 53 General structure of the NBA graph and other collaboration networks .84 54 Number of vertices in the NBA graph with different values of Jordan num ber . . . . . .. . ... 85 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy OPTIMIZATION AND INFORMATION RETRIEVAL TECHNIQUES FOR COMPLEX NETWORKS By Vladimir L. Boginski August 2005 C('! Ii: Panagote M. Pardalos Major Department: Industrial and Systems Engineering This study develops novel approaches to modeling realworld datasets arising in diverse application areas as networks and information retrieval from these datasets using network optimization techniques. Networkbased models allow one to extract information from datasets using various concepts from graph theory. In many cases, one can investigate specific properties of a dataset by detecting special formations in the corresponding graph (for instance, connected components, spanning trees, cliques, and independent sets). This process often involves solving computationally challenging combinatorial optimization problems on graphs (maximum independent set, maximum clique, minimum clique partition, etc.). These problems are especially difficult to solve for large graphs. However, in certain cases, the exact solution of a hard optimization problem can be found using a special structure of the considered graph. A significant part of the dissertation focuses on developing networkbased models of realworld complex systems, including the stock market and the human brain, which have ahvl been of special interest to scientists. These systems gen erate huge amounts of data and are especially hard to ain &v. This dissertation demonstrates that networkbased models can be successfully applied to information retrieval from datasets, providing new insight into the structural properties and patterns underlying the corresponding complex systems. The developed network representations of the considered datasets are in many cases nontrivial and include certain statistical preprocessing techniques. In particular, the U.S. stock market is represented as a network based on cross correlations of price fluctuations of the financial instruments, which are calculated over a certain number of trading div This model (market p'jiq'l) allows one to analyze the structure and dynamics of the stock market from an alternative perspective and obtain useful information about the global structure of the market, classes of similar stocks, and diversified portfolios. Similarly, a macroscopic network model of the human brain is constructed based on the statistical measures of entrainment between electroencephalographic (EEG) signals recorded from different functional units of the brain. Studying the evolution of the properties of these networks revealed some interesting facts about brain disorders, such as epilepsy. CHAPTER 1 INTRODUCTION Now,1dl, the process of studying reallife complex systems often deals with large datasets arising in diverse applications including government and military systems, telecommunications, biotechnology, medicine, finance, astrophysics, ecol ogy, geographical information systems, etc. [3, 25]. Understanding the structural properties of a certain dataset is in many cases the task of crucial importance. To get useful information from these data, one often needs to apply special techniques of summarizing and visualizing the information contained in a dataset. An appropriate mathematical model can simplify the analysis of a dataset and even theoretically predict some of its properties. Thus, a fundamental problem that arises here is modeling the datasets characterizing realworld complex systems. In this dissertation, we concentrate on one aspect of this problem: network representation of realworld datasets. According to this approach, a certain dataset is represented as a p',h'l (network) with certain attributes associated with its vertices and edges. Studying the structure of a graph representing a dataset is often important for understanding the internal properties of the application it represents, as well as for improving storage organization and information retrieval. One can visualize a graph as a set of dots and links connecting them, which often makes this representation convenient and easily understandable. The main concepts of graph theory were founded several centuries ago, and many network optimization algorithms have been developed since then. However, graph models have been applied only recently to representing various reallife massive datasets. Graph theory is quickly becoming a practical field of science. Expansion of graphtheoretical approaches in various applications gave birth to the terms 3,i 1h! practice" and ,i 111 1(1 ii ,, ii1, [63]. Networkbased models allow one to extract information from realworld datasets using various standard concepts from graph theory. In many cases, one can investigate specific properties of a dataset by detecting special formations in the corresponding graph, for instance, connected components, a ur.'.':'ij trees, cliques and independent sets. In particular, cliques and independent sets can be used for solving the important clustering problem arising in data mining, which essentially represents partitioning the set of elements of a certain dataset into a number of subsets (clusters) of objects according to some similarity (or dissimilarity) criterion. These concepts are associated with a number of network optimization problems discussed later. Another aspect of investigating network models of realworld datasets is studying the degree distribution of the constructed graphs. The degree distribution is an important characteristic of a dataset represented by a graph. It represents the largescale pattern of connections in the graph, which reflects the global properties of the dataset. One of the important results discovered during the last several years is the observation that many graphs representing the datasets from diverse areas (Internet, telecommunications, biology, sociology) obey the powerlaw model [9]. The fact that graphs representing completely different datasets have a similar welldefined powerlaw structure has been widely reflected in the literature [10, 19, 20, 25, 63, 116, 117]. It indicates that global organization and evolution of datasets arising in various spheres of life 1 i., .I1 ,is follow similar laws and patterns. This fact served as a motivation to introduce a concept of "selforganized networks." Later we discuss in more detail various aspects of modeling realworld datasets as networks, and retrieving useful information from these networks. The practical importance of graphtheoretic techniques is shown by several examples of applying these approaches associated with datasets arising in telecommunications, internet, sociology, etc. The 1i ii' ', part of the dissertation devoted to novel networkbased techniques and models that allow one to obtain important nontrivial information from datasets arising in finance and biomedicine. 1.1 Basic Concepts from Graph Theory and Data Mining Interpretation To facilitate further discussion, we present several basic definitions and notations from graph theory and discuss the interpretation of the introduced concepts from the perspective of data mining and information retrieval. Let G = (V, E) be an undirected graph with the set of n vertices V and the set of edges E = {(i,j) : i,j E V}. Directed graphs, where the head and tail of each edge are specified, are considered in some applications. The concept of a mi"ill.:raph is also sometimes introduced. A multigraph is a graph where multiple edges connecting a given pair of vertices may exist. One of the important characteristics of a graph is its edge /,. ,.:1;I: the ratio of the number of edges in the graph to the maximum possible number of edges.1 1.1.1 Connectivity and Degree Distribution The graph G = (V, E) is connected if there is a path from any vertex to any vertex in the set V. If the graph is disconnected, it can be decomposed into several connected subgraphs, which are referred to as the connected components of G. The degree of a vertex is the number of edges emanating from it. For every integer k one can calculate the number of vertices n(k) with a degree equal to k, and then get the probability that a vertex has the degree k as P(k) = n(k)/n, where n is the total number of vertices. The function P(k) is referred to as the 1 The maximum possible number of edges in a graph is equal to n(n 1)/2 (n is the number of vertices). degree distribution of the graph. In the case of a directed graph, the concept of degree distribution is generalized: one can distinguish the distribution of indegrees and outdegrees, which deal with the number of edges ending at and starting from a vertex, respectively. Degree distribution is an important characteristic of a dataset represented by a graph. It reflects the overall pattern of connections in the graph, which in many cases reflects the global properties of the dataset this graph represents. As mentioned above, many realworld graphs representing the datasets coming from diverse areas (Internet, telecommunications, finance, biology, sociology) have degree distributions that follow the powerlaw model, which states that the probability that a vertex of a graph has a degree k (i.e., there are k edges emanating from it) is P(k) oc k. (11) Equivalently, one can represent it as logP oc log k, (12) which demonstrates that this distribution forms a straight line in the logarithmic scale, and the slope of this line equals the value of the parameter 7. An important characteristic of the powerlaw model is its scalefree property. This property implies that the powerlaw structure of a certain network should not depend on the size of the network. Clearly, realworld networks dynamically grow over time, therefore, the growth process of these networks should obey certain rules in order to satisfy the scalefree property. The necessary properties of the evolution of the realworld networks are growth and preferential attachment [20]. The first property implies the obvious fact that the size of these networks grows continuously (i.e., new vertices are added to a network, which means that new elements are added to the corresponding dataset). The second property represents the idea that 5 new vertices are more likely to be connected to old vertices with high degrees. It is intuitively clear that these principles characterize the evolution of many realworld complex networks i.v, '. Il, ivs. From another perspective, some properties of graphs that follow the powerlaw model can be predicted theoretically. Aiello et al. [9] studied the properties of the powerlaw graphs using the theoretical powerlaw random ','rll, model representing the the class of random graphs obeying the power law (see C'! plter 2). Among their results, one can mention the existence of a giant connected component in a powerlaw graph with 7 < 7o a 3.47875, and the fact that a giant connected component does not exist otherwise.2 Emergence of a giant connected component at the point 7o m 3.47875 is often called phase transition. The size of connected components of the graph may provide useful information about the structure of the corresponding dataset, as the connected components would normally represent groups of "!~iI! i1 objects. In some applications, decomposing the graph into a set of connected components can provide a reason able solution to the clustering problem (i.e., partitioning the graph into several subgraphs, each of which corresponds to a certain cluster). 1.1.2 Cliques and Independent Sets Given a subset S C V, we denote G(S) as the subgraph induced by S. A subset C C V is a clique if G(C) is a complete graph (i.e., it has all possible edges). The maximum clique problem is to find the largest clique in a graph. The following definitions generalize the concept of clique. Instead of cliques one can consider dense subgraphs, or quasicliques. A 7clique C., also called a 2 These results are valid i ;,,,;///. i//.;/rl almost surely (a.a.s.), which means that the probability that a given property takes place tends to 1 as the number of ver tices n goes to infinity. quasiclique, is a subset of V such that G(C.) has at least L[q(q 1)/2] edges, where q is the cardinality (i.e., number of vertices) of C.. An independent set is a subset I C V such that the subgraph G(I) has no edges. The maximum independent set problem can be easily reformulated as the maximum clique problem in the complementary graph G(V, E), defined as follows. If an edge (i,j) E E, then (i,j) E; and if (i,j) E, then (i,j) E E. Clearly, a maximum clique in G is a maximum independent set in G, so the maximum clique and maximum independent set problems can be easily reduced to each other. Locating cliques (quasicliques) and independent sets in a graph representing a dataset provides important information about this dataset. Intuitively, edges in such a graph would connect vertices corresponding to "!!!!i! 1i elements of the dataset. Therefore, cliques (or quasicliques) would naturally represent dense clusters of similar objects. On the contrary, independent sets can be treated as groups of objects that differ from every other object in the group. This information is also important in some applications. Clearly, it is useful to find a maximum clique or independent set in the graph, since it would give the maximum possible size of the groups of mid! 11 or "dil, I, i objects. The maximum clique problem (as well as the maximum independent set prob lem) is known to be NPhard [59]. Moreover, it turns out that these problems are difficult to approximate [18, 62]. This makes these problems especially challenging in large graphs. 1.1.3 Clustering via Clique Partitioning The problem of locating cliques and independent sets in a graph can be naturally extended to finding an optimal partition of a graph into a minimum number of distinct cliques or independent sets. These problems are referred to as minimum clique partition and u',jl' coloring, respectively. Pardalos et al. [102] give various mathematical programming formulations of these problems. Clearly, as in the case of maximum clique and maximum independent set problems, minimum clique partition and graph coloring are reduced to each other by considering the complimentary graph, and both of these problems are NPhard [59]. Solving these problems for graphs representing reallife datasets is important from a data mining perspective; especially for solving the clustering problem. The essence of clustering is partitioning the elements in a certain dataset into several distinct subsets (clusters) grouped according to an appropriate .;,ii,..,li/ criterion [34]. Identifying the groups of objects that are "iidI! ,o to each other but "dill, i il from other objects in a given dataset is important in many practical applications. The clustering problem is challenging because the number of clusters and the similarity criterion are usually not known a priori. If a dataset is represented as a graph, where each data element corresponds to a vertex, the clustering problem essentially deals with decomposing this graph into a set of subgraphs subsetss of vertices), so that each of these subgraphs correspond to a specific cluster. Since the data elements assigned to the same cluster should be !r!!! 1" to each other, the goal of clustering can be achieved by finding a clique partition of the graph, and the number of clusters will equal the number of cliques in the partition. Similar arguments hold for the case of the graph coloring problem which should be solved when a dataset needs to be decomposed into the clusters of "dil!, i, il objects (i.e., each object in a cluster is different from all other objects in the same cluster), that can be represented as independent sets in the corresponding graph. The number of independent sets in the optimal partition is referred to as the chromatic number of the graph. Instead of cliques and independent sets one can consider quasicliques, and quasiindependent sets and partition the graph on this basis. As mentioned, 8 quasicliques are subgraphs that are dense enough (i.e., they have a high edge density). Therefore, it is often reasonable to relate clusters to quasicliques, since they represent sufficiently dense clusters of similar objects. Obviously, in the case of partitioning a dataset into clusters of "dl!1i i, i, objects, one can use quasi independent sets (i.e., subgraphs that are sparse enough) to define these clusters. CHAPTER 2 REVIEW OF NETWORKBASED MODELING AND OPTIMIZATION TECHNIQUES IN MASSIVE DATA SETS In this chapter, we review current developments in studying massive graphs used as models of certain realworld datasets.1 Massive data sets arise in a broad spectrum of scientific, engineering and commercial applications [3]. Some of the wide range of problems associated with massive data sets are data warehousing, compression and visualization, information retrieval, clustering and pattern recognition, and nearest neighbor search. Handling these problems requires special interdisciplinary efforts to develop novel sophisticated techniques. The pervasiveness and complexity of the problems brought by massive data sets make it one of the most challenging and exciting areas of research for years to come. In many cases, a massive data set can be represented as a very large graph with certain attributes associated with its vertices and edges. These attributes may contain specific information characterizing the given application. Studying the structure of this graph is important for understanding the structural properties of the application it represents, as well as for improving storage organization and information retrieval. 2.1 Modeling and Optimization in Massive Graphs In this section we discuss recent advances in modeling and optimization for massive graphs. As examples, Call, Internet, and Web graphs will be used. 1 This chapter is based on the joint publication with Butenko and Pardalos [25]. As before, by G = (V, E) we will denote a simple undirected graph with the set of n vertices V and the set of edges E. A multigraph is an undirected graph with multiple edges. The distance between two vertices is the number of edges in the shortest path between them (it is equal to infinity for vertices representing different connected components). The diameter of a graph G is usually defined as the maximal distance between pairs of vertices of G. In a disconnected graph, the usual definition of the diameter would result in the infinite diameter, so the following definition is in order. By the diameter of a disconnected graph we will mean the maximum finite shortest path length in the graph (the same as the largest of the diameters of the graph's connected components). 2.1.1 Examples of Massive Graphs 2.1.1.1 Call Graph Here we discuss an example of a massive graph representing telecommunica tions traffic data presented by Abello, Pardalos and Resende [2]. In this call p',j'l, the vertices are telephone numbers, and two vertices are connected by an edge if a call was made from one number to another. Abello et al. [2] experimented with data from AT&T telephone billing records. To give an idea of how large a call graph can be we mention that a graph based on one 20dv period had 290 million vertices and 4 billion edges. The analyzed oned(v call graph had 53,767,087 vertices and over 170 million edges. This graph appeared to have 3,667,448 connected components, most of them tiny; only 302,468 (or '.) components had more than 3 vertices. A giant connected component with 44,989,297 vertices was computed. It was observed that the existence of a giant component resembles a behavior si l i l by the random graphs theory of Erdos and Rinyi [47, 48], which will be mentioned below, but by the pattern of connections the call graph obviously does not fit into this theory(Subsection 2.1.3). The maximum clique problem and problem of finding large quasicliques with prespecified density were considered in this giant component. These problems were attacked using a greedy randomized adaptive search procedure (GRASP) [51, 52]. In short, GRASP is an iterative method that at each iteration constructs, using a greedy function, a randomized solution and then finds a locally optimal solution by searching the neighborhood of the constructed solution. This is a heuristic approach which gives no guarantee about quality of the solutions found, but proved to be practically efficient for many combinatorial optimization problems. To make application of optimization algorithms in the considered large component possible, the authors use some suitable graph decomposition techniques employing external memory algorithms (see Subsection 2.1.2). 1000 freq 100 10 1__ 5 10 15 20 25 30 clique size Figure 2 1. Frequencies of clique sizes in the call graph found by Abello et al. [2]. Abello et al. [2] ran 100,000 GRASP iterations taking 10 parallel processors about one and a half dv4 to finish. Of the 100,000 cliques generated, 14,141 appeared to be distinct, although rn r: of them had vertices in common. Abello et al. i::I. 1 that the graph contains no clique of a size greater than 32. Figure 21 shows the number of detected cliques of various sizes. Finally, large 12 quasicliques with density parameters 7 = 0.9, 0.8, 0.7, and 0.5 for the giant connected component were computed. The sizes of the largest quasicliques found were 44, 57, 65, and 98, respectively. le+07 le+06 le+05 le+04 Sle+03 Sle+02 le+01 le+01 1le+ 00 .. . ......, ...... le+00 le+01 l+02 le+03 Outdegree le+06j le+05 le+ Sle+ le+ &* le+ (a) le+04 1 05 le+07 le+06 le+05 le+04 " le+03 le+02 ie+03 le+01 le+00 le+00 (b) le+01 le+02 le03 le+04 le05 Indegree (c) 04 03 02 01 : <* le+00 le+01 le+02 le+03 le+04 le+05 le+06 le+07 Component size Figure 22. Pattern of connections in the call graph: number of vertices with various outdegrees (a) and indegrees (b); number of connected com ponents of various sizes (c) in the call graph [8]. Aiello et al. [8] used the same data as Abello et al. [2] to show that the considered call graph fits to their powerlaw random graph model (Section 2.1.3). The plots in Figure 22 demonstrate some connectivity properties of the call graph. Summarizing the results presented in this subsection, one can w that graph based techniques proved to be rather useful in the analysis and revealing the global S M patterns of the telecommunications traffic dataset. In the next subsection, we will consider another example of a similar type of dataset associated with the WorldWide Web. 2.1.1.2 Internet and Web Graphs The role of the Internet in the modern world is difficult to overestimate; its invention changed the way people interact, learn, and communicate like nothing before. Alongside with increasing significance, the Internet itself continues to grow at an overwhelming rate. Figure 23 shows the dynamics of growth of the number of Internet hosts for the last 13 years. As of January 2002 this number was estimated to be close to 150 million.2 The number of web pages indexed by large search engines exceeds 2 billion, and the number of web sites is growing by thousands daily. 160,000,000 140,000,000 120,000,000 100,000,000 80,000,000 60,000,000 40,000,000 20,000,000 0 Figure 23. Number of Internet hosts for the period 01/199101/2002. Data by Internet Software Consortium. 2 According to Internet Software Consortium, http://www.isc.org/ds/hostcount history.html le+lO I I I !e 7 i le+O  le+ 4 le 1+ +e+ 1e405 le40 le le+00 I I \ le'o0 1e4O I 1 10 100 1000 10 100 100000 outdegee iz of coponta Figure 24. Pattern of connections in the Web graph: number of vertices with var ious outdegrees (left) and distribution of sizes of strongly connected components (right) in Web graph [37]. The highly dynamic and seemingly unpredictable structure of the World Wide Web attracts more and more attention of scientists representing many diverse disciplines, including graph theory. In a graph representation of the World Wide Web, the vertices are documents and the edges are hyperlinks pointing from one document to another. Similarly to the call graph, the Web is a directed multigraph, although often it is treated as an undirected graph to simplify the analysis. Another graph is associated with the physical network of the Internet, where the vertices are routers navigating packets of data or groups of routers (domains). The edges in this graph represent wires or cables in the physical network. Graph theory has been applied for web search [36, 78], web mining [96, 97] and other problems arising in the Internet and World Wide Web. In several recent studies, there were attempts to understand some structural properties of the Web graph by investigating large Web crawls. Adamic and Huberman [6, 65] used crawls which covered almost 260,000 pages in their studies. Barabdsi and Albert [20] analyzed a subgraph of the Web graph approximately 325,000 nodes representing nd.edu pages. In another experiment, Kumar et al. [82] examined a data set containing about 40 million pages. In a recent study, Broder et al. [37] used two Altavista crawls, each with about 200 million pages and 1.5 billion links, thus significantly exceeding the scale of the preceding experiments. This work yielded several remarkable observations about local and global properties of the Web graph. All of the properties observed in one of the two crawls were validated for the other as well. Below, by the Web graph we will mean one of the crawls, which has 203,549,046 nodes and 2130 million arcs. The first observation made by Broder et al. confirms a property of the Web graph i... i. 1 in earlier works [20, 82] claiming that the distribution of degrees follows a power law. Interestingly, the degree distribution of the Web graph resembles the powerlaw relationship of the Internet graph topology, which was first discovered by Faloutsos et al. [50]. Broder et al. [37] computed the in and out degree distributions for both considered crawls and showed that these distributions agree with power laws. Moreover, they observed that in the case of indegrees the constant 7 m 2.1 is the same as the exponent of power laws discovered in earlier studies [20, 82]. In another set of experiments conducted by Broder et al., directed and undirected connected components were investigated. It was noticed that the distribution of sizes of these connected components also obeys a power law. Figure 24 illustrates the experiments with distributions of outdegrees and connected component sizes. The last series of experiments discussed by Broder et al. [37] aimed to explore the global connectivity structure of the Web. This led to the discovery of the so called BowTie model of the Web [38]. Similarly to the call graph, the considered Web graph appeared to have a giant connected component, containing 186,771,290 nodes, or over 9,n'. of the total number of nodes. Taking into account the directed nature of the edges, this connected component can be subdivided into four pieces: ,i,..('i/, connected component (SCC), In and Out components, and "T i,..i/".. Overall, the Web graph in the BowTie model is divided into the following pieces: /i Tendrils \ 43,797,944 , \. tubes>' / \ _.. *' SSCC / In 43318 )  Out .43,343,168 ,, \43,166,185 /56 463 993  SDisc.) 16,777,756 Figure 25. Connectivity of the Web (BowTie model) [37]. * Strongly connected component: the part of the giant connected component in which all nodes are reachable from one another by a directed path. * In component: nodes which can reach any node in the SCC but cannot be reached from the SCC. * Out component: contains the nodes that are reachable from the SCC, but cannot access the SCC through directed links. * Tendrils component: accumulates the remaining nodes of the giant connected component, i.e., the nodes which are not connected with the SCC. * Disconnected component: the part of the Web which is not connected with the giant connected component. Figure 25 shows the connectivity structure of the Web, as well as sizes of the considered components. As one can see from the figure, the sizes of SCC, In, Out and Tendrils components are roughly equal, and the Disconnected component is significantly smaller. % Broder et al. [37] have also computed the diameters of the SCC and of the whole graph. It was shown that the diameter of the SCC is at least 28, and the diameter of the whole graph is at least 503. The average connected distance is defined as the pairwise distance averaged over those directed pairs (i,j) of nodes for which there exists a path from i to j. The average connected distance of the whole graph was estimated as 16.12 for inlinks, 16.18 for outlinks, and 6.83 for undirected links. Interestingly, it was also found that for a randomly chosen directed pair of nodes, the chance that there is a directed path between them is only about 2!' . 2.1.2 External Memory Algorithms In many cases, the data associated with massive graphs is too large to fit entirely inside the fast computer's internal memory, therefore a slower external memory (for example disks) needs to be used. The input/output communication (I/O) between these memories can result in an algorithm's slow performance. External memory (EM) algorithms and data structures are designed with aim to reduce the I/O cost by exploiting the locality. Recently, external memory algorithms have been successfully applied for solving batched problems involving graphs, including connected components, topological sorting, and shortest paths. The first EM graph algorithm was developed by Ullman and Yannakakis [112] in 1991 and dealt with the problem of transitive closure. ,ii;: other researchers contributed to the progress in this area ever since [1, 15, 16, 39, 42, 83, 115]. Chi ang et al. [42] proposed several new techniques for design and analysis of efficient EM graph algorithms and discussed applications of these techniques to specific problems, including minimum spanning tree verification, connected and biconnected components, graph drawing, and visibility representation. Abello et al. [1] proposed a functional approach for EM graph algorithms and used their methodology to develop deterministic and randomized algorithms for computing connected com ponents, maximal independent sets, maximal matching, and other structures in the graph. In this approach each algorithm is defined as a sequence of functions, and the computation continues in a series of scan operations over the data. If the produced output data, once written, cannot be changed, then the function is said to have no side effects. The lack of side effects enables the application of standard checkpointing techniques, thus increasing the reliability. Abello et al. presented a semiexternal model for graph problems, which assumes that only the vertices fit in the computer's internal memory. This is quite common in practice, and in fact this was the case for the call graph described in Subsection 2.1.1, for which efficient EM algorithms developed by Abello et al. [1] were used in order to compute its connected components [2]. For more detail on external memory algorithms see the book [4] and the extensive review by Vitter [115] of EM algorithms and data structures. 2.1.3 Modeling Massive Graphs The size of reallife massive graphs, many of which cannot be held even by a computer with several gigabytes of main memory, vanishes the power of classical algorithms and makes one look for novel approaches. External memory algorithms and data structures discussed in the previous subsection represent one of the research directions aiming to overcome difficulties created by data sizes. But in some cases not only is the amount of data huge, but the data itself is not completely available. For instance, one can hardly expect to collect complete information about the Web graph; in fact, the largest search engines are estimated to cover only 3'. of the Web [84]. Therefore, to investigate reallife massive graphs, one needs to use the available information in order to construct proper theoretical models of these graphs. One of the earliest attempts to model real networks theoretically goes back to the late 1950's, when the foundations of random graph theory had been developed. In this subsection we will present some of the results produced by this and other (more realistic) graph models. 2.1.3.1 Uniform Random Graphs The classical theory of random graphs founded by Erd6s and R6nyi [47, 48] deals with several standard models of the socalled ;, iu .[rm random graphs. Two of such models are G(n, m) and g(n,p) [30]. The first model assigns the same probability to all graphs with n vertices and m edges, while in the second model each pair of vertices is chosen to be linked by an edge randomly and independently with probability p. In most cases for each natural n a probability space consisting of graphs with exactly n vertices is considered, and the properties of this space as n i 0 are studied. It is said that a typical element of the space or almost every (a.e.) graph has property Q when the probability that a random graph on n vertices has this property tends to 1 as n  oo. We will also v that the property Q holds i;/,i,'..1. 'i ll1i almost surely (a.a.s.). Erd6s and R6nyi discovered that in many cases either almost every graph has property Q or almost every graph does not have this property. Many properties of uniform random graphs have been well studied [29, 30, 73, 80]. Below we will summarize some known results in this field. Probably the simplest property to be considered in any graph is its connec S.: .:/;, It was shown that for a uniform random graph G(n,p) E G(n,p) there is a I !i, ! 1'" value of p that determines whether a graph is almost surely connected or not. More specifically, a graph G(n,p) is a.a.s. disconnected ifp < lon. Fur thermore, it turns out that ifp is in the range I < p < ,gn the graph G(n,p) a.a.s. has a unique .i.:rl connected component [30]. The emergence of a giant connected component in a random graph is very often referred to as the p!i .i transition". The next subject of our discussion is the diameter of a uniform random graph G(n,p). Recall that the diameter of a disconnected graph is defined as the maximum diameter of its connected components. When dealing with random graphs, one usually speaks not about a certain diameter, but rather about the distribution of the possible values of the diameter. Intuitively, one can i, that this distribution depends on the interrelationship of the parameters of the model n and p. However, this dependency turns out to be rather complicated. It was discussed in many papers, and the corresponding results are summarized below. It was proved by Klee and Larman [77] that a random graph ..imptotically almost surely has the diameter d, where d is a certain integer value, if the following conditions are satisfied d1 d p p  0 and oo, 00. n n Bollobds [30] proved that if np log n i o then the diameter of a random graph is a.a.s. concentrated on no more than four values. Luczak [87] considered the case np < 1, when a uniform random graph a.a.s. is disconnected and has no giant connected component. Let diamT(G) denote the maximum diameter of all connected components of G(n,p) which are trees. Then if (1 np)n1/3  o the diameter of G(n,p) is a.a.s. equal to diamr(G). Chuing and Lu [43] investigated another extreme case: np  o. They showed that in this case the diameter of a random graph G(n,p) is a.a.s. equal to log n (1 + o(1)) . log(up) Moreover, they considered the case when np > c > 1 for some constant c and got a generalization of the above result: 10c 1 log n logn (C + 1) logn (1 + o(1)) < diam(G n, p)) < + 2 ) + 1. log(np) log(np) c log(2c) np Also, they explored the distribution of the diameter of a random graph with respect to different ranges of the ratio np/log n. They obtained the following results: * For np/logn = c > 8 the diameter of G(n,p) is a.a.s. concentrated on at most two values at log n/ log(np). For 8 > np/logn c > 2 the diameter of G(n,p) is a.a.s. concentrated on at most three values at log n/ log(np). For 2 > np/logn c > 1 the diameter of G(n,p) is a.a.s. concentrated on at most four values at log n/log(np). * For 1 > np/logn c > co the diameter of G(n,p) is a.a.s. concentrated on a finite number of values, and this number is at most 2 L + 4. More specifically, in this case the following formula can be proved: ( f33C 122 rlog((,,/11) log 0 nlognT 1 i \< diam(G(n,p)) < [ g p)) L + 2 +2. l og(np) l og(np) co As pointed out above, a graph G(n,p) a.a.s. has a giant connected component for 1 < np < log n. It is natural to assume that in this case the diameter of G(n,p) is equal to the diameter of this giant connected component. However, it was strictly proved by Chung and Lu [43] that it is a.a.s. true only if np > 3.5128. 2.1.3.2 Potential Drawbacks of the Uniform Random Graph Model There were some attempts to model the reallife massive graphs by the uniform random graphs and to compare their behavior. However, the results of these experiments demonstrated a significant discrepancy between the properties of real graphs and corresponding uniform random graphs. The further discussion analyzes the potential drawbacks of applying the uniform random graph model to the reallife massive graphs. Though the uniform random graphs demonstrate some properties similar to the reallife massive graphs, many problems arise when one tries to describe the real graphs using the uniform random graph model. As it was mentioned above, a giant connected component a.a.s. emerges in a uniform random graph at a certain threshold. It looks very similar to the properties of the real massive graphs discussed in Subsection 2.1.3. However, after deeper insight, it can be seen that the giant connected components in the uniform random graphs and the reallife massive graphs have different structures. The fundamental difference between them is as follows: it was noticed that in almost all the real massive graphs the property of socalled clustering takes place [116, 117]. It means that the probability of the event that two given vertices are connected by an edge is higher if these vertices have a common neighbor (i.e., a vertex which is connected by an edge with both of these vertices). The probability that two neighbors of a given vertex are connected by an edge is called the clustering coefficient. It can be easily seen that in the case of the uniform random graphs, the clustering coefficient is equal to the parameter p, since the probability that each pair of vertices is connected by an edge is independent of all other vertices. In reallife massive graphs, the value of the clustering coefficient turns out to be much higher than the value of the parameter p of the uniform random graphs with the same number of vertices and edges. Adamic [5] found that the value of the clustering coefficient for some part of the Web graph was approximately 0.1078, while the clustering coefficient for the corresponding uniform random graph was 0.00023. PastorSatorras et al. [103] got similar results for the part of the Internet graph. The values of the clustering coefficients for the real graph and the corresponding uniform random graph were 0.24 and 0.0006 respectively. Another significant problem arising in modeling massive graphs using the uniform random graph model is the difference in degree distributions. It can be shown that as the number of vertices in a uniform random graph increases, the distribution of the degrees of the vertices tends to the wellknown Poisson distribution with the parameter np which represents the average degree of a vertex. However, as it was pointed out in Subsection 2.1.3, the experiments show that in the real massive graphs degree distributions obey a power law. These facts demonstrate that some other models are needed to better describe the properties of real massive graphs. Next, we discuss two of such models; namely, the random graph model with a given degree sequence and its most important special case the powerlaw model. 2.1.3.3 Random Graphs with a Given Degree Sequence Besides the uniform random graphs, there are more general viv of modeling massive graphs. These models deal with random '.irl,' with a given degree sequence. The main idea of how to construct these graphs is as follows. For all the vertices i 1 ... n the set of the degrees {ki} is specified. This set is chosen so that the fraction of vertices that have degree k tends to the desired degree distribution pk as n increases. It turns out that some properties of the uniform random graphs can be generalized for the model of a random graph with a given degree sequence. Recall the notation of socalled 1ph  transition" (i.e., the phenomenon when at a certain point a giant connected component emerges in a random graph) which happens in the uniform random graphs. It turns out that a similar thing takes place in the case of a random graph with a given degree sequence. This result was obtained by Molloy and Reed [98]. The essence of their findings is as follows. Consider a sequence of nonnegative real numbers po, pi, ..., such that ~ pk 1. Assume that a graph G with n vertices has approximately pkn vertices k of degree k. If we define Q k>1 k(k 2)pk then it can be proved that G a.a.s. has a giant connected component if Q > 0 and there is a.a.s. no giant connected component if Q < 0. As a development of the analysis of random graphs with a given degree se quence, the work of Cooper and Frieze [45] should be mentioned. They considered a sparse directed random graph with a given degree sequence and analyzed its strong connectivity. In the study, the size of the giant strongly connected compo nent, as well as the conditions of its existence, were discussed. The results obtained for the model of random graphs with a given degree sequence are especially useful because they can be implemented for some important special cases of this model. For instance, the classical results on the size of a connected component in uniform random graphs follow from the aforementioned fact presented by Molloy and Reed. Next, we present another example of applying this general result to one of the most practically used random graph models the powerlaw model. 2.1.3.4 PowerLaw Random Graphs One of the most important special cases of the model of random graphs with a given degree sequence is the powerlaw random j,'li model, which represents the class of random graphs with a powerlaw degree sequence. This models theoretically describes the properties of powerlaw graphs that were mentioned above. Some important results for this model were obtained by Aiello, Chung and Lu [8, 9]. The powerlaw random graph model (also referred to P(a, 3) assigns two parameters characterizing a powerlaw random graph. If we define y to be the number of nodes with degree x, then according to this model y = ec/X/ (21) Equivalently, we can write logy = a logrx. (22) Similarly to formulas in Chapter 1, the relationship between y and x can be plotted as a straight line on a loglog scale, so that (3) is the slope, and a is the intercept. The following properties of a graph described by the powerlaw random graph model [8] are valid: * The maximum degree of the graph is e''. * The number of vertices is a (( )et, >l, n P j ,a,& 1, (23) X 1 e /(1 3),0 < 3< 1, where ((t) is the Riemann Zeta function. n= * The number of edges is 1((0 1)ea, > 2, 1 I 1(/32 ,)e /3 > 2, xE 1e 2, (2 4) i_ I e'C /(2/3),0 Since the powerlaw random graph model is a special case of the model of a random graph with a given degree sequence, the results discussed above can be applied to the powerlaw graphs. We need to find the threshold value of 3 in which the 1i! .i transition" (i.e., the emergence of a giant connected component) occurs. In this case Q = Y> x(x 2)p4 is defined as Ca C3 C3 Q x(x 2)LYj 23: ' [((0 2) 26((0 1)]1e for X1 X1 X1 /3>3. Hence, the threshold value Oo can be found from the equation (( / 2) 2(( 1) 0, which yields 3o 3.47875. The results on the size of the connected component of a powerlaw graph were presented by Aiello et al [8]. These results are summarized below. * If 0 < 3 < 1, then a powerlaw graph is a.a.s. connected (i.e., there is only one connected component of size n). * If 1 < 3 < 2, then a powerlaw graph a.a.s. has a giant connected component (the component size is O(n)), and the second largest connected component a.a.s. has a size 0(1). * If 2 < 3 < /o = 3.47875, then a giant connected component a.a.s. exists, and the size of the second largest component a.a.s. is O (log n). * = 2 is a special case when there is a.a.s. a giant connected component, and the size of the second largest connected component is 0(log n/log log n). * If 3 > o = 3.47875, then there is a.a.s. no giant connected component. The powerlaw random graph model was developed for describing reallife massive graphs. So the natural question is how well it reflects the properties of these graphs. Though this model certainly does not reflect all the properties of real massive graphs, it turns out that the massive graphs such as the call graph or the Internet graph can be fairly well described by the powerlaw model. The following example demonstrates it. Aiello, Clhuii and Lu [8] investigated the same call graph that was analyzed by Abello et al. [2]. This massive graph was already discussed in Subsection 2.1.3, so it is interesting to compare the experimental results presented by Abello et al. [2] with the theoretical results obtained in [8] using the powerlaw random graph model. Figure 22 shows the number of vertices in the call graph with certain in degrees and outdegrees. Recall that according to the powerlaw model the dependency between the number of vertices and the corresponding degrees can be plotted as a straight line on a loglog scale, so one can approximate the real data shown in Figure 22 by a straight line and evaluate the parameter a and 3 using the values of the intercept and the slope of the line. The value of 3 for the indegree data was estimated to be approximately 2.1, and the value of e" was approximately 30 x 106. The total number of nodes can be estimated using formula (23) as ((2.1) x e" = 1.56 x e" 47 x 106 (compare with Subsection 2.1.3). According to the results for the size of the largest connected component presented above, a powerlaw graph with 1 < 3 < 3.47875 a.a.s. has a giant connected component. Since j3 w 2.1 falls in this range, this result exactly coincides with the real observations for the call graph (see Subsection 2.1.3). Another aspect that is worth mentioning is how to generate powerlaw graphs. The methodology for doing it was discussed in detail in the literature [9, 44]. These papers use a similar approach, which is referred to as a random .,j'l, evolution process. The main idea is to construct a powerlaw massive graph "stepbyl p : at each time step, a node and an edge are added to a graph in accordance with certain rules in order to obtain a graph with a specified indegree and outdegree powerlaw distribution. The indegree and outdegree parameters of the resulting powerlaw graph are functions of the input parameters of the model. A simple evolution model was presented by Kumar et al. [81]. Aiello, Chuing and Lu [9] developed four more advanced models for generating both directed and undirected powerlaw graphs with different distributions of indegrees and outdegrees. As an example, we will briefly describe one of their models. It was the basic model developed in the paper, and the other three models actually were improvements and generalizations of this model. The main idea of the considered model is as follows. At the first time moment a vertex is added to the graph, and it is assigned two parameters the inweight and the outweight, both equal to 1. Then at each time step t + 1 a new vertex with inweight 1 and outweight 1 is added to the graph with probability 1 a, and a new directed edge is added to the graph with probability a. The origin and destination vertices are chosen according to the current values of the inweights and outweights. More specifically, a vertex u is chosen as the origin of this edge with the probability proportional to its current outweight which is defined as wt = 1 + 6t where 6Pt is the outdegree of the vertex u at time t. Similarly, a vertex v is chosen as the destination with the probability proportional to its current inweight '. = 1 + 6^ where 6' is the indegree of v at time t. From the above description it can be seen that at time t the total inweight and the total outweight are both equal to t. So for each particular pair of vertices u and v, the probability that an edge going from u to v is added to the graph at time t is equal to (1 67)D(1 (t ) t2 In the above notations, the parameter a is the input parameter of the model. The output of this model is a powerlaw random graph with the parameter of the degree distribution being a function of the input parameter. In the case of the considered model, it was shown that it generates a powerlaw graph with the distribution of indegrees and outdegrees having the parameter 1 + 1. The notion of the socalled scale invariance [20, 21] must also be mentioned. This concept arises from the following considerations. The evolution of massive graphs can be treated as the process of growing the graph at a time unit. Now, if we replace all the nodes that were added to the graph at the same unit of time by only one node, then we will get another graph of a smaller size. The bigger the time unit is, the smaller the new graph size will be. The evolution model is called scalefree (scaleinvariant) if with high probability the new (scaled) graph has the same powerlaw distribution of indegrees and outdegrees as the original graph, for any choice of the time unit length. It turns out that most of the random evolution models have this property. For instance, the models of Aiello et al. [9] were proved to be scaleinvariant. 2.1.4 Optimization in Random Massive Graphs Recent random graph models of reallife massive networks, some of which were mentioned in Subsection 2.1.3 increased interest in various properties of random graphs and methods used to discover these properties. Indeed, numerical characteristics of graphs, such as clique and chromatic numbers, could be used as one of the steps in validation of the proposed models. In this regard, the expected clique number of powerlaw random graphs is of special interest due to the results by Abello et al. [2] and Aiello et al. [9] mentioned in Subsections 2.1.1 and 2.1.3. If computed, it could be used as one of the points in verifying the validity of the model for the call graph proposed by Aiello et al. [9]. In this subsection we present some wellknown facts regarding the clique and chromatic numbers in uniform random graphs. 2.1.4.1 Clique Number The earliest results describing the properties of cliques in uniform random graphs are due to Matula [93], who noticed that for a fixed p almost all graphs G E G(n,p) have about the same clique number, if n is sufficiently large. Bollobas and Erdbs [32] further developed these remarkable results by proving some more specific facts about the clique number of a random graph. Let us discuss these results in more detail by presenting not only the facts but also some reasoning behind them. For more detail see books by Bollobas [29, 30] and Janson et al. [73]. Assume that 0 < p < 1 is fixed. Then instead of the sequence of spaces {(n, p),n > 1} one can work with the single probability space g(N,p) containing graphs on N with the edges chosen independently with probability p. In this way, (n, p) becomes an image of g(N,p), and the term "almost ( iy is used in its usual measuretheory sense. For a graph G E g(N,p) we denote by G, the subgraph of G induced by the first n vertices {1, 2,..., n}. Then the sequence w(G,) appears to be almost completely determined for a.e. G E (N,p). For a natural 1, let us denote by ki(G,) the number of cliques spanning I vertices of G,. Then, obviously, w(G) = max{l: ki(G) > 0}. When I is small, the random variable ki(G,) has a large expectation and a rather small variance. If I is increased, then for most values of n there exists some number lo for which the expectation of ko (G,) is fairly large (> 1) and k0o+l(G,) is much smaller than 1. Therefore, if we find this value l0 then w(G,) = lo with a high probability. The expectation of kl(G,) can be calculated as E(k(G,)) =( p Denoting by f(l1) E(k,(G,)) and replacing (n) by its Stirling approximation we obtain n n+1/2 f(1) ] 2 1(11)/2 V/(n 1)nl+1/211+1/2 Solving the equation f(1) = 1 we get the following approximation l0 of the root: lo 2log/ n 2log/plog/pn + 21og/,(e/2) + + o(1) (25) S2log/ n + O(log log n). Using this observation and the second moment method, Bollobds and Erdos [32] proved that if p = p(n) satisfies n' < p < c for every c and some c < 1, then there exists a function cl : N N such that a.a.s. cl(n) < w(G,) < cl(n) + l, i.e., the clique number is .i mptotically distributed on at most two values. The sequence cl(n) appears to be close to lo(n) computed in (25). Namely, it can be shown that for a.e. G E g(N,p) if n is large enough then [lo(n) 2 log log nlog n < (G,) < Llo(n) + 2 log log n/ log n] and w(G,) 21og/ n + 21og/p logpn n 21og/p(e/2) 1 < . Frieze [56] and Janson et al. [73] extended these results by showing that for c > 0 there exists a constant ce, such that for c < pp(n) < log2 n a.a.s. [21og /n 2log,/p log,/ n + 21og/ (e/2) + 1 c/p] < w(G,) < L2 log/p n 2 log/p log/p n +21log,/p(e/2)+1+ c/p]. 2.1.4.2 Chromatic Number Grimmett and McDiarmid [61] were the first to study the problem of coloring random graphs. Many other researchers contributed to solving this problem [12, 31]. We will mention some facts emerged from these studies. Luczak [85] improved the results about the concentration of X(G(n,p)) previously proved by Shamir and Spencer [110], proving that for every sequence p = p(n) such that p < n6/7 there is a function ch(n) such that a.a.s. ch(n) < x(G(n,p)) < ch(n) + 1. Alon and Krivelevich [12] proved that for any positive constant 6 the chromatic number of a uniform random graph G(n,p), where p = n2, is a.a.s. concentrated in two consecutive values. Moreover, they proved that a proper choice of p(n) may result in a onepoint distribution. The function ch(n) is difficult to find, but in some cases it can be characterized. For example, Janson et al. [73] proved that there exists a constant co such that for any p p(n) satisfying C0 < p < log7 n a.a.s. np V np 2 log np 2 log log np + 1 2 log np 40 log log np In the case when p is constant Bollob&s' method utilizing martingales [30] yields the following estimate: xG(n ) 2logbn 2logblogbn+ O(1)' where b 1/(1 p). 2.1.5 Remarks We discussed advances in several research directions dealing with massive graphs, such as external memory algorithms and modeling of massive networks as random graphs with powerlaw degree distributions. Despite the evidence that uniform random graphs are hardly suitable for modeling the considered reallife graphs, the classical random graphs theory still may serve as a great source of ideas in studying properties of massive graphs and their models. We recalled some wellknown results produced by the classical random graphs theory. These include results for concentration of clique number and chromatic number of random graphs, which would be interesting to extend to more complicated random graph models (i.e., powerlaw graphs and graphs with arbitrary degree distributions). External memory algorithms and numerical optimization techniques could be applied to find an approximate value of the clique number (as it was discussed in Subsection 2.1.1). On the other hand, probabilistic methods similar to those discussed in Subsection 2.1.4 could be utilized in order to find the .,vmptotical distribution of the clique number in the same network's random graph model, and therefore verify this model. CHAPTER 3 NETWORKBASED APPROACHES TO MINING STOCK MARKET DATA One of the most important problems in the modern finance is finding efficient v,I of summarizing and visualizing the stock market data that would allow one to obtain useful information about the behavior of the market. Nowad, , a great number of stocks are traded in the US stock market; moreover, this number steadily increases. The amount of data generated by the stock market every dv is enormous. This data is usually visualized by thousands of plots reflecting the price of each stock over a certain period of time. The analysis of these plots becomes more and more complicated as the number of stocks grows. It turns out that the stock market data can be effectively represented as a network, although this representation is not so obvious as in the case of telephone traffic or internet data. We have developed the networkbased model of the market referred to as the market pi,''l, This chapter is based on the results described in [26, 27, 28]. A natural graph representation of the stock market is based on the cross correlations of price fluctuations. A market graph can be constructed as follows: each financial instrument is represented by a vertex, and two vertices are connected by an edge if the correlation coefficient of the corresponding pair of instruments (calculated for a certain period of time) exceeds a specified threshold 0, 1 < 0 < 1. Nowadl, a great number of different instruments are traded in the US stock market, so the market graph representing them is very large. The market graph that we construct has 6546 vertices and several million edges. In this chapter, we present a detailed study of the properties of this graph. It turns out that the market graph can be rather accurately described by the power law model. We an iv. .. the distribution of the degrees of the vertices in this graph, the edge density of this graph with respect to the correlation threshold, as well as its connectivity and the size of its connected components. Furthermore, we look for maximum cliques and maximum independent sets in this graph for different values of the correlation threshold. Analyzing cliques and independent sets in the market graph gives us a very valuable knowledge about the internal structure of the stock market. For instance, a clique in this graph represents a set of financial instruments whose prices change similarly over time (a change of the price of any instrument in a clique is likely to affect all other instruments in this clique), and an independent set consists of instruments that are negatively correlated with respect to each other; therefore, it can be treated as a diver.:l;. portfolio. Based on the information obtained from this analysis, we will be able to classify financial instruments into certain groups, which will give us a deeper insight into the stock market structure. 3.1 Structure of the Market Graph 3.1.1 Constructing the Market Graph The market graph that we study in this chapter represents the set of financial instruments traded in the US stock markets. More specifically, we consider 6546 instruments and analyze daily changes of their prices over a period of 500 consecutive trading d,, in 20002002. Based on this information, we calculate the crosscorrelations between each pair of stocks using the following formula [92]: j (R Rj) (R)(R ) (R2f \(R)2) (R (Rj2) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 31. Distribution of correlation coefficients in the stock market where Ri(t) In Pt) defines the return of the stock i for d t. Pi(t) denotes the price of the stock i on di t. The correlation coefficients Ci can vary from 1 to 1. Figure 31 shows the distribution of the correlation coefficients based on the prices data for the years 20002002. It can be seen that this plot has a shape similar to the normal distribution with the mean 0.05. The main idea of constructing a market graph is as follows. Let the set of financial instruments represent the set of vertices of the graph. Also, we specify a certain threshold value 0, 1 < 0 < 1 and add an undirected edge connecting the vertices i and j if the corresponding correlation coefficient Ci is greater than or equal to 0. Obviously, different values of 0 define the market graphs with the same set of vertices, but different sets of edges. It is easy to see that the number of edges in the market graph decreases as the threshold value 0 increases. In fact, our experiments show that the edge density 0.07 0.06 0.05 0.04 36 60.00% 50.00% 40.00% S30.00% C 20.00% 10.00% 0.00%  0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 correlation threshold Figure 32. Edge density of the market graph for different values of the correlation threshold. of the market graph decreases exponentially w.r.t. 0. The corresponding graph is presented on Figure 37. 3.1.2 Connectivity of the Market Graph In Subsection 2.1.3 we mentioned the connectivity thresholds in random graphs. The main idea of this concept is finding a threshold value of the parameter of the model that will define if the graph is connected or not. A similar question arises for the market graph: what is its connectivity threshold? Since the number of edges in the market graph depends on the chosen correlation threshold 0, we should find a value 00 that determines the connectivity of the graph. As it was mentioned above, the smaller value of 0 we choose, the more edges the market graph will have. So, if we decrease 0, after a certain point, the graph will become connected. We have conducted a series of computational o 7000 S6000  0 6 5000 T 4000  o E 3000 0 S 2000 0 0 1 0.90.80.70.6 0.5 0.40.3 0.20.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 correlation threshold Figure 33. Plot of the size of the largest connected component in the market graph as a function of correlation threshold 0. experiments for checking the connectivity of the market graph using the breadth first search technique, and we obtained a relatively accurate approximation of the connectivity threshold: 00 0.14382. Moreover, we investigated the dependency of the size of the largest connected component in the market graph w.r.t. 0. The corresponding plot is shown in Figure 33. 3.1.3 Degree Distribution of the Market Graph The next important subject of our interest is the distribution of the degrees of the vertices in the market graph. We have conducted several computational experiments with different values of the correlation threshold 0, and these results are presented below. It turns out that if a small (in absolute value) correlation threshold 0 is spec ified, the distribution of the degrees of the vertices does not have any welldefined structure. Note that for these values of 0 the market graph has a relatively high edge density (i.e. the ratio of the number of edges to the maximum possible number of edges). However, as the correlation threshold is increased, the degree Table 31. Leastsquares estimates of the parameter 7 in the market graph for different values of correlation threshold (* complementary graph) 0 7 0.25* 1.2922 0.2* 1.4088 0.15* 1.4072 0.2 0.4931 0.25 0.5820 0.3 0.6793 0.35 0.7679 0.4 0.8269 0.45 0.8753 0.5 0.9054 0.55 0.9331 0.6 0.9743 distribution more and more resembles a power law. In fact, for 0 > 0.2 this distri bution is approximately a straight line in the logarithmic scale, which represents the powerlaw distribution, as it was mentioned above. Figure 34 demonstrates the degree distributions of the market graph for some positive values of the correla tion threshold, along with the corresponding linear approximations. The slopes of the approximating lines were estimated using the leastsquares method. Table 31 summarizes the estimates of the parameter 7 of the powerlaw distribution (i.e., the slope of the line) for different values of 0. From this table, it can be seen that the slope of the lines corresponding to positive values of 0 is rather small. According to the powerlaw model, in this case a graph would have many vertices with high degrees, therefore, one can intuitively expect to find large cliques in a powerlaw graph with a small value of the parameter 7. We also analyze the degree distribution of the complement of the market graph, which is defined as follows: an edge connects instruments i and j if the correlation coefficient between them Ci < 0. Studying this complementary graph is important for the next subject of our consideration finding maximum independent Figure 34. Degree distribution of the market graph for 0 = 0.4 (left); 0 = 0.5 (right) (logarithmic scale) sets in the market graph with negative values of the correlation threshold 0. Obviously, a maximum independent set in the initial graph is a maximum clique in the complement, so the maximum independent set problem can be reduced to the maximum clique problem in the complementary graph. Therefore, it is useful to investigate the degree distributions of the complementary graphs for different values of 0. As it can be seen from Figure 31, the distribution of the correlation coefficients is nearly symmetric around 0 = 0.05, so for the values of 0 close to 0 the edge density of both the initial and the complementary graph is high enough. For these values of 0 the degree distribution of a complementary graph also does not seem to have any welldefined structure, as in the case of the corresponding initial graph. As 0 decreases (i.e., increases in the absolute value), the degree distribution of a complementary graph starts to follow the power law. Figure 35 shows the degree distributions of the complementary graph, along with the leastsquares linear regression lines. However, as one can see from Table 31, the slopes of these lines are higher than in the case of the graphs with positive values of 0, which implies that there are fewer vertices with a high degree in these graphs, so intuitively, the size of a cliques in a complementary graph (i.e., the size 2 3 4 Degree Figure 35. Degree distribution of the complementary market graph for = 0.15 (left); 0 = 0.2 (right) (logarithmic scale) of independent sets in the original graph) should be significantly smaller than in the case of the market graph with positive values of the correlation threshold (see Section 3.2). 3.1.4 Instruments Corresponding to HighDegree Vertices Up to this point, we studied the properties of the market graph as one big system, and did not consider the characteristics of every vertex in this graph. However, an important practical issue is to look at the degree of each vertex in the market graph and to find the vertices with high degrees, i.e. the stocks that are highly correlated with many other instruments in the market. Clearly, this information will help us to answer the question: which instruments most accurately reflect the behavior of the market? For this purpose, we chose the market graph with a high correlation threshold (8 = 0.6), calculated the degrees of each vertex in this graph and sorted the vertices in the decreasing order of their degrees. 0 1 2 3 4 Degree 0 1 2 3 4 Degree Interestingly, even though the edge density of the considered graph is only 0.0 !'. (only highly correlated instruments are connected by an edge), there are many vertices with degrees greater than 100. According to our calculations, the vertex with the highest degree in this market graph corresponds to the NASDAQ 100 Index Tracking Stock. The degree of this vertex is 216, which means that there are 216 instruments that are highly correlated with it. An interesting observation is that the degree of this vertex is twice higher than the number of companies whose stock prices the NASDAQ index reflects, which means that these 100 companies greatly influence the market. In Table 32 we present the "top 25" instruments in the U.S. stock mar ket, according to their degrees in the considered market graph. The corre sponding symbols definitions can be found on several websites, for example http://www.nasdaq.com. Note that most of them are indices that incorporate a number of different stocks of the companies in different industries. Although this result is not surprising from the financial point of view, it is important as a practical justification of the market graph model. 3.1.5 Clustering Coefficients in the Market Graph Next, we calculate the clustering coefficients in the original and complemen tary market graphs for different values of 0. The clustering coefficient is defined as the probability that for a given vertex its two neighbors are connected by an edge. Interestingly, clustering coefficients in the original market graph are large even for high correlation thresholds, however, in the complementary graphs with a negative correlation threshold the values of the clustering coefficient turned out to be very close to 0. These results are summarized in Table 33. For instance, as one can see from this table, the market graph with 0 = 0.6 has almost the same edge density as the complementary market graph with 0 0.15, however, their clustering coefficients differ dramatically. This fact also intuitively explains the Table 32. Top 25 instruments with highest degrees in the market graph (0 = 0.6) symbol vertex degree QQQ IWF IWO IYW XLK IVV MDY SPY IJH IWV IVW IAH IYY IWB IYV BDH MKH IWM IJR SMH STM IIH IVE DIA IWD 216 193 193 193 181 175 171 162 159 158 156 155 154 153 150 144 143 142 134 130 118 116 113 106 106 results presented in the next section, which deals with cliques and independent sets in the market graph. 3.2 Analysis of Cliques and Independent Sets in the Market Graph In this section, we discuss the methods of finding maximum cliques and maximum independent sets in the market graph and analyze the obtained results. The maximum clique problem (as well as the maximum independent set problem) is known to be NPhard [59]. Moreover, it turns out that the maximum clique is difficult to approximate [18, 62]. This makes these problems especially challenging in large graphs. However, as we will see in the next subsection, even Table 33. Clustering coefficients of the market graph (* complementary graph) 0 edge density clustering coef. 0.15* 0.0005 2.64 x 105 0.1* 0.0050 0.0012 0.3 0.0178 0.4885 0.4 0.0047 0.4458 0.5 0.0013 0.4522 0.6 0.0004 0.4872 0.7 0.0001 0.4886 though the maximum clique problem is generally very hard to solve in large graphs, the special structure of the market graph allows us to find the exact solution relatively easily. 3.2.1 Cliques in the Market Graph In this subsection, we consider cliques in the market graph, which have a clear interpretation in terms of finance. Since a clique is a set of completely interconnected vertices, any stock that belongs to the clique is highly correlated with all other stocks in this clique; therefore, a stock is assigned to a certain group only if it demonstrates a behavior similar to all other stocks in this group. Clearly, the size of the maximum clique is an important characteristic of the stock market, since it represents the maximum possible group of similar objects (i.e., mutually correlated stocks). A standard integer programming formulation [33] was used to compute the exact maximum clique in the market graph, however, before solving this problem, we applied a greedy heuristic for finding a lower bound of the clique number, and a special preprocessing technique which reduces the problem size. To find a large clique, we apply the "bestin" greedy algorithm based on degrees of vertices. Let C denote the clique. Starting with C = 0, we recursively add to the clique a vertex vmax of largest degree and remove all vertices that are not .,.i i: ent to vmax from the graph. After running this algorithm, we applied the following preprocessing procedure [2]. We recursively remove from the graph all of the vertices which are not in C and whose degree is less than ICI, where C is the clique found by the greedy algorithm. Denote by G' = (V', E') the graph induced by remaining vertices. Then the maximum clique problem can be formulated and solved for G'. The following integer programming formulation was used [33]: Iv'I maximize x i= 1 s.t. xi + x < (i,j) E' xi C {0, 1} It should be noted that in the case of market graph instances with a high positive correlation threshold, the aforementioned preprocessing procedure is very efficient and significantly reduces the number of vertices in a graph [26]. This can be intuitively explained by the fact that these instances of the market graph are clustered (i.e. two vertices in a graph are more likely to be connected if they have a common neighbor), so the clustering coefficient, which is defined as the probability that for a given vertex its two neighbors are connected by an edge, is much higher than the edge density in these graphs (see Table 38). This characteristic is also typical for other powerlaw graphs arising in different applications. After reducing the size of the original graph, the resulting integer programming problem for finding a maximum clique can be relatively easily solved using the CPLEX integer programming solver [71]. Table 34 summarizes the exact sizes of the maximum cliques found in the market graph for different values of 0. It turns out that these cliques are rather large, which agrees with the analysis of degree distributions and clustering coefficients in the market graphs with positive values of 0. Table 34. Sizes of the maximum cliques in the market graph with positive values of the correlation threshold (exact solutions) 0 edge density clique size 0.35 0.0090 193 0.4 0.0047 144 0.45 0.0024 109 0.5 0.0013 85 0.55 0.0007 63 0.6 0.0004 45 0.65 0.0002 27 0.7 0.0001 22 These results show that in the modern stock market there are large groups of instruments whose price fluctuations behave similarly over time, which is not surprising, since 1i. ,1 is different branches of economy highly affect each other. 3.2.2 Independent Sets in the Market Graph Here we present the results of solving the maximum independent set problem in the market graphs with nonpositive values of the correlation threshold 0. As it was pointed out above, this problem is equivalent to the maximum clique problem in a complementary graph. However, the preprocessing procedure that was very helpful for finding maximum cliques in the original graph could not eliminate any vertices in the case of the complement, and we were not able to find the exact solution of the maximum independent set problem in this case. Recall that the clustering coefficients in the complementary graph were very small, which intuitively explains the failure of the preprocessing procedure. Therefore, solving the maximum independent set in the market graph is more challenging than finding the maximum clique. Table 35 presents the sizes of the independent sets found using the greedy heuristic that was described in the previous section. Table 35. Sizes of independent sets in the complementary market graph found using the greedy algorithm (lower bounds) 0 edge density indep. set size 0.05 0.4794 45 0.0 0.2001 12 0.05 0.0431 5 0.1 0.005 3 0.15 0.0005 2 This table demonstrates that the sizes of computed independent sets are rather small, which is in agreement with the results of the previous section, where we mentioned that in the complementary graph the values of the parameter of the powerlaw distribution are rather high, and the clustering coefficients are very small. The small size of the computed independent sets means that finding a large "completely diversified" portfolio (where all instruments are negatively correlated to each other) is not an easy task in the modern stock market. Moreover, it turns out that one can make a theoretical estimation of the maximum size of a diversified portfolio, where all stocks are strictly negatively correlated with each other. Intuitively, the lower (higher by the absolute value) threshold 0 we set, the smaller diversified portfolio one would expect to find. These considerations are confirmed by the following theorem. Theorem 3.1. Consider a market ',jI'l,' with the correlation threshold 0 < 0. Assume that each stock's return has a finite variance. Then there is no independent set (diver'il; portfolio) of a size greater than 1 + 1 Proof. Let a random variable Xi denote the return of stock i at some time moment, of denote the variance of Xi, and jmax = maxi ui. Suppose that there are m stocks, which are pairwise negatively correlated, i.e., C, < 0,Vi,j =1,... m, and the maximum correlation is 0 = rnr:: i Ci < 0. Consider the variance of the sum of these variables: Var(y X,) = Var(Xi) + Y Cov(Xi, X)  i= 1 i 1 i j S+ max + (m )0 n x(1 + 1)0) i= 1 ij Note that if 0 < 0, ma l(1 + (m 1)0) < 0 for m > 1 + 1. Consequently, Var(Z X,) < 0 for m > 1 + . Therefore, the number of stocks with pairwise correlations Cij < 0 < 0 cannot be greater than m = 1 + which completes the proof. Another natural question now arises: how many completely diversified portfolios can be found in the market? In order to find an answer, we have calculated maximal independent sets starting from each vertex, by running 6546 iterations of the greedy algorithm mentioned above. That is, for each of the considered 6546 financial instruments, we have found a completely diversified portfolio that would contain this instrument. Interestingly enough, for every vertex in the market graph, we were able to detect an independent set that contains this vertex, and the sizes of these independent sets were rather close. Moreover, all these independent sets were distinct. Figure 36 shows the frequency of the sizes of the independent sets found in the market graphs corresponding to different correlation thresholds. These results demonstrate that it is alvi possible for an investor to find a group of stocks that would form a completely diversified portfolio with any given stock, and this can be efficiently done using the technique of finding independent sets in the market graph. 4500 1400 3500 3000 1000 2000 600 1500 400 100500 8 S 200 1000 200 500 0 0 32 33 343536 3 383Z0 Ind. Set Size 12 Ind. Set Size 44 45 Figure 36. Frequency of the sizes of independent sets found in the market graph with 0 = 0.00 (left), and 0 = 0.05 (right) 3.3 Data Mining Interpretation of the Market Graph Model As we have seen, the analysis of the market graph provides a practically useful methodology of extracting information from the stock market data. In this subsection, we discuss the conceptual interpretation of this approach from the data mining perspective. An important aspect of the proposed model is the fact that it allows one to reveal certain patterns underlying the financial data, therefore, it represents a structured data mining approach. Nontrivial information about the global properties of the stock market is obtained from the analysis of the degree distribution of the market graph. Highly specific structure of this distribution si . I that the stock market can be analyzed using the powerlaw model, which can theoretically predict some characteristics of the graph representing the market. On the other hand, the analysis of cliques and independent sets in the mar ket graph is also useful from the data mining point of view. As it was pointed out above, cliques and independent sets in the market graph represent groups of ",ii i and "dIl. i. i l financial instruments, respectively. Therefore, informa tion about the size of the maximum cliques and independent sets is also rather important, since it gives one the idea about the trends that take place in the stock market. Besides analyzing the maximum cliques and independent sets in the mar ket graph, one can also divide the market graph into the smallest possible set of distinct cliques (or independent sets). Partitioning a dataset into sets (clusters) of elements grouped according to a certain criterion is referred to as clustering, which is one of the wellknown data mining problems [34]. As discussed above, the main difficulty one encounters in solving the clustering problem on a certain dataset is the fact that the number of desired clusters of similar objects is usually not known a priori, moreover, an appropriate ii,.lr,.ii:, criterion should be chosen before partitioning a dataset into clusters. Clearly, the methodology of finding cliques in the market graph provides an efficient tool of performing clustering based on the stock market data. The choice of the grouping criterion is clear and natural: o~ .I w'" financial instruments are determined according to the correlation between their price fluctuations. Moreover, the minimum number of clusters in the partition of the set of financial instruments is equal to the minimum number of distinct cliques that the market graph can be divided into (the minimum clique partition problem). Similar partition can be done using independent sets instead of cliques, which would represent the partition of the market into a set of distinct diversified portfolios. In this case the minimum possible number of clusters is equal to a partition of vertices into a minimum number of distinct independent sets. This problem is called the tlji', coloring problem, and the number of sets in the optimal partition is referred to as the chromatic number of the graph. We should also mention another in i, i type of data mining problems with many applications in finance. They are referred to as /1.i.:7 ,l.:.>n problems. Although the setup of this type of problems is similar to clustering, one should clearly understand the difference between these two types of problems. In classification, one deals with a predefined number of classes that the data elements must be assigned to. Also, there is a socalled tr ':.' :' dataset, i.e., the set of data elements for which it is known a priori which class they belong to. It means that in this setup one uses some initial information about the classification of existing data elements. A certain classification model is constructed based on this information, and the parameters of this model are 1iii, 1 to classify new data elements. This procedure is known as 11 ,iiiig the classifier". An example of the application of this approach to classifying financial instruments can be found in [40]. The main difference between classification and clustering is the fact that unlike classification, in the case of clustering, one does not use any initial information about the class attributes of the existing data elements, but tries to determine a classification using appropriate criteria. Therefore, the methodology of classifying financial instruments using the market graph model is essentially different from the approaches commonly considered in the literature in the sense that it does not require any apriori information about the classes that certain stocks belong to, but classifies them only based on the behavior of their prices over time. 3.4 Evolution of the Market Graph In the previous sections, we have discussed the properties of the market graph constructed for one 500d4v period. We have revealed a number of important properties of this model; however, another crucial question that needs to be answered is how these characteristics change over time. This analysis would provide more information about the patterns underlying the stock market dynamics. We address these issues in this section. In order to investigate the dynamics of the market graph structure, we chose the period of 1000 trading d4,va in 19982002 and considered eleven 500d4iv shifts within this period. The starting points of every two consecutive shifts are separated Table 36. Dates and mean correlations corresponding to each considered 500div shift Period # Starting date Ending date Mean correlation 1 09/24/1998 09/15/2000 0.0403 2 12/04/1998 11/27/2000 0.0373 3 02/18/1999 02/08/2001 0.0381 4 04/30/1999 04/23/2001 0.0426 5 07/13/1999 07/03/2001 0.0444 6 09/22/1999 09/19/2001 0.0465 7 12/02/1999 11/29/2001 0.0545 8 02/14/2000 02/12/2002 0.0561 9 04/26/2000 04/25/2002 0.0528 10 07/07/2000 07/08/2002 0.0570 11 09/18/2000 09/17/2002 0.0672 by the interval of 50 d,,i Therefore, every pair of consecutive shifts had 450 d,i in common and 50 d , different. Dates corresponding to each shift and the corresponding mean correlations are summarized in Table 36. This procedure allows us to accurately reflect the structural changes of the market graph using relatively small intervals between shifts, but at the same time one can maintain sufficiently large sample sizes of the stock prices data for calculating crosscorrelations for each shift. We should note that in our analysis we considered only stocks which were among those traded as of the last of the 1000 trading d,, i.e. for practical reasons we did not take into account stocks which had been withdrawn from the market. 3.4.1 Dynamics of Global Characteristics of the Market Graph In this subsection, we analyze the evolution of the basic characteristics of the market graph model that were considered above for one trading period: the distribution of the correlation coefficients in the market, the degree distribution, and the edge density. As we will see, some properties of the market graph remain stable; however, there are certain trends that can be observed in the stock market development. The first subject of our consideration is the distribution of correlation coeffi cients between all pairs of stocks in the market. As it was mentioned above, this distribution on [1, 1] had a shape similar to a part of normal distribution with mean close to 0.05 for the sample data considered in [26, 27]. One of the interpre tations of this fact is that the correlation of most pairs of stocks is close to zero, therefore, the structure of the stock market is substantially random, and one can make a reasonable assumption that the prices of most stocks change independently. As we consider the evolution of the correlation distribution over time, it turns out that the shape of this distribution remains stable, which is illustrated by Figure 37. 0.08 0.08 0.067 0.05 0.04 0.03 0.02 0.01 0 ( 5. I.@ 5. 4. 5. Q Z Q Q) Q). Q) Q Q) Q).  period 1  period 3 period 5 period7 period 9 period 11 Figure 37. Distribution of correlation coefficients in the US stock market for sev eral overlapping 500d4v periods during 20002002 (period 1 is the earliest, period 11 is the latest). The stability of the correlation coefficients distribution of the market graph intuitively motivates the hypothesis that the degree distribution should also remain stable for different values of the correlation threshold. To verify this assumption, 53 we have calculated the degree distribution of the graphs constructed for all considered time periods. The correlation threshold 0 = 0.5 was chosen to describe the structure of connections corresponding to significantly high correlations. Our experiments show that the degree distribution is similar for all time intervals, and in all cases it is well described by a power law. Figure 38 shows the degree distributions (in the logarithmic scale) for some instances of the market graph (with 0 = 0.5) corresponding to different intervals. (a) period 1 1 o00  100  1000 10000 1 10 100 degree (b) period 4 10000 1000 om m 100 10 10 1000 10000 1 10 100 degree (c) period 7 (d) period 11 Figure 38. Degree distribution of the market graph for different 500d4v periods in 20002002 with 0 = 0.5: (a) period 1, (b) period 4, (c) period 7, (d) period 11. The crosscorrelation distribution and the degree distribution of the market graph represent the general characteristics of the market, and the aforementioned 10C00 o10  Io  100  10 1 1 140 I 10 100 degree 10000 100 om m 100 10 N. 1 10 100 degree ~*4SiO 1000 10000 1000 10000 results lead us to the conclusion that the global structure of the market is stable over time. However, as we will see now, some global changes in the stock market structure do take place. In order to demonstrate it, we look at another characteris tic of the market graph its edge density. In our analysis of the market graph dynamics, we chose a relatively high correlation threshold 0 = 0.5 that would ensure that we consider only the edges corresponding to the pairs of stocks, which are significantly correlated with each other. In this case, the edge density of the market graph would represent the proportion of those pairs of stocks in the market, whose price fluctuations are similar and influence each other. The subject of our interest is to study how this proportion changes during the considered period of time. Table 37 summarizes the obtained results. As it can be seen from this table, both the number of vertices and the number of edges in the market graph increase as time goes. Obviously, the number of vertices grows since new stocks appear in the market, and we do not consider those stocks which ceased to exist by the last of 1000 trading di,4 used in our analysis, so the maximum possible number of edges in the graph increases as well. However, it turns out that the number of edges grows faster; therefore, the edge density of the market graph increases from period to period. As one can see from Figure 39(a), the greatest increase of the edge density corresponds to the last two periods. In fact, the edge density for the latest interval is approximately 8.5 times higher than for the first interval! This dramatic jump ii:: 1 that there is a trend to the "globalization" of the modern stock market, which means that nowad1, more and more stocks significantly affect the behavior of the others. It should be noted that the increase of the edge density could be predicted from the analysis of the distribution of the crosscorrelations between all pairs of stocks. From Figure 37, one can observe that even though the distributions corresponding to different periods have a similar shape and the same mean, Table 37. Number of vertices and number of edges in the ent periods (0 = 0.5) market graph for differ Number of Vertices 5430 5507 5593 5666 5768 5866 6013 6104 6262 6399 6556 Number of Edges 2258 2614 3772 5276 6841 7770 10428 12457 12911 19707 27885 Edge density 0.015' 0.017' 0.02!' . 0.0; 0.041 0.045' (I II ' 0.01 ,' . 0.0 I . (I I II . O. 1i i' the I il!" of the distribution corresponding to the latest period (period 11) is somewhat "heavier" than for the earlier periods, which means that there are more pairs of stocks with higher values of the correlation coefficient. 90 80 70 60 ~50 40 U 0 30 20 10 0 1 2 3 4 5 6 7 time period 8 9 10 11 Figure 39. Dynamics of edge density and maximum clique size in the market graph: Evolution of the edge density (a) and maximum clique size (b) in the market graph (0 = 0.5) 3.4.2 Dynamics of the Size of Cliques and Independent Sets in the Market Graph In this subsection we ,in iv. .. the evolution of the size of the maximum clique in the market graph over the considered period of time. Period 1 2 3 4 5 6 7 8 9 10 11 0.14% 0.12% 0.10% 2 0.08% % 0.06% 0.04% S0.02% o 000% 12 3 4 5 6 7 8 9 10 11 ine period Table 38 presents the sizes of the maximum cliques found in the market graph for different time periods. As in the previous subsection, we used a relatively high correlation threshold 0 = 0.5 to consider only significantly correlated stocks. As one can see, there is a clear trend of the increase of the maximum clique size over time, which is consistent with the behavior of the edge density of the market graph discussed above (see Figure 39(b)). This result provides another confirmation of the globalization hypothesis discussed above. Another related issue to consider is how much the structure of maximum cliques is different for the various time periods. Table 39 presents the stocks included into the maximum cliques for different time periods. It turns out that in most cases stocks that appear in a clique in an earlier period also appear in the cliques in later periods. There are some other interesting observations about the structure of the maximum cliques found for different time periods. It can be seen that all the cliques include a significant number of stocks of the companies representing the "hightech" industry sector. As the examples, one can mention wellknown com panies such as Sun Mi. i ', i. ini Inc., Cisco Systems, Inc., Intel Corporation, etc. Moreover, each clique contains stocks of the companies related to the semi conductor industry (e.g., Cypress Semiconductor Corporation, Cree, Inc., Lattice Semiconductor Corporation, etc.), and the number of these stocks in the cliques increases with the time. These facts ii: 1 that the corresponding branches of industry expanded during the considered period of time to form a in i ri cluster of the market. In addition, we observed that in the later periods (especially in the last two periods) the maximum cliques contain a rather large number of exchange traded funds, i.e., stocks that reflect the behavior of certain indices representing various groups of companies. It should be mentioned that all maximum cliques contain Table 38. Greedy clique size and the clique number for different time periods ( = 0.5) Period IV Edge Dens. C!i,. iii C1 IV' Edge Dens. Clique in G Coefficient in G' Number 1 5430 0.00015 0.505 15 76 0.286 18 2 5507 0.00017 0.504 18 43 0.731 19 3 5593 0.00024 0.499 26 49 0.817 27 4 5666 0.00033 0.517 34 70 0.774 34 5 5768 0.00041 0.550 42 82 0.787 42 6 5866 0.00045 0.558 45 86 0.804 45 7 6013 0.00058 0.553 51 110 0.769 51 8 6104 0.00067 0.566 60 114 0.819 60 9 6262 0.00066 0.553 62 107 0.869 62 10 6399 0.00096 0.486 77 134 0.841 77 11 6556 0.00130 0.452 84 146 0.844 85 N 11 1 100 tracking stock (QQQ), which was also found to be the vertex with the highest degree (i.e., correlated with the most stocks) in the market graph [26]. Another natural question that one can pose is how the size of independent sets (i.e., diversified portfolios in the market) changes over time. As it was pointed out in [26, 27], finding a maximum independent set in the market graph turns out to be a much more complicated task than finding a maximum clique. In particular, in the case of solving the maximum independent set problem (or, equivalently, the maximum clique problem in the complementary graph), the preprocessing procedure described above does not reduce the size of the original graph. This can be explained by the fact that the clustering coefficient in the complementary market graph with 0 = 0 is much smaller than in the original graph corresponding to 0 = 0.5 (see Table 310). Similarly to Section 3.2, we calculate maximal independent sets (a maximal independent set is an independent set that is not a subset of another independent set) in the market graph using the above greedy algorithm. As one can see from Table 310, the sizes of independent sets found in the market graph for 0 = 0 are rather small, which is consistent with the results of Section 3.2. Table 39. Structure of maximum cliques in the market graph for different time periods (0 = 0.5) Period Stocks included into maximum clique 1 BK, EMC, FBF, HAL, HP, INTC, NCC, NOI, NOK, PDS, PMCS, QQQ, RF, SII, SLB, SPY, TER, WM 2 ADI, ALTR, AMAT, AMCC, ATML, CSCO,KLAC, LLTC, LSCC, MDY, MXIM, NVLS, PMCS, QQQ, SPY, SUNW, TXN, VTSS, XLNX 3 AMAT, AMCC, CREE, CSCO, EMC, JDSU, KLAC, LLTC, LSCC, MDY, MXIM, NVLS, PHG, PMCS, QLGC, QQQ, SEBL, SPY, STM, SUNW, TQNT, TXCC, TXN, VRTS, VTSS, XLK, XLNX 4 AMAT, AMCC, ASML, ATML, BRCM, CHKP, CIEN, CREE, CSCO, EMC, FLEX, JDSU, KLAC, LSCC, MDY, MXIM, NTAP, NVLS, PMCS, QLGC, QQQ, RFMD, SEBL, SPY, STM, SUNW, TQNT, TXCC, TXN, VRSN, VRTS, VTSS, XLK, XLNX 5 ALTR, AMAT, AMCC, ASML, ATML, BRCM, CIEN, CREE, CSCO, EMC, FLEX, IDTI, IRF, JDSU, JNPR, KLAC, LLTC, LRCX, LSCC, LSI, MDY, MXIM, NTAP, NVLS, PHG, PMCS, QLGC, QQQ, RFMD, SEBL, SPY, STM, SUNW, SWKS, TQNT, TXCC, TXN, VRSN, VRTS, VTSS, XLK, XLNX 6 ADI, ALTR, AMAT, AMCC, ASML, ATML, BEAS, BRCM, CIEN, CREE, CSCO, CY, ELX, EMC, FLEX, IDTI, ITWO, JDSU, JNPR, KLAC, LLTC, LRCX, LSCC, LSI, MDY, MXIM, NTAP, NVLS, PHG, PMCS, QLGC, QQQ, RFMD, SEBL, SPY, STM, SUNW, TQNT, TXCC, TXN, VRSN, VRTS, VTSS, XLK, XLNX 7 ALTR, AMAT, AMCC, ATML, BEAS, BRCD, BRCM, CHKP, CIEN, CNXT, CREE, CSCO, CY, DIGL, EMC, FLEX, HHH, ITWO, JDSU, JNPR, KLAC, LLTC, LRCX, LSCC, MDY, MERQ, MXIM, NEWP, NTAP, NVLS, ORCL, PMCS, QLGC, QQQ, RBAK, RFMD, SCMR, SEBL, SPY, SSTI, STM, SUNW, SWKS, TQNT, TXCC, TXN, VRSN, VRTS, VTSS, XLK, XLNX 8 ALTR, AMAT, AMCC, AMKR, ARMHY, ASML, ATML, AVNX, BEAS, BRCD, BRCM, CHKP, CIEN, CMRC, CNXT, CREE, CSCO, CY, DIGL, ELX, EMC, EXTR, FLEX, HHH, IDTI, ITWO, JDSU, JNPR, KLAC, LLTC, LRCX, LSCC, MDY, MERQ, MRVC, MXIM, NEWP, NTAP, NVLS, ORCL, PMCS, QLGC, QQQ, RFMD, SCMR, SEBL, SNDK, SPY, SSTI, STM, SUNW, SWKS, TQNT, TXCC, TXN, VRSN, VRTS, VTSS, XLK, XLNX 9 ADI, ALTR, AMAT, AMCC, ARMHY, ASML, ATML, AVNX, BDH, BEAS, BHH, BRCM, CHKP, CIEN, CLS, CREE, CSCO, CY, DELL, ELX, EMC, EXTR, FLEX, HHH, IAH, IDTI, IIH, INTC, IRF, JDSU, JNPR, KLAC, LLTC, LRCX, LSCC, LSI, MDY, MXIM, NEWP, NTAP, NVLS, PHG, PMCS, QLGC, QQQ, RFMD, SCMR, SEBL, SNDK, SPY, SSTI, STM, SUNW, SWKS, TQNT, TXCC, TXN, VRSN, VRTS, VTSS, XLK, XLNX 10 ADI, ALTR, AMAT, AMCC, AMD, ASML, ATML, BDH, BHH, BRCM, CIEN, CLS, CREE, CSCO, CY, CYMI, DELL, EMC, FCS, FLEX, HHH, IAH, IDTI, IFX, IIH, IJH, IJR, INTC, IRF, IVV, IVW, IWB, IWF, IWM, IWV, IYV, IYW, IYY, JBL, JDSU, KLAC, KOPN, LLTC, LRCX, LSCC, LSI, LTXX, MCHP, MDY, MXIM, NEWP, NTAP, NVDA, NVLS, PHG, PMCS, QLGC, QQQ, RFMD, SANM, SEBL, SMH, SMTC, SNDK, SPY, SSTI, STM, SUNW, TER, TQNT, TXCC, TXN, VRTS, VSH, VTSS, XLK, XLNX 11 ADI, ALA, ALTR, AMAT, AMCC, AMD, ASML, ATML, BDH, BEAS, BHH, BRCM, CIEN, CLS, CNXT, CREE, CSCO, CY, CYMI, DELL, EMC, EXTR, FCS, FLEX, HHH, IAH, IDTI, IIH, IJH, IJR, INTC, IRF, IVV, IVW, IWB, IWF, IWM, IWO, IWV, IWZ, IYV, IYW, IYY, JBL, JDSU, JNPR, KLAC, KOPN, LLTC, LRCX, LSCC, LSI, LTXX, MCRL, MDY, MKH, MRVC, MXIM, NEWP, NTAP, NVDA, NVLS, PHG, PMCS, QLGC, QQQ, RFMD, SANM, SEBL, SMH, SMTC, SNDK, SPY, SSTI, STM, _SUNW, TER, TQNT, TXN, VRTS, VSH, VTSS, XLK, XLNX Table 310. Size of independent sets in the market graph found using the greedy heuristic (8 = 0.0). Edge density and clustering coefficient are given for the complementary graph. Period Number of Edge Clustering Independent vertices density coefficient set size 1 5430 0.258 0.293 11 2 5507 0.275 0.307 11 3 5593 0.281 0.307 10 4 5666 0.265 0.297 11 5 5768 0.260 0.292 11 6 5866 0.254 0.288 11 7 6013 0.228 0.269 11 8 6104 0.227 0.268 10 9 6262 0.238 0.277 12 10 6399 0.228 0.269 12 11 6556 0.201 0.245 11 3.4.3 Minimum Clique Partition of the Market Graph Besides analyzing the maximum cliques in the market graph, one can also divide the market graph into the smallest possible set of distinct cliques. As it was pointed out above, the partition of a dataset into sets (clusters) of elements grouped according to a certain criterion is referred to as clustering. For finding a clique partition, we choose the instance of the market graph with a low correlation threshold 0 = 0.05 (the mean of the correlation coefficients distribution shown in Figure 37), which would ensure that the edge density of the considered graph is high enough and the number of isolated vertices (which would obviously form distinct cliques) is small. We use the standard greedy heuristic to compute a clique partition in the market graph: recursively find a maximal clique and remove it from the graph, until no vertex remain. Cliques are computed using the previously described greedy algorithm. The corresponding results for the market graph with threshold 0 = 0.05 are presented in Table 311. Note that the size of the largest clique in the partition is increasing from one period to another, with the largest clique in the last period Table 311. The largest clique size and partitions (0 = 0.05) Period Number of Edge vertices density 1 5430 0.400 2 5507 0.377 3 5593 0.379 4 5666 0.405 5 5768 0.413 6 5866 0.425 7 6013 0.469 8 6104 0.475 9 6262 0.456 10 6399 0.474 11 6556 0.521 the number of cliques in computed clique Largest clique in the partition 469 552 636 743 789 824 929 983 997 1159 1372 Sof cliques in the partition 494 517 513 503 501 496 471 470 509 501 479 containing about three times as many vertices as the corresponding clique in the first partition. At the same time, the number of cliques in the partition is comparable for different periods, with a slight overall trend towards decrease, whereas the number of vertices is increasing as time goes. 3.5 Concluding Remarks Graph representation of the stock market data and interpretation of the properties of this graph gives a new insight into the internal structure of the stock market. In this paper, we have studied different characteristics of the market graph and their evolution over time and came to several interesting conclusions based on our analysis. It turns out that the powerlaw structure of the market graph is quite stable over the considered time intervals; therefore one can v that the concept of selforganized networks, which was mentioned above, is applicable in finance, and in this sense the stock market can be considered as a "selfor5, i... i system. Another important result is the fact that the edge density of the market graph, as well as the maximum clique size, steadily increase during the last several years, which supports the wellknown idea about the globalization of economy which has been widely discussed recently. 61 We have also indicated the natural way of dividing the set of financial instru ments into groups of similar objects (clustering) by computing a clique partition of the market graph. This methodology can be extended by considering quasicliques in the partition, which may reduce the number of obtained clusters. Moreover, finding independent sets in the market graph provides a new approach to choosing diversified portfolios where all stocks are pairwise uncorrelated, which is potentially useful in practice. CHAPTER 4 NETWORKBASED TECHNIQUES IN ELECTROENCEPHALOGRAPHIC (EEG) DATA ANALYSIS AND EPILEPTIC BRAIN MODELING Human brain is one of the most complex systems ever studied by scientists. Enormous number of neurons and the dynamic nature of connections between them makes the analysis of brain function especially challenging. One of the most important directions in studying the brain is treating disorders of the central nervous system. For instance, /'.:/ I/,; is a common form of such disorders, which affects approximately 1 of the human population. Essentially, epileptic seizures represent excessive and hypersynchronous activity of the neurons in the cerebral cortex. During the last several years, significant progress in the field of epileptic seizures prediction has been made. The advances are associated with the extensive use of electr ... '., p1,l.'u11.'I (EEG) which can be treated as a quantitative repre sentation of the brain function. Rapid development of computational equipment has made possible to store and process huge amounts of EEG data obtained from recording devices. The availability of these massive datasets gives a rise to another problem utilizing mathematical tools and data mining techniques for extracting useful information from EEG data. Is it possible to construct a ilp!l." mathe matical model based on EEG data that would reflect the behavior of the epileptic brain? In this chapter, we make an attempt to create such a model using a network based approach. In the case of the human brain and EEG data, we apply a relatively simple networkbased approach. We represent the electrodes used for obtaining the EEG readings, which are located in different parts of the brain, as the vertices of the constructed graph. The data received from every single electrode is essentially a time series reflecting the change of the EEG signal over time. Later in the chapter we will discuss the quantitative measure characterizing statistical relationships between the recordings of every pair of electrodes so called Tindex. The values of the Tindex Ti measured for all pairs of electrodes i and j enable us to establish certain rules of placing edges connecting different pairs of vertices i and j depend ing on the corresponding values of Tij. Using this technique, we develop several graphbased mathematical models and study the dynamics of the structural prop erties of these graphs. As we will see, these models can provide useful information about the behavior of the brain prior to, during, and after an epileptic seizure. 4.1 Statistical Preprocessing of EEG Data 4.1.1 Datasets. The datasets consisting of continuous longterm (3 to 12 d iv) multichannel intracranial EEG recordings that had been acquired from 4 patients with medically intractable temporal lobe epilepsy. Each record included a total of 28 to 32 intracranial electrodes (8 subdural and 6 hippocampal depth electrodes for each cerebral hemisphere). A diagram of electrode locations is provided in Figure 41. 4.1.2 Tstatistics and STLmax In this subsection we give a brief introduction to nonlinear measures and statistics used to analyze EEG data (for more information see [67, 69, 101]). Since the brain is a nonstationary system, algorithms used to estimate measures of the brain dynamics should be capable of automatically identifying and appropriately weighing existing transients in the data. In a chaotic system, orbits originating from similar initial conditions (nearby points in the state space) diverge exponentially (expansion process). The rate of divergence is an important aspect of the system dynamics and is reflected in the value of Lyapunov exponents. The R '43 2'1 R\I2 3 AL BR L CR CL BL1 Figure 41. Electrode placement in the brain: (A) Inferior transverse and (B) lateral views of the brain, illustrating approximate depth and subdu ral electrode placement for EEG recordings are depicted. Subdural electrode strips are placed over the left orbitofrontal (AL), right or bitofrontal (AR), left subtemporal (BL), and right subtemporal (BR) cortex. Depth electrodes are placed in the left temporal depth (CL) and right temporal depth (CR) to record hippocampal activity. method used for estimation of the short time largest Lyapunov exponent STLmax, an estimate of Lmx for nonstationary data, is explained in detail in [66, 68, 118]. By splitting the EEG time series recorded from each electrode into a sequence of nonoverlapping segments, each 10.24 sec in duration, and estimating STLma, for each of these segments, profiles of STLmx, over time are generated. Having estimated the STLma, temporal profiles at an individual cortical site, and as the brain proceeds towards the ictal state, the temporal evolution of the stability of each cortical site is quantified. The spatial dynamics of this transition are captured by consideration of the relations of the STLmax between different cortical sites. For example, if a similar transition occurs at different cortical sites, the STLmax of the involved sites are expected to converge to similar values prior to the transition. Such participating sites are called "critical i. and such a convergence dynamicall ( 1i i i i:n. i i More specifically, in order for the dynamical entrainment to have a statistical content, we allow a period over which the difference of the means of the STLma, values at two sites is estimated. We use periods of 10 minutes (i.e. moving windows including approximately 60 STLmx, values over time at each electrode site) to test the dynamical entrainment at the 0.01 statistical significance level. We employ the Tindex (from the wellknown paired Tstatistics for comparisons of means) as a measure of distance between the mean values of pairs of STLma, profiles over time. The Tindex at time t between electrode sites i and j is defined as: Ti,,(t) = N x E{STLmx,i STLm }x,j} /ai,j(t) (41) where E{} is the sample average difference for the STLma,i STLma,,j estimated over a moving window wt(A) defined as: S1 if AE [t t] 0 if A [t N t], where N is the length of the moving window. Then, ai,j(t) is the sample standard deviation of the STLmax differences between electrode sites i and j within the moving window wt(A). The Tindex follows a tdistribution with N1 degrees of freedom. For the estimation of the Tij(t) indices in our data we used N = 60 (i.e., average of 60 differences of STLmax exponents between sites i and j per moving window of approximately 10 minute duration). Therefore, a twosided ttest with N 1(= 59) degrees of freedom, at a statistical significance level a should be used to test the null hypothesis, Ho: "brain sites i and j acquire identical STLmax values at time t". In this experiment, we set the probability of a type I error a = 0.01 (i.e., the probability of falsely rejecting Ho if Ho is true, is 1 .). For the Tindex to pass this test, the Tij(t) value should be within the interval [0, 2.662]. We will refer to the upper bound of this interval as Tcritical. 4.2 Graph Structure of the Epileptic Brain 4.2.1 Key Idea of the Model If we model the brain (with epilepsy) by a graph (where nodes are "functional units" of the system and edges are connections between them) we need to answer the following questions: what properties the model has, i.e. what the properties of this graph are; how the properties of the graph change prior to, during, and after epileptic seizures. We try to answer this question using the following idea we study the system of the electrodes as a weighted graph where nodes are electrodes and weights of the edges between nodes are values of the corresponding Tindex. More specifically, we consider three types of graphs constructed using this principle: * GRAPHI is a complete graph, i.e., it has all possible edges, * GRAPHII is obtained from the complete graph by removing all the edges (i,j) for which the corresponding value of Ti is greater than Tcritical, * GRAPHIII is obtained from the complete graph by removing all the edges (i,j) for which the corresponding value of Ti is less than Tcritical 10 minutes after the seizure point and greater than Tcritical at the seizure point. 4.2.1.1 Interpretation of the Considered Graph Models Before proceeding with the further discussion, we need to give a conceptual interpretation of the ideas lying behind introducing the aforementioned graphs. * GRAPHI contains all the edges connecting the considered brain sites, and it is considered in order to reflect the general distribution of the values of Tindices between each pair of vertices (i.e., the weights of the corresponding edges). * GRAPHII contains only the edges connecting the brain sites (electrodes) that are statistically entrained at a certain time, which means that they exhibit a similar behavior. Recall that a pair of electrodes is considered to be entrained if the value of the corresponding Tindex between them is less than Tcrutcal, that is why we remove all the edges with the weights greater than Tcritcal. The main point of our interest is studying the evolution of the properties of this graph over time. As we will see in the next subsections, this analysis can help in revealing the /;;,:, ,,i.., / patterns underlying the functioning of the brain during preictal, ictal, postictal, and interictal states. Therefore, this graph can be used as a basis for the mathematical model describing some characteristics of the epileptic brain. * GRAPHIII is constructed to reflect the connections only between those electrodes that are entrained during the seizure, but are not entrained 10 minutes after the seizure. The motivation for introducing this graph is the existence of i. . 11 'ig" of the brain after the seizure [70, 108, 111], which is essentially the divergence of the profiles of the STLma, time series. As it was indicated above, this divergence is characterized by the values of Tindex greater than Tcritica. 4.2.2 Properties of the Graphs In this subsection, we investigate the properties of the considered graph models and give an intuitive explanation of the observed results. As we will see, there are specific tendencies in the evolution of the properties of the considered graphs prior to, during, and after epileptic seizures, which indicates that the proposed models capture certain trends in the behavior of the epileptic brain. 4.2.2.1 Edge Density Recall that GRAPHII was introduced to reflect the connections between brain sites that are statistically entrained at a certain time moment. Figure 42 0) "o 4 200 E Z 150 100 50 I I I II I I i I I I I 7900 7950 8000 8050 8100 8150 8200 8250 8300 8350 MINUTES Figure 42. Number of edges in GRAPHII illustrates the typical evolution of the number of edges in GRAPHI over time. As it was indicated above, edge density of the graph is proportional to the number of edges in a graph. It is easy to notice that the number of edges in GRAPHII dramatically increases at seizure points (represented by dashed vertical lines), and it decreases immediately after seizures. It means that the global structure of the graph significantly changes during the seizure and after the seizure, i.e. the density of increases during ictal state and decreases in postictal state, which supports the idea that the epileptic brain (and GRAPHII as the model of the brain) experiences a "phase transition" during the seizure. 4.2.2.2 Connectivity Another important property of GRAPHII that we are interested in is its .., ,.. /.'; :/ We need to check if this graph is connected prior to, during, and after epileptic seizures, and if not, find the size of its largest connected component. Clearly, this information will also be helpful in the analysis of the structural properties of the brain. If GRAPHII is connected (i.e., the size of the largest connected component is equal to the number of vertices in the graph), then all the functional units of the brain are "linked" with each other by a path, and in this case the brain can be treated as an i i, i ii, 1 system, however, if the size of the largest connected component in GRAPHII is significantly smaller than the total number of the vertices, it means that the brain becomes 1' 1 iied" into smaller dii.iil subsystems. The size of the largest connected component of the GRAPHII is presented in Figure 43. One can see that GRAPHII is connected during the interictal period (i.e., the brain is a connected system), however, it becomes disconnected after the seizure (during the postical state): the size of the largest connected component significantly decreases. This fact is not surprising and can be intuitively explained, since after the seizure the brain needs some time to "reset" [70, 108, 111] and restore the connections between the functional units. 4.2.2.3 Minimum Spanning Tree The next subject of our discussion is the analysis of minimum ',','.:,':,'; trees of GRAPHI, which was defined as the graph with all possible edges, where each edge (i, j) has the weight equal to the value of Tindex Tij corresponding to brain sites i and j. The definition of Minimum Sr I !',!'.:,t Tree was given in Section 2. Studying minimum spanning trees in GRAPHI is motivated by the hypothesis that the seizure signal in the brain propagates to all functional units according to the minimum y'u~r...':: tree, i.e. along the edges with small values of Ti. This 70 II 32  30 28 S26 o 0 4 24 E 22 N / 20  18 16 14 I I I 9700 9800 9900 10000 10100 10200 10300 MINUTES Figure 43. The size of the largest connected component in GRAPHII. Number of nodes in the graph is 30. hypothesis is partially supported by the behavior of the average Tindex of the edges corresponding to the Minimum Spanning Tree of GRAPHI, which is shown in Figure 44. However, this hypothesis cannot be verified using the considered data, since the values of average Tindices are calculated over a 10minute interval, whereas the the seizure signal propagates in a fraction of a second. Therefore, in order to check if the seizure signal actually spreads along the minimum spanning tree, one needs to introduce other nonlinear measures to reflect the behavior of the brain over short time intervals. 1 .1 ' 1  0.9 0.8 o 0 0.7 0.6 0.5 0.4 0.3 0.2 I I I I I I I I i I 0.995 1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 MINUTES x104 Figure 44. Average value of Tindex of the edges in Minimum Spanning Tree of GRAPHI. Also, note that the average value of the T index in the Minimum Spanning Tree is less than Tcritical, which also supports the above statement about the connectivity of the system. 4.2.2.4 Degrees of the Vertices Another important issue that we analyze here is the degrees of the vertices in GRAPHII. Recall that the degree of a vertex is defined simply as the number of edges emanating from it. We look at the behavior of the average degree of the vertices in GRAPHII over time. Clearly, this plot is very similar to the behavior of the edge density of GRAPHII (see Figure 45). 72 101 9 8 7 S6 5 4 3 2 1 7800 7900 8000 8100 8200 8300 8400 MINUTES Figure 45. Average degree of the vertices in GRAPHII. We are also particularly interested in highdegree vertices, i.e., the functional units of the brain that are at a certain time moment connected (entrained) with many other brain sites. Interestingly enough, the vertex with a maximum degree in GRAPHH usually corresponds to the electrode which is located in RTD (right temporal depth) or RST (right subtemporal cortex), in other words, the vertex with the maximum degree is located near the epileptogenic focus. 4.2.2.5 Maximum Cliques In the previous works in the field of epileptic seizure prediction, a quadratic 01 programming approach based on EEG data was introduced [69]. In fact, this approach utilizes the same preprocessing technique (i.e., calculating the values of Tindices for all pairs of electrode sites) as we apply in this chapter. In this subsection, we will briefly describe this quadratic programming technique and relate it to the graph models introduced above. The main idea of the considered quadratic programming approach is to construct a model that would select a certain number of socalled "critical" electrode sites, i.e., those that are the most entrained during the seizure. According to Section 3, such group of electrode sites should produce a minimal sum of T indices calculated for all pairs of electrodes within this group. If the number of critical sites is set equal to k, and the total number of electrode sites is n, then the problem of selecting the optimal group of sites can be formulated as the following quadratic 01 problem [69]: min xTAx (42) s.t. Eix = k. (43) i e{0,1} Vie {1,..., n} (44) In this setup, the vector x = (xl, x2, ..., ,) consists of the components equal to either 1 (if the corresponding site is included into the group of critical sites) or 0 (otherwise), and the elements of the matrix A = [aij],j= 1...,n are the values of Tij's at the seizure point. However, as it was shown in the previous studies, one can observe the "re setting of the brain after seizures' onset [111, 70, 108], that is, the divergence of STLmax profiles after a seizure. Therefore, to ensure that the optimal group of critical sites shows this divergence, one can reformulate this optimization problem by adding one more quadratic constraint: xTBx > Tcritical k (k 1), (45) where the matrix B = r1 .]ij1,...,, is the Tindex matrix of brain sites i and j within 10 minute windows after the onset of a seizure. This problem is then solved using standard techniques, and the group of k critical sites is found. It should be pointed out that the number of critical sites k is predetermined, i.e., it is defined empirically, based on practical observations. Also, note that in terms of GRAPHI model this problem represents finding a subgraph of GRAPHI of a fixed size, satisfying the properties specified above. Now, recall that we introduced GRAPHIII using the same principles as in the formulation of the above optimization problem, that is, we considered the connections only between the pairs of sites i,j satisfying both of the two conditions: Ti < Tcritical at the seizure point, and Tj > Tcritical 10 minutes after the seizure point, which are exactly the conditions that the critical sites must satisfy. A natural way of detecting such a groups of sites is to find cliques in GRAPHIII. Since a clique is a subgraph where all vertices are interconnected, it means that all pairs of electrode sites in a clique would satisfy the aforementioned conditions. Therefore, it is clear that the size of the maximum clique in GRAPH III would represent the upper bound on the number of selected critical sites, i.e., the maximum value of the parameter k in the optimization problem described above. Computational results indicate that the maximum clique sizes for different instances of GRAPHIII are close to the actual values of k empirically selected in the quadratic programming model, which shows that these approaches are consistent with each other. 4.3 Graph as a Macroscopic Model of the Epileptic Brain Based on the results obtained in the sections above, we now can formulate the graph model which describes the behavior of the epileptic brain at the macroscopic level. The main idea of this model is to use the properties of GRAPHI, GRAPH II, and GRAPHIII as a characterization of the behavior of the brain prior to, during, and after epileptic seizures. According to this graph model, the graphs reflecting the behavior of the epileptic brain demonstrate the following properties: * Increase and decrease of the edge density and the average degree of the vertices during and after the seizures respectively; * The graph is connected during the interictal state, however, it becomes disconnected right after the seizures (during the postictal state); * The vertex with the maximum degree corresponds to the epileptogenic focus. Moreover, one of the advantages of the considered graph model is the possi bility to detect special formations in these graphs, such as cliques and minimum spanning trees, which can be used for further studying of various properties of the epileptic brain. 4.4 Concluding Remarks and Directions of Future Research In this chapter, we have made the initial attempt to analyze EEG data and model the epileptic brain using networkbased approaches. Despite the fact that the size of the constructed graphs is rather small, we were able to determine specific patterns in the behavior of the epileptic brain based on the information obtained from statistical analysis of EEG data. Clearly, this model can be made more accurate by considering more electrodes corresponding to smaller functional units. Among the directions of future research in this field, one can mention the possibility of developing directed graph models based on the analysis of EEG data. Such models would take into account the natural ivnii.i i i'1 of the brain, where certain functional units control the other ones. Also, one could apply a similar approach to studying the patterns underlying the brain function of the patients with other types of disorders, such as Parkinson's disease, or sleep disorder. 76 Therefore, the methodology introduced in this chapter can be generalized and applied in practice. CHAPTER 5 COLLABORATION NETWORKS IN SPORTS In this chapter, we will discuss one of the most interesting reallife graph applications socalled "social ii. i l: where the vertices are real people [63, 116]. The main idea of this approach is to consider the ." .I1 l iii o:eship graph" connecting the entire human population. In this graph, an edge connects two given vertices if the corresponding two persons know each other. Social networks are associated with a famous i i illworld" hypothesis, which claims that despite the large number of vertices, the distance between any two vertices (or, the diameter of the graph) is small. More specifically, the idea of "six degrees of separation" has been introduced. It states that any two persons in the world are linked with each other through a sequence of at most six people [63, 116, 117]. Clearly, one cannot verify this hypothesis for the graph incorporating more than 6 billion people living on the Earth, however, smaller subgraphs of the acquaintanceship graph connecting certain groups of people can be investigated in detail. One of the most wellknown graphs of this type is the scientific collaboration ,jir1,' reflecting the information about the joint works between all scientists. Two vertices are connected by an edge if the corresponding two scientists have a joint research paper. Another graph of this type is known as the "H .//;/;, ...../ Il,,ll, : it links all the movie actors, and an edge connects two actors if they ever appeared in the same movie. Wellknown concepts associated with these graphs are so called "Erdos number" (in the scientific collaboration graph) and "Bacon number" (in the Hollywood graph), which are assigned to every vertex and characterize the distance from this vertex to the vertex denoting the "center" of the graph. In the collaboration graph, the central vertex corresponds to the famous graph theoretician Paul Erdis, whereas in the Hollywood graph the same position is assigned to Kevin Bacon. In this chapter, we discuss graphs of a similar type arising in sports, that represent the pll li rs' "collaboration". In these graphs, the pl li rs are the vertices, and an edge is added to the graph if the corresponding two pl ivrs ever pll li d together in the same team. One of the examples of this type of graphs is the graph representing baseball pl i, rs. For any two baseball pl li, rs who ever pll li d in the Major League Baseball(j\l .), a path connecting them can be found in this graph. As another instance of social networks in sports, we study the "NBA graph" where the vertices represent all the basketball pl li, rs who are currently pl viing in the NBA. We apply standard graphtheoretical algorithms for investigating the properties of this graph, such as its connectivity and diameter (i.e., the maximum distance between all pairs of vertices in the graph). As we will see later in the chapter, this study also confirms the 1!! illworld hypotl! Moreover, we introduce a distance measure in the NBA graph similar to the Erdis number and the Bacon number. The central role in this graph is given to Michael Jordan, the greatest basketball pl liv r of all times, and we refer to this measure as the Jordan number. 5.1 Examples of Social Networks In this section, we give a more detailed description of the examples of social networks mentioned in the introduction the scientific collaboration graph, the Hollywood graph, and the baseball graph. 5.1.1 Scientific Collaboration Graph and Erdis Number As it was mentioned above, the vertices of the scientific collaboration graph are scientists, and the edges in this graph connect the scientists who have ever collaborated with each other (i.e., had a joint paper). In order to measure the distances in this graph, the "central v, i I :; is introduced. This vertex corresponds to Paul Erd6s, the father of the theory of random graphs. This vertex is assigned Erdos number equal to 0. For all other vertices in the graph, the Erd6s number is defined as the distance (i.e., the shortest path length) from the central vertex. For example, those scientists who had a joint paper with Erd6s have Erdis number 1, those who did not collaborate with Erd6s, but collaborated with Erd6s' collaborators have Erd6s number 2, etc. Following this logic, one can construct the connected component of the collaboration graph with "concentric circles", which would incorporate almost all scientists in the world, except those who never collaborate with anybody. This connected component is expected to have a relatively small diameter. The idea of constructing collaboration graphs encompassing people in different areas gave a rise to several other applications. Next, we discuss the Hollywood graph and the baseball graph, where the number of vertices is significantly smaller than in the scientific collaboration graph, which allows one to study their structure in more detail. 5.1.2 Hollywood Graph and Bacon Number The Hollywood graph is constructed using the same principles as the scientific collaboration graph, however, the number of Hollywood actors is much smaller than the number of scientists, therefore, one can investigate the characteristics of every vertex in this graph. This information is maintained at the "Oracle of B ..I  website.1 The most recent Hollywood graph contains 595,578 vertices (actors). The central vertex in this graph represents the famous actor Kevin Bacon, and this vertex obviously has Bacon number 0. Since the number of vertices in this graph is small enough, one can explicitly calculate the Bacon number for every 1 http://www.cs.virginia.edu/oracle/ Average Bacon number= 2.946 400000 364066 350000 300000 S250000 S200000  o 133856 W 150000  88058 E 100000 50000 1686 6960 854 94 3 0  0 1 2 3 4 5 6 7 8 Bacon number Figure 51. Number of vertices in the Hollywood graph with different values of Bacon number. Average Bacon number = 2.946. actor. It turns out that most of the actors have Bacon numbers equal to 2 or 3, and the maximum possible Bacon number is equal to 8, which is the case only for 3 vertices. The distribution of Bacon numbers in the Hollywood graph is shown in Figure 51. The average Bacon number (i.e., the average path length from a given actor to Bacon) is equal to 2.946. As one can see, both the average and the maximum Bacon numbers of the Hollywood graph are very small, which provides an argument in favor of the 11 i 11 world hypot : mentioned above. 5.1.3 Baseball Graph and Wynn Number Collaboration networks similar to the ones mentioned above can also be constructed in sports. One example of such a network is the "baseball graph" representing all baseball p1 li, rs who ever p .li', d in the MLB. In this graph, two pl i rs are connected if they ever were teammates. The most recent baseball graph Average Wynn number = 2.901 7000 6663 6000 5286 0 5000 > 4000 o 3000 2472 3 2000 E 899 = 1000 408 88 0 7 0 1 2 3 4 5 6 Wynn number Figure 52. Number of vertices in the baseball graph with different vaues of Wynn number. Average Wynn number = 2.901 has 15817 vertices. Links between any pair of baseball p1 li. rs can be found at the "Oracle of Baseball" website.2 One can assign the central role in this graph to Early Wynn, a member of the Hall of Fame who spent 23 seasons in the MLB. Figure 52 shows the distribution of Wynn numbers in the baseball graph. The maximum Wynn number is 6, which is smaller than the maximum Bacon number since total number of baseball p1 li. rs is less than the number of Hollywood actors. 5.1.4 Diameter of Collaboration Networks Another aspect that should be mentioned here is that the maximum from the central vertex in the collaboration graphs certainly depends on the choice of this central vertex. The reason for choosing Kevin Bacon as the center of the 2 http://www.baseballreference.com/oracle/ Hollywood graph, and Early Wynn as the center of the baseball graph is the fact that it is reasonable to expect them to be connected to many vertices: Bacon appeared in many movies, and Wynn p1l li, d in several baseball teams had a lot of teammates during his long career. However, one can choose less "connected" centers of these graphs, and in this case the maximum distance from the new center of the graph may significantly increase. For example, if one chooses Barry Bonds as the center of the baseball graph, the maximum Bonds number will be 9 instead of 6. Moreover, in the Hollywood graph, it is possible to choose the center so that the maximum distance from it is equal to 14, and the average distance is greater than 6 (instead of 2.946). Therefore, in order to have a more complete information about the structure of these graphs, one should calculate the maximum possible distance among all pairs of vertices in the graph. Recall that this quantity is referred to as the diameter of the graph. Clearly, the diameter can be found by considering each vertex as the center of the graph, calculating corresponding maximal distances, and then choosing the maximum among them. In the next section, we study the properties of the NBA graph incorporating basketball p1l i, rs p1 giving in the world's best basketball league. In a similar fashion, we introduce the Jordan number, investigate its values corresponding to different vertices, and calculate the diameter of this graph. 5.2 NBA Graph The NBA graph considered in this section is constructed using the same idea as the graphs described above. Here we provide a detailed description of the structural properties of this graph. As we will see, its properties are rather similar to the properties of other social networks, which confirms the smallworld hypothesis. 5.2.1 General Properties of the NBA Graph The instance of the NBA graph that we consider in this section is relatively small and contains only those 1pl ,i rs who are curr n i/,ll pl1 ,iing in the NBA (as of the season of 20022003). However, this information is sufficient to reveal that the NBA graph follows similar patterns as other social networks. As of May 2003, the total number of p1 li, rs in the rosters of all the NBA teams is equal to 404 (pl irs picked in the 2003 NBA draft and transfers that occurred after the end of the 20022003 season are not taken into account). An edge connects two given pl!i,rs if they ever p1l li, d in the same team. Consequently, the constructed NBA graph has 404 vertices, and 5492 edges connecting them. Note that the maximum possible number of edges is equal to 404 x (404 1)/2 = 81406, therefore, the edge /. ,.:1;, of this graph (i.e., the ratio of the number of edges to the maximum possible number of edges) is rather small: 5492/81406 = 6.7.'. As one can easily see, this graph has a highly specific structure: the p1 li. rs of every team form a clique in the graph (i.e., the set of completely interconnected vertices), because all the vertices corresponding to the p1 li, rs of the same team must be interconnected. Since many 1p ii rs change teams during or between the seasons, there are edges connecting the vertices from different cliques (teams). Note that this type of structure is common for all "collaboration 1. I . i I: (see Figure 53). It should be pointed out that the number of p1 li. rs in a basketball team is relatively small, and the pl liv rs' transfers between different teams occur rather often, therefore, it would be logical to expect that the NBA graph should be connected, i.e., there is a path from every vertex to every vertex, moreover, the length of this path must be small enough. As we will see below, calculations confirm these assumptions. Figure 53. General structure of the NBA graph and other collaboration networks First, we used a standard breadthfirst search technique for checking the connectivity of the considered graph. Starting from an arbitrary vertex, we were able to locate all other vertices in the graph, which means that every vertex is reachable from another, therefore, the graph is connected. In the next subsection, we will also see that every pair of vertices in this graph are connected by a short path, which is in agreement with the i,, illworld hypot !.  . Average Jordan number = 2.270 300 244 u 250 2 200 135 "6 150 0 I ~ 100 E 5 50 24 co 1 0 1 2 3 Jordan number Figure 54. Number of vertices in the NBA graph with different values of Jordan number. Average Jordan number = 2.270 5.2.2 Diameter of the NBA Graph and Jordan Number The next subject of our interest is verifying if the NBA graph follows the smallworld hypothesis. We need to answer the question, what is the distance between any two vertices in this graph? Similarly to the social graphs mentioned above, we define the "central v iI : in the NBA graph corresponding to Michael Jordan, who p1 i, 1 for Washington Wizards during his final NBA season. Obviously, all other pl i ,v rs in the Wizards' roster for 20022003, as well as all the pl .i, rs who have plin .1 with Jordan during at least one season in the past, have Jordan number 1. It should be noted that Michael Jordan p1l li, d only for two teams (Chicago Bulls and Washington Wizards) through his entire career, therefore, one can expect that the number of pl vrs with Jordan number 1 is rather small. In fact, only 24 pl li, rs currently pl giving in the NBA have Jordan number 1. Table 51. Jordan numbers of some NBA stars (end of the 20022003 season). PlIiyr Kobe Bryant Vince Carter Vlade Divac Tim Duncan Michael Finley Steve Francis Kevin Garnett Pau Gasol Richard Hamilton Allen Iverson Jason Kidd Toni Kukoc Karl Malone Stephon Marbury Shawn Marion K. ion Martin Jamal Mashburn Tracy McGrady R. ._i Miller Yao Ming Dikembe Mutombo Steve Nash Dirk Nowitzki Jermaine O'Neal Shaquille O'Neal Gary Payton Paul Pierce Scottie Pippen David Robinson Arvydas Sabonis Jerry Stackhouse Predrag Stojakovic Antoine Walker Ben Wallace C'!hin Webber Team Los Angeles Lakers Toronto Raptors Sacramento Kings San Antonio Spurs Dallas Mavericks Houston Rockets Minnesota Timberwolves Memphis Grizzlies Detroit Pistons Philadelphia 76ers New Jersey Nets Milwaukee Bucks Utah Jazz Phoenix Suns Phoenix Suns New Jersey Nets New Orleans Hornets Orlando Magic Indiana Pacers Houston Rockets New Jersey Nets Dallas Mavericks Dallas Mavericks Indiana Pacers Los Angeles Lakers Milwaukee Bucks Boston Celtics Portland Trail Blazers San Antonio Spurs Portland Trail Blazers Washington Wizards Sacramento Kings Boston Celtics Detroit Pistons Sacramento Kings Jordan Number 2 2 2 2 2 3 3 3 1 2 2 1 2 2 2 3 2 2 3 3 2 2 2 2 2 2 2 1 2 2 1 2 2 2 2 Following similar logic, the p1l li, rs who have pl li, d with Jordan's "collabora tors" have Jordan number 2, and so on. However, it turns out that the maximum Jordan number in this instance of the NBA graph is only 3, i.e., all the p1l l, rs are linked with Jordan through at most two vertices, which is certainly not surprising: with 29 teams and only around 15 pl.' rs in each team, NBA is really a in, ,1 v i !. 1[ Figure 54 shows the distribution of Jordan numbers in the NBA graph. The average Jordan number is equal to 2.27, which is smaller than the average Bacon number in the Hollywood graph, and the average Wynn number in the baseball graph, due to smaller number of vertices. Table 51 presents Jordan numbers corresponding to some wellknown NBA pli rs. Not surprisingly, most of them have Jordan number 2, except for several p!i rs with Jordan number 3: those who joined this league recently, and therefore did not have many teammates through their career, as well as R... : Miller who spent 16 seasons in the same team (Indiana Pacers), and Kevin Garnett who p1l li d in Minnesota for 8 years. Scottie Pippen, Toni Kukoc, and Jerry Stackhouse were Jordan's teammates at different times, therefore, they have Jordan number 1. Furthermore, we calculated the diameter of the NBA graph, i.e., the maximum possible distance between any two vertices in the graph. Since the maximum Jordan number in the NBA graph is equal to 3, one would expect that the value of the diameter to be of the same order of magnitude. As it was mentioned in the previous section, the diameter of the NBA graph can be found as follows: for every given vertex, we calculate the distances between this vertex and all others. In this approach, we need to repeat this procedure 404 times, and every time a different vertex is considered to be the "center" of the graph. Our calculations show that the diameter of the NBA graph (the maximum distance between all pairs of vertices) is equal to 4. Therefore, one can claim that the NBA graph actually follows the smallworld hypothesis, since its diameter is small enough. Table 52. Degrees of the Vertices in the NBA graph degree interval number of vertices 1120 134 2130 116 3140 103 4150 42 5160 8 61+ 2 5.2.3 Degrees and "Connectedness" of the Vertices in the NBA Graph As it was pointed out above, the maximum and the average distance from the center of the graph actually depend on the choice of this center. One can easily guess that Michael Jordan is not the most "connected" central vertex of the NBA graph, since he pll li d only for two teams and the number of his former teammates among currently active pl ,i rs is rather small. In fact, the degree of the vertex (i.e., the number of edged starting from it, or, the number of teammates) corresponding to Jordan is only 24. Table 52 presents the number of vertices in the NBA graph corresponding to different intervals of the degree values. It would be reasonable to assume that if one picks a vertex with a high degree as the center of the NBA graph, the average distance in the graph corresponding to this vertex would be smaller than the average Jordan number. We have found the most "connected" pl li, rs in the NBA graph with the smallest corresponding aver age distances. Table 53 presents five pl .i, rs who could be the most "connected" centers of the NBA graph. As one can notice, all of them are "bench p! .i, i who have changed many teams during their career, therefore, they have high degrees in the NBA graph. Also, an interesting observation is that although Corie Blount's vertex is degree smaller than Jim Jackson's, the average connectivity is higher for Corie Blount, which could be explained by the fact that his teammates were highly "connected" themselves. 