universitätsverlag karlsruhe

Analysis of AsymmetricCommunication Patternsin Computer MediatedCommunication Environments

Bettina Hoser

Hoser // 28.02.05

Bettina Hoser

Analysis of Asymmetric Communication Patterns

in Computer Mediated Communication Environments

Analysis of Asymmetric Communication Patterns in Computer Mediated Communication Environments

vonBettina Hoser

Dissertation, genehmigt von der Fakultät für Wirtschaftswissenschaften

der Universität Fridericiana zu Karlsruhe, 2004

Referenten: Prof. Dr. Andreas Geyer-Schulz, Prof. Dr. Karl-Heinz Waldmann

Impressum

Universitätsverlag Karlsruhe c/o Universitätsbibliothek Straße am Forum 2 D-76131 Karlsruhe www.uvka.de � Universitätsverlag Karlsruhe 2005 Print on Demand ISBN 3-937300-49-X

Acknowledgement

To DorisHoser.

But fromtheSage it is sohard at anyprice to get a singlewordThatwhenhis taskis accomplished,his workdone,Throughoutthecountryeveryonesays:“It happenedof its ownaccord”.(Lao-zi,TheDaoDe Jing)

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Contents

1 Intr oduction 11.1 Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Guideto theReader. . . . . . . . . . . . . . . . . . . . . . . . . 6

2 RelatedWork 72.1 SocialNetwork Analysis . . . . . . . . . . . . . . . . . . . . . . 72.2 DegreebasedMeasurements. . . . . . . . . . . . . . . . . . . . 82.3 EigensystemAnalysisin SocialNetwork Analysis. . . . . . . . . 102.4 OtherApplicationsof EigenvectorCentrality . . . . . . . . . . . 13

3 Theoretical Foundationsfor Pattern Analysis in SocialNetworks 153.1 SocialNetworks . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 RankandStatus. . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.1 DegreebasedRank . . . . . . . . . . . . . . . . . . . . . 183.2.2 RankPrestigeor Statusindex andeigenvectorcentrality . 213.2.3 CommunicationPatterns . . . . . . . . . . . . . . . . . . 22

3.3 LinearOperatorsin Hilbert Space . . . . . . . . . . . . . . . . . 233.3.1 Complex NumbersandHermitianMatrices . . . . . . . . 233.3.2 Hilbert SpaceasaspecialVectorSpace . . . . . . . . . . 263.3.3 HermitianOperatorsin Hilbert Space . . . . . . . . . . . 293.3.4 Eigensystemanalysisof HermitianOperators. . . . . . . 303.3.5 OrthogonalProjectors . . . . . . . . . . . . . . . . . . . 353.3.6 PerturbationTheoryfor HermitianOperators . . . . . . . 36

3.4 Eigensystemof realadjacency matricesfor non-directionalgraphs 37

4 Analysis of GeneralBidir ectional Communication Patterns 414.1 CommunicationBehavior . . . . . . . . . . . . . . . . . . . . . . 414.2 Complex HermitianAdjacency Matrix . . . . . . . . . . . . . . . 424.3 Interpretationof theEigensystem. . . . . . . . . . . . . . . . . . 45

4.3.1 Eigenspectrum. . . . . . . . . . . . . . . . . . . . . . . 46

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4.3.2 Eigenvectorsandorthogonalprojectors . . . . . . . . . . 464.4 Examplesof Eigensystems. . . . . . . . . . . . . . . . . . . . . 47

4.4.1 StarGraph . . . . . . . . . . . . . . . . . . . . . . . . . 484.4.2 CompleteGraph . . . . . . . . . . . . . . . . . . . . . . 614.4.3 Two subgroupsconnectedby onelink . . . . . . . . . . . 63

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5 Data sets 675.1 EIESdataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.2 Subsetof EIESdataset . . . . . . . . . . . . . . . . . . . . . . . 73

6 Resultsand Discussion 796.1 EIESDataSet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.1.1 Eigensystemof theEIESDataSet . . . . . . . . . . . . . 806.1.2 Comparisonwith dichotomizedandsymmetrisizedeigen-

system . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.2 Subsetof EIES . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.2.1 Eigensystemof theEIESSubset . . . . . . . . . . . . . . 1036.2.2 Comparisonof subsetresultswith dichotomizedandsym-

metriziedeigensystem. . . . . . . . . . . . . . . . . . . 114

7 Conclusionand Future Applications 1177.1 FutureApplications . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.1.1 Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187.1.2 OrganizationAnalysis . . . . . . . . . . . . . . . . . . . 1207.1.3 Viral Marketing . . . . . . . . . . . . . . . . . . . . . . . 120

7.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Appendix 123

Notation

�setof realnumbers�setof complex numbers�matrix: capitalletters����� matrixentryin thek-th row andl-th column�vectorspace:boldcapitalletters vector:bold letters��� : l-th componentof vector ���eigenvalue:greekletters ��������������graph

with vertices��� � , edges��� � andadjacency matrix

�� �"!$# innerproductof vectors and !� %�� absolutevalueof %& & normof vector &(')&normof matrix

�

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Chapter 1

Intr oduction

Many predictionsabout the waysnew technologieswill transformsocietyfadequickly - the telegraph, radio, movies,and television did createrevolutions,butnot the oneswe expected. Hence, it is especiallyimportant that we turn fromopinionsandpredictionsto theseriousanalysisanddescriptionof onlinegroups.(M. SmithandP. Kollock, 1999) [SK99,p.23]

As peopletransferpartof their sociallife to theInternet(seefor exampleon-line communitieslikefriendster[Fri04], linkedin[Lin04], orkut [Ork04], . . . ), thequestionarisesif theuseof computermediatedcommunication(CMC) changestheway communitiesarebuilt andkeptalive,or if theknown dynamicsof sociallife arejust beingadaptedto thenew technology [Wel02]. Onefocusof interestis thenotionof a centerof a network. Thequestionis how theway peoplebuildgroupsaroundcertain“very importantpeople”is transferedto CMC. Not only isthis questionof interestto the sociologist(seefor example [Wel01]), but it hasbecomeincreasinglyinterestingfor securityreasonson theonehand,andfor itspossibleeconomicimpacton theother.

Someapplicationshave gainedwide publicity, e.g. thehunt for terroristnet-works, or the useof network dynamicsfor political goals. Sincethe terroristattackson New York on september11th,2001the ideaof usingmethodsdrawnfrom socialnetwork analysishave gaineda wide interest[Kre02]. For the sui-cideattackonNew York theterroristshadusedmoderncommunicationchannels,e.g.emailandmobilephones,to planandexecutetheir dreadfulattack.Thusin-terestin thedevelopmentandenhancementof methodsto detectcommunicationpatternsof suchnetworkswell in advancehasgrown dramatically.

In the 2004 U.S. presidentialelectionsthe campaignusedthe possibilitiesof the Internet to reachout to communitiesby “snowballing ” (see [Ker04],

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[Bus04]). The aim is to motivatepeopleto participatein the elections,and tousethesocialnetworksof supportersto win overevenmorepotentialvoters.

In the world of economicsHayek [Hay45, p.519-520]points to one of themajor problemswhich researcherstry to solve by using network analysis. Heexplicitly states,thatthepeculiarcharacterof theproblemof a rationaleconomicorder is determinedpreciselyby thefact that theknowledge of thecircumstancesof which we mustmake usenever exists in concentratedor integratedform, butsolelyasthedispersedbitsof incompleteandfrequentlycontradictoryknowledgewhich all the separate individualspossess. Thuscommunicationis at the heartof economicsuccess,aswell as the efficient allocationof informationwithin anetwork. To know thepatternsof suchcommunicationoffersthepossibilityto actaccordingto theneedsof theorganization.Hayek´sexampleis anillustrationthattheknowledgegainedaboutsocialnetworksandhow they areanalyzedmight beusefulin understandingeconomicprocesseslikemarkets.

Thuson the businesssidethe opportunityto view customer, employee,sup-plier or traderbehavior via theiruseof new mediabecomesevermoreinteresting,temptingandrevealing.Theavailability of hugedatasets,e.g. emailserver dataor market log files, hasopenedthe possibility to analyzebehavior without thedrawbackof having to explicitly askthe observed subjectsabouttheir behavior.Thediscussionaboutprivacy in thiscontext andthelegalaspectsof theuseof datawithout theexplicit consentof theobservedsubjectsis still on going. The legalsystemsof Germany andtheUnitedStatesfor examplehavedifferentapproachesto privacy, which hastheeffect, thatwhile in Germany emailsin a company en-vironmentfall understrict laws protectingthe privacy of the author, this is notthecasein theU.S.Thusdatalike this is not easilyaccessiblein Germany. Mes-sagespostedto newsgroupsor any otheropencommunicationplattform arenotprotectedby law neitherin the U.S. nor in Germany. The scientificcommunitythoughtendsto eitheraskpermissionof any suchgroupbeforeusingthedata,orafterwards.

Thewayemail,for examplein anorganization,is beingusedcanrevealif theorganizationalstructureof thecompany is reflectedin thegroupsthatcanbede-tectedoutof theCMC [KH93]. On thecustomersideon theotherhandtheanal-ysis of customernewsgroupsor chatroomscouldhelp identify thosecustomerswho canswingopinions,andthushelp in marketingefforts. An applicationhereis for exampleviral marketing. Anotherexampleof a network is givenwhenonelooks at supplychainmanagement.Herea network of suppliershasto performthe difficult taskof bringing partsandcomponentstogetherat the right time, intherequiredamountto themanufacturer. In this context informationalefficiencyplaysa major role to cut costsor to stayon budget. If onenow looks at evenlargerstructures,onecansee,thatinformationefficiency is highly correlatedwithefficient markets.And theseagaincanbeviewedassocialnetworksin thesense,

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thatagentsinteractvia communicationto tradegoodsor information.As canbeseenfrom thesefew examplestheimpactof CMC on communities

and social interactionis strong. Consequentlythe interestof researchers,andsubsequently, theuseof their resultsfor businessapplications,is growing rapidly.

Thereare many different anglesfrom which theseinteractionscan be ana-lyzed. Theonethatwill bein thefocusin thecourseof this bookis thequestionof centralityof a memberwithin his or hersocialnetwork. Centralitycharacter-izesthefact,thatamemberplaysaveryimportantrolewithin agivennetwork andwith respectto theinteractionobserved.Themethodsthathavebeendevelopedinsuchuntil now unrelatedfieldsassociology, computerscience,psychology, math-ematics,operationsresearch,physics,andsignalintelligencehave beenbroughttogetherto yield insight into subjectbehavior in the Internetageby analyzingcommunicationstructuresanddynamicsin virtual communitiesusingCMC.

1.1 Moti vation

It´s whoyouknow. Really.(D. Gross,2004) [Gro04]

To find structurein “uncharted”communitieshasbeenof interestfor sociol-ogistsandintelligenceserviceresearchersfor many yearsand,dueto the effecttheInternethasonmarketingit hasgainedgrowing interestin thebusinessworld.Theinterestof sociologistsis to find out,whois “important” within acommunity,who“leads”,whocorrespondswith whom,thusputtingthefocusonrelationshipswithin a groupin a sociologicalsense.The interestof intelligenceservicesis toidentify the links within a communicationnetwork, that, if shutdown, mostef-fectively hurt the targetednetwork. Lastly the interestof companieslies on theonehandin the knowledgeaboutthe behavior of their customers,which canbean integral partof customerrelationshipmanagementefforts. On theotherhandthe knowledgeof the existenceof so called“hidden organizations”is of impor-tancefor the businessprocesseswithin the company whenprocessoptimizationis the goal. Thusthe knowledgeof communicationstructurehasbeenof majorimportancefor thesefields.

Makingall communicationstructureswith respectto socialinteractionvisiblehasbeenmadepossibleby applying different methodsto different “views” ofthe communicationof a givencommunity. Someof theseviews are: hierarchy,friendship,communicationwith, etc. Someof theemployedmethodsare: graphtheory, hubsandauthorityalgorithmsandstochasticprocesses.

4 CHAPTER1

But onemajoraspectof communicationhasbeenverydifficult to analyzedueto difficulties which arisefrom the mathematicaltools usedto explain commu-nicationstructurethroughcommunicationbehavior. Communicationis a processbywhich informationis exchangedbetweenindividualsthrougha commonsystemof symbols,signs,or behavior [Dic04]. Thusby definitionit is bilateral.Severalquestionsarisefrom this fact.For onethereis thelargefield of signalprocessing,which looksat theway signalscanbetransmittedfrom onepoint, thesource,toanother, the destination.Whensignalsarecombinedto createmeaningfulmes-sages,the questionof semanticandinformationarise. Informationhasbeenthefocusof mathematicalinformationtheorywhichwasfoundedby Shannonin 1948[SW63].

Communicationin thecourseof thisbookwill beseenasthebilateralprocessof exchangeof messagesbetweento membersof a network. Neither will thequestionbe discussedwhetherthe informationtransmittedhasbeenunderstoodby the recipient,nor will therebe any discussionaboutthe technicalmeasuresof information. The focus will be on the questionhow to identify patternsofcommunicationin a given network if all exchangesare includedin the analysissimultaneously. This hasuntil now posedseveral problemswhen trying to usestandardproceduresasfor exampleeigensystemanalysis. The underlyingdatahadto berepresentedin a realvaluedsocalledadjacency matrix.

Real numberscan only storedataon one communicationdirection insteadof two when looking at bilateral communication. To encodetwo components,datahadto be for exampledichotomizedto representthe dominantdirectionofcommunication.This led to asymmetricmatrices. Thesematricesare difficultto usein eigensystemanalysis,sincetheir spectrumcanyield negative or evenimaginaryeigenvalues,which aredifficult to interpret.

To circumventthis,matricesarebeingpreprocessedin differentwaysto makethemsymmetric,becausein this casetheeigensystemis easierto interpret. Oneway is to multiply thesematriceswith their transposedmatrices,which makesthe resultsymmetric,andthusbetterbehaved. Anothermethodis to usematrixdecompositionfor squarematricesto constructasymmetricmatrix.

Anotherproblemarisesfrom the fact that realvaluedmatricesuseEuclideanspace. Thus distancesare interpretedin a way, that is often inappropriateforthesubject.Especiallyin casesof non-transitivity, wheredistancesin Euclideanspacesuggestthat thereshouldbe transitivity, give rise to somedifficulties ininterpretation.

All thesemethodshave one problemin common. Information is lost or isproneto noise-inducederrors. Anotherproblemis the fact, that someof thesemethodsonly usethe principal eigenpairfor interpretation,thusneglecting theinformation that the rest of the eigensystemmight reveal aboutthe underlyingstructure.

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The objective of this book is to introducea generalizedmethodto analyzecompletelythe structureof an asymmetriccommunicationnetwork, to validatethismethodusingstandarddatasetsandto pointoutsomeapplicationsthatmightbenefitfrom thesefindings.Thedatasetsusedconsistof two way emailcommu-nicationwithin acommunity. Theresultswill beobtainedby usingcomplex Her-mitian adjacency matricesinsteadof realvaluedadjacency matrices.A complexnumbercanstorebothcommunicationdirectionsin onenumber. Theeigensystemof complex Hermitianmatricesyieldsa powerful tool to explain thecommunica-tion structureof a communitybasedon its oftenasymmetriccommunicationpat-ternstructure,dueto themathematicalpropertiesof thisclassof matrices.Severalpointsarenoteworthy, andwill bediscussedin thecourseof thiswork:

* The Hermitian matrix canbe decomposedinto a Fourier sum of its sub-spacesweightedwith therespectiveeigenvalue.

* All eigenvaluesof Hermitianmatricesarerealanddependontheamountofcommunicationwithin thegroup:Thismakesinterpretationof theseeigen-valuesas“energy levels” intuitive.

* Theeigenvaluescanalsobeusedto interprettherelevanceof patternsdueto the fact that they explain the part of the total variancecontainedin asubspace.

* The eigenvaluesare at the sametime also coordinatesin Hilbert space,showing therelevanceof thesubspacein referenceto theoriginalmatrix.

* The eigenvectorsform an orthogonal(e.g. pairwiseindependent)system:all eigenvectorscanbeusedfor interpretationpurposes.

* Theorthogonalprojector, derivedfrom theeigenvector, revealsthesimilar-ity patternbetweendifferentmembersin respectof thesubspacepattern.

* To show stability or instability of a matrix eigensystemwhendisturbancestake place- for exampleflamewars - perturbationtheorycanbe usedtoexplainwhathappensandwhy.

* No ex antenotion hasto be usedor filters appliedto preprocessthe data:No lossof informationhasto beaccepted.

Thesecharacteristicsgive rise to a clearerview on communicationpatternsin substructureswithin virtual communities(VC) basedon the communicationtraffic volumeandcommunicationbehavior within thegroup.

6 CHAPTER1

1.2 Guide to the Reader

In chapter 2 of this booka review of theresearchdonein this field will begivenand the relevant literaturewill be surveyed to introducethe readerto the topicof theuseof eigensystemanalysisin communicationstructureanalysis.A shortexcursionat theendof thatchapterwill show how communicationis analyzedus-ing graph-theoreticalapproachesor toolsusedin physics.Also someapplicationsfrom economicswill bepresented.

After thisintroductionto thefield,chapter3will establishthetheoreticalbasison which this book is grounded. Complex eigensystemanalysisof Hermitianmatriceswill bethemainsubject.

Chapter 4 will presentthe main ideaof this book. The constructionof thecomplex Hermitianadjacency matrix will beexplainedandsomeexamplesof ar-tificially constructedcommunicationpatternswill bediscussedto makethereaderfamiliarwith theapproach.

Chapter 5 describesthe datasetsthat have beenused. In orderto comparetheresultsof theproposedmethodto thosealreadyexisting, theEIES(ElectronicInformationExchangeSystem)datasetof L. andS. Freeman [FF79] hasbeenusedaswell asa subsetthereof [Fre97].

After thattheresultsobtainedwith theproposedmethodwill bepresentedandresultswill bediscussedextensively in chapter6. Heretheeigensystemsgainedfrom the dataareshown andinterpretationswill be given basedon the insightsgainedin chapter4. In additiona comparative interpretationwill begiven,whichpoints out the advantagesof the proposedmethodwith regard to the standardmethods.

Chapter 7 summarizesthe resultsof this book, refers to someinterestingapplicationsandpointsoutproblemsthatmightbeof interestfor furtherresearch.

Chapter 2

RelatedWork

Theaim of this chapteris to provide anoverview of researchdonein thefield ofcommunicationstructureanalysiswith thehelpof eigensystemanalysisof socialnetworks,especiallyin emailnetworks. To achieve that,a shortreview of litera-tureon rank,prestigeandstatusindicesaswell aseigenvectorcentralitywill begiven.Thiswill providethebackgroundfor thegeneralizedapproachfor commu-nicationstructureanalysispresentedin this book.

In section 2.2 degreebasedmeasurementswill be discussed,which give anintuitive introductioninto section 2.3 abouteigensystemanalysis,which startswith rank andprestigeor statusmeasurementsandgoeson to othereigenvectorcentralitymethods.Asasidetrackashortoverview of approachestowardnetworkanalysisfrom aphysicspointof view will begiven.Section2.4will thepointoutsomeapplicationsof themethodsdescribedin theprecedingsections.

2.1 SocialNetwork Analysis

A recentstandardtext bookfor SNA is by WassermanandFaust [WF94]. Socialnetwork analysis(SNA) offersmany toolsto describeand/orevaluaterelationshipwithin a given network or groupof people. It focuseson patternsof relationsamongpeople, organizations,states,etc. [GHW97]. This tool setallows to testtheoriesabouttheconceptsthatdescribetheserelationships.For amoreelaboratedefinition of SNA see [WF94, p.16]. This tool set hasover time grown andbecomemoreelaborate.

Thebasicconcept,thatis investigated,is thatof choice,meaning,thattheun-derlyingsociomatrixis built upfrom thecommunicationchoicesthatthememberswithin a network make. The importanceof choicein an economicalsettinghasfirst beenemphasizedin A. Samuelson´sRevealedPreferenceTheory [Sam48].Choicecanbe interpretedin a broadsense.As early as1984Freeman [Fre84]

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showedhow CMC, andthus“choice” of email partners,couldhelp to form andstrengthentieswithin ascientificcommunity. Also GartonandWellman [GW95]give a review on how theuseof electronicmail hasan impacton organizations.In 2001Wellman [Wel01,p.2032]pointsout, thatthefactthatonepersonwritesanemailto anotherpersoncanalsobeseenasa form of choice.

2.2 DegreebasedMeasurements

Theintuitiveexplanationof centralityandprestigeof anetwork memberis basedon thenumberanddirectionof choicesmembersin a network make. In additionto intuitivenessit is alsoeasyto encode.A network canbeviewedasa graph

wherea line/edgeconnectstwo vertices+ and , if thosetwo verticesarerelatedinany sense.If theconnectionhasa direction,thentheedgeis transformedinto anarrow startingat thevertex thatmadethechoiceandendsat thevertex thatwaschosen.

Oneof thefirst to introduceacentralitymeasurewasBavelas [Bav48], whointhefield of psychologytransferedideasfrom topologyto describegroupstructureandpossiblebehavior.

In graphtheoryit is known thatgraphscanbeencodedin socalledadjacencymatrices [Jun99]. The entriesin that matrix areeither - if a connectionexistsbetweentwovertices,or . if it doesnotexist. Thusthismatrixis eithersymmetric,if no directionis apparent,or asymmetric,if thechoicehasaninherentdirection.

A standardapproachfor the measurementof centrality is basedon degreemeasurements(seesection.3.2.1). The degreeis the numberof edges(choicesmade),that go out of (out-degree)or comeinto (in-degree)a vertex of a graph.It focuseson thequestionwho (or which group)canbeseenasmostcentralwithrespectto their in- or out-degreewith respectto their capacityto transferfor ex-amplemessageswithin a network. Thesemeasurementscaneitherbe usedasastandalonecharacterizationof a network, or in conjunctionwith othertools. Ev-erettandBorgatti [EB99] give an overview of thesemeasuresandeven extendit further to helpanalyzegroups.To usethesetools, measurementslike degree,closeness,betweennesscentralityandflow betweennesshave beendefined.Themajordrawbacklies in thefactthateitherin-degreeor out-degreecanbeused,butnotbothsimultaneously.

A branchof thedegreebasedapproachlooksat largenetworksandtheir evo-lution. Erdos andRenyi describedin their 1959 paperon randomgraphshowsuchsystemscanbedescribed.They proposed,thatnetworks(or graphs)arebuiltby the attachmentof new verticesto an existing graph,andthat the probabilitywith which thenew vertex attachesto anexistingvertex is equallydistributed.In1999BarabasiandAlbert found [ALB99], that for real world networks,suchas

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theWorld WideWebor thenervoussystem,theassumptionof equallydistributedattachmentprobability doesnot hold. They found that suchnetworks show ascale-freedegreedistribution in contrastto the distribution expectedby the ran-domgraphtheoryof ErdosandRenyi. Scale-freenetworksaredescribedby thefactthattheprobability /103254 thata vertex is connectedto 2 otherverticesfollowsapowerlaw distribution /603254�782:9�; . Thismeansthatverticeswith almostany de-greecanbefoundin suchanetwork. A specialinterestof thisbranchof researchisin thevulnerabilityof networks,sincein alargescale-freenetwork therearenodesthathaveaveryhighdegreethat,whenattacked,couldcauseacompletenetworkbreakdown. Large in this context meansthat thenumberof nodes/membersin anetworksexceeds-<.>= . As anexamplesee,Ebelet al. [EMB02]. They describeanemailnetwork of size ?A@CB�D � D:-"E nodes.They useemailaddressesasnodesandexchangedmessagesaslinks. For instantmessagingSmith [Smi02] showsalsothescale-freecharacteristicandpointstowardsaspectsof vulnerability.

Basedon degreemeasurementsthereis alsoresearchin thefield of commu-nity analysis.Therehave beentwo approaches.Oneapproachlooks for denselyknit groupswithin a network, which translatesinto high degreecentrality, whilethe other looks for thosememberswho have a high betweennesscentrality(seesection.3.2.1)andthencutthenetwork atthatpoint,repeatingthisprocedureuntilthenetwork decomposes.Seefor example [ADDG F 04].

A slightly differentstandardapproachis basedon themoregraph-theoreticalapproaches,even thoughit is closelyconnectedto the degreemeasurements.Itfocusesmainlyon theidentificationof subgroupswithin agivennetwork/graph.

Recentresearchby Tyler et al. [TWH03] showed that they could identifysubgroupsin emailnetworksby analyzingbetweennesscentralityin the form ofinter-communityedgeswith a largebetweennessvalue.Theseedgesarethenre-moveduntil thegraphdecomposesinto separatecommunities,thusre-organizingthe graphstructure. This is a min-cut-max-flow approachas discussedfor ex-amplein Ford andFulkerson [FF62]. The graphrepresentingthe network wasgeneratedby definingacut-off numberof emailsthatmustbeexchangedbetweenmembers,anda minimum numberof emailsthathadto be exchanged,so thatamemberwasincludedinto the graph. As discussedabove, this could meanlossof informationFor example,it is not possiblelater on to find out who werethereal “lurkers” in that network, sincethe thresholdeliminatedthese.Lurkersarethosenetwork memberswho standat thesidelinesandobserve whathappens.Ina newsgroupthis couldmeanthat thesearemembersthat readall messages,butonly rarelyparticipate,or donotgetanswersto their postings.

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2.3 EigensystemAnalysis in SocialNetwork Analy-sis

Theuseof eigensystemanalysisin SNA startedin thelate1940sandearly1950swhensocial relationshipswere encodedin so calledsociomatriceswhich werecloselyrelatedto adjacency or incidencematricesalreadywell known from graphtheoryat thattime.

In 1953Katz [Kat53] presentedan index basedon this idea,namelythat therank of a groupmemberdependson the rank of the membershe or sheis con-nectedto by choice. Thesechoiceswererepresentedin a binary socalledadja-cency matrix. This index doesnot only take into accountthedirectpathbetweentwo members,but alsolongerpathsthatconnecttwo membersvia othermembers.Statedin mathematicaltermsthisyieldstheeigenvalueequation(for aneigenvalueequalto 1). Thecomponentsof theprincipaleigenvectorarethestatusindicesofeachgroupmember. Thehighestrankedmember, theonewith the largeststatusindex, is theonewho is eitherchosenby few but high rankingco-members,or bymany relatively low rankedco-members.This approachthoughdid not presentasolutionto theproblemthatchoiceneednotbemutual.In 1965Hubbell [Hub65]extendedthis approachbasedon theLeontiefInput-Outputmodelto alsoincludeweightedor unilateralchoices. Thus the ideaof usingeigensystemanalysistodefinetherelevanceof groupmembersby comparingtheir individual statuswiththoseof othermembersthey areconnectedto is in itself nota new idea.

Sincethe early 1950sthe interpretationof the resultsof suchmethodshavebeenenhancedand its applicationsbroadened. Richardsand Seary [RS00],[SR03a] givea verygoodoverview of thedifferentapproachesusedin theeigen-systemanalysisof socialnetworks. Goh et al. [GKK01] show how the largesteigenvaluedependsonsystemsize,andthattheeigenfunction(eigenvector)pointstowardthenodewith thehighestdegree.They useeigensystemanalysisto furtherinvestigatescale-freenetworks,but notasa tool by itself.

Onefinding in thecourseof researchwasthatdistancesin networks(betweenmembers)arevery difficult to describe,sinceit may be the casethat memberGknowsmembersH and + , but themembersH and + haveabsolutelyno knowledgeabouteachother, thusdenying transitivity of the relationof knowing eachother.Transitivity, however, is a key factor in the conceptof distances.This leadstotheassumptionof non-Euclideandistancesin networks.BarnettandRice[BR85]showedthatif thematrix is for exampledefinedasadistancematrix,triads(trian-glesof members)formedbetweenthreenetwork membersneednotbeEuclidean.If sucha matrix is convertedby multi dimensionalscaling(non-linearprincipalcomponentanalysis)into Cartesiancoordinates,eigenvaluesmay becomenega-tive andcomplex eigenspacescanemerge. To avoid this, suchasymmetricbe-

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havior is normallyexcludedby designingthematrix in sucha way thatdistancesaredichotomized.But this evidentlymeanslossof information.BarnettandRiceshow thatthesenegativeeigenvalues

�JI . yield information,for exampleaboutthe homogeneityof the structureof the group. For this purposethey definethewarp astheratio of therealvariance(all positiveeigenvalues)andthetotal vari-ance(all eigenvalues). If this warp equals - , then the spaceis Euclidean. Thelargerit gets,themorenon-Euclideanthespacebecomes,whichmeans,thatmoreandmoreinteractionis donethroughlessandlessnodes.Thusthey founda wayto interpretthenegativeeigenvaluesthathadbeenaproblemin thefield.

Multi dimensionalscalingis alsothestartingpoint of Chino[Chi98], He wasthefirst to introducetheuseof Hilbert spacetheoryin theanalysisof preferencedatain psychology. He foundthatwith thehelpof Hilbert spacetheoryasymmet-ric relationshipsbetweensubjectscouldbebetterdescribed.Theinterpretationofnegative eigenvalueshasalsobeendiscussedin his work. He proposesthat thesigngivesahint on symmetricandasymmetricrelationsbetweensubjects.

Eigensystemanalysishasthusbeenpushedconsiderablytowardsthe expla-nationof negative eigenvalues.On the otherhandtherewereproblems,alreadydiscussedby Katz,aboutthegeneralquestionof whatstatusis supposedto mean,andwhereit comesfrom.

BonacichandLloyd [BL01] presentanintroductionof theuseof eigenvector-like measurementsof centralityfor unpreprocessedasymmetricdatathat broad-enstheeigensysteminterpretationin thatdirection.For asymmetricrelationstheyshow thatfor somenetworksthestandardeigenvectormeasuresfor network cen-trality do not leadto meaningfulresults. As a solution they suggestthe notionof K -centralitywhich combinesanindividual’sown inherentstatuswith a weightK for therelative importanceof theperceivedstatusanda vector � of exogenoussourcesof status.In their conclusionthey find thatthis K -centralitycouldbeseenasageneralizationof thestandardeigenvectorcentralitymeasure.

Thisapproach,to useanexternalor nodeinherentstatuseffect,wasalsotakenupby physicistswhouseIsingmodelsto explainthebehavior of networks.Onthebasisof theIsingmodelwhichdescribetheinteractionbetweenspinsoftenusedincondensedmatterphysics,someresearchers(for exampleHerrero [Her02]) havealsoachievedsomegoodinsightsinto interactionswithin a socialnetwork usingthismodel.Thesemethodsfocusontransitionof anetwork from onestateinto an-otherandthey try to determinetheexactsystemcharacteristics(e.g.temperature)thatleadto aphasetransitionof thesystem.

Therehavebeenotherapproachesto theinterpretationof asymmetriccommu-nicationbehavior with theuseof suchindices.Freeman[Fre97]usedit to showhierarchieswithin a group.He, likeChino[Chi98] andBarnettandRice[BR85],generatesthematricesthatarethenthebasisfor theeigenvalueanalysisby split-ting a givenasymmetricrealsociomatrixinto its symmetricandskew-symmetric

12 CHAPTER2

part. While BarnettandRiceandFreemanremainin the real space,Chinousesthis approachto shift into Hilbert space.

Onelastaspectis thatof structuralstability whenlooking at thecommunica-tion patternstructure.This problemis at thecenterof perturbationtheory. Thereis no literatureaboutthis topic in SNA. Still it is of interestto find out how muchperturbationa network can“digest” without changingits structure.In thecourseof thisbookthereforegeneralliteratureaboutperturbationof Hermitianeigensys-temswill beintroduced.Therearetwo aspectsto keepin mind:

* Perturbationof theeigenvalues.This is closelyrelatedto traffic volumeandrelevanceof pattern.

* Perturbationof theeigenspaces.This is closelyrelatedto a changein rele-vanceof thememberswithin thestructure.

A generaland extensive introduction into perturbationtheory of linear op-eratorsis given in Kato [Kat95] (originally publishedin 1966). Kato givesanintroductioninto BanachandHilbert space,andon this basisgoeson to describeperturbationtheory. As anapplicationof specialinteresthegivesexamplesfromquantummechanics.Heretheresultsof perturbationtheoryin Hilbert spacehavean applicationfor examplewhen looking at Dirac operatorsor the Schrodingerequation.

A modern,moremathematicalapproachto perturbationboundsof eigensys-temsof Hermitianmatricescanbefoundfor examplein Dopicoet al. [DMM00],Mathias [Mat97], Barlow and Slapnicar [BS00], Fonseca [dF00] and Ipsen[Ips03] .

In agraph-theoreticalsettingCvetkovic etal. [CS97] giveanintroductionintothe effectsof perturbationon the eigenvectorsandeigenvaluesof graphsrepre-sentedby real valuedadjacency matrices.AppendixB [CS97, p.235-238]pro-videsa tableof eigenvalueandeigenvectorconfigurationsfor differentkinds ofgraphsup to B vertices.

As a lastsetof methodswe presentthoseusedto describetopologyandevo-lution of large networks. For a good overview of thesemethodsseeBarabasiandAlbert [BA02] andBarabasiandBianconi[BB01]. Heredifferentmethodsarebeingdescribedrangingfrom randomgraphsto percolationtheoryandBose-Einsteincondensation.Thesemethodshave beentaken from differentfields inphysicssuchascondensedmatterphysics,solid statephysicsandstatisticalme-chanics. The idea is that in all of thesefields thereexist extensive methodstodescribethe evolution andbehavior of either latticesor particleaggregates. Itis only a shortstepfrom theseconceptsto the ideathat humannetworks mightfollow similar lawsasthosedescribedin physics.

CHAPTER2 13

A field of physicsthatalsohasmuchto offer whenlooking at thebehavior ofnetworks is nonlineardynamics.SmaleandHirsch [HS74] or Strogatz [Str01]giveanintroductionto thefield. Thesemethodsplay animportantrole for exam-ple in theapplicationto viral marketing,wherethenonlinearbehavior on nearlyscale-freenetworksis anecessaryconditionfor success.

2.4 Other Applications of Eigenvector Centrality

Oneof thosemany applicationsnot within SNA, which will not be further dis-cussedin the courseof this work, has beenpresentedby Kleinberg [Kle99],[GKR98] who showed how web siteson the internetare linked to form groupsin view of their commoncontext. In a hypertext context the fact that onewebsite featuresa link to anotherweb site canbe definedasa choice,andthusthelink definesanentryin theadjacency matrix. In this context thematrixexplainsadirectedgraph,at thatpointwithoutweightsattachedto theedges.Kleinberg usestheconceptof hubandauthority, whichhederivesfrom incrementallycomputingandapproximatingtheprincipaleigenvalueandtheleft andright handeigenvectorbelongingto theprincipaleigenvalue.Theinterpretationis limited dueto thefactthat it identifiesonly the major hubandthe major authority, but doesnot revealmuchaboutthe substructure.Pageet al. [PBMW98] usedthis approachto cre-atethePageRankalgorithmwhich is now thebasisof theGooglesearchengine.A goodreview of thesemethodsandresultscanbe found in Park andThalwall[PT03]. A collectionof algorithmsthat arebeingusedin that field is given inBorodinetal. [BRRT01].

AnotherInternetrelatedresearchwasdonebyYaltaghianandBehnak[YC02],whousedifferentmethodsto analyzetheresultsof theGooglesearchengine.

Applicationsin organizationresearchhave alreadybeenpointedout in 1945by Simon(see [Sim00]). He suggeststhat theuseof email in organizationswillchangethe way information is being distributed, and that it may well causeadifferentway of informationdiffusion thanbefore. An examplethat this is thecaseis given in a paperby Krackhardt [KH93], who usesmethodsof SNA touncover hiddenstructureswithin anorganizationcomparedto thosegivenby theformal organization. In managementsiencethe evolution of organizationsis ofsomeimportance.This behavior canalsobe analyzedusingmethodsdescribedabove asfor examplein Guimeraet al. [GDDGF 02]. A goodoverview of theapplicationof socialnetwork analysisin organizationalresearchcanbefound inBorgatti andFoster[BF03].

In a morecommunicationmethodbasedarticleSmith [Smi02] appliessomeof theabovementionedmethodsto instantmessagingsystemsjustasEbel [EMB02]appliedthemto emailnetworks. The largestemailnetwork is mostprobablythe

14 CHAPTER2

USENET. A thoroughanalysisof this large network canbe found in Choi andDanowski [CD02].

The ideato to usethe communicationflow betweendifferentpoints(or ver-tices)to detectpatternsandactaccordingto theseresultsplayeda major role innaval warfareduringthesecondWorld War whensignalintelligencewasusedtofind from thepatternsof communicationthewhereaboutsof targets.The ideatoanalyzecommunicationstructureswith mathematicalmethodsdatesbackto the1950sandhassincegrown in relevanceandcapability. Yet thesimultaneousanal-ysis of inboundand outoundcommunicationbetweentwo members/nodeshasposeddifficulties,aswell asthefactthattheeigensystemanalysisusedup to nowdoesnot provide a completeinterpretationof the eigenspacethat could yield adetailedview on thepatternstructure.

Chapter 3

Theoretical Foundationsfor PatternAnalysis in SocialNetworks

In this chapterthe theoreticaland methodologicalbackgroundfor the analysisof communicationpatternsin socialnetworks will be presented.In section 3.1notationsanddefinitionsnecessaryfor SNA will be given. Section 3.2.1 willexplain theconceptsof rankandstatusandhow they canbemeasuredin a givennetwork. Thelastsection3.3will presentthemathematicalandgraph-theoreticalconceptsthatwill be needed.The main focusof this last sectionis to make thereaderfamiliarwith complex Hilbert spaceandthebehavior of linearoperatorsinthatspace.

3.1 SocialNetworks

3.1.1 Definitions

A socialnetworkwill bedefinedasin WassermanandFaust [WF94, p.20] asafinite setor setsof actors and the relation or relationsdefinedon them. (SocialNetwork.)

Therelationin thecasesdiscussedherewill bethesendingof andansweringto emails.Thenext smallerunit is thegroup.

Wellman[Wel01]definesagroupasaspecialtypeof socialnetwork;onethatis heavilyinterconnectedandclearlybounded.(Group.)

A subgroupis asubstructureof agroup,thatis definedby therelationtowardsits anchor (seebelow).(Subgroup.)

A virtual community(VC) following Rheingold [Rhe00,p.xvii] is a groupofpeoplewho usewordson screensto exchange pleasantriesandargue, engage inintellectualintercourse.... (Virtual Community.)

15

16 CHAPTER3

A moreformalizeddefinitionof anonlinecommunitycanbefoundin Preece[Pre00,p.10].Suchacommunityconsistsof

* peoplewho haveany kind of socialinteraction,

* haveacommongoal,asfor exampleinformationexchange,

* performtheir exchangeswithin certainrulesor socialnorms,

* andusecomputersto communicate.

ThusaVC is aspecialkind of asocialnetwork.In thecontext of thiswork ananchor is definedasthemostprominentmember

of asubgroup.(Anchor.)Thisprominenceis expressedby thefactthatthecomplex rankprestigeindex

hasthehighestabsolutevaluein theeigenvectorunderinspection.The terman-chor canbe usedarbitrarily eitherfor the mostprominentmemberof the wholegroup,or themostprominentmemberof asubgroup.Thenetwork‘s anchoris thememberwith thehighestabsoluteeigenvectorcomponentvaluebelongingto thehighestabsolutevalueeigenvalue.

Networks,andthuscommunicationnetworks,will berepresentedasweighteddirectedgraphs

@ ���L��������� wherethe M membersof the network are thevertices�N� � . Connectionsbetweenmemberswill bemappedasweightededges�O� � . Thegraph

will bedescribedby a weightedMQPJM adjacency matrix

�.

An entryof this matrix �����SR@T. if thereis a connectionbetweentwo vertices� �and � � , which is equivalentto �U0V+ � ,W4X� � . In theunweightedandundirectedcasetheadjacency matrixhasentries����� @Y- if �U0V+ � ,W4Z� � and . otherwise.

3.2 Rank and Status

Thissectionwill introducerankandstatusastermsandconceptsthatarerelevantto this book. After a discussionof several definitionsof thesetermsdifferentmeasurementsfor theseconceptswill begiven.

In Tab.( 3.1) definitionsof the conceptsby differentauthorsaregiven. Thegermantextshavebeentranslatedfor betterunderstanding.No standarddefinitioncanbe given for eitherof the concepts.In additionTab.( 3.2) shows how someof the conceptsare the foundationsfor otherconcepts(representedby “ [ ”) orarederivedon thebasisof anotherconcept(‘” \ ”), or areusedin a similar wayor evensynonymical (“ 7 ”). Only theuppertriangularof this matrix is filled forclearervisibility.

CHAPTER3 17C

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18 CHAPTER3

socialposition Role Status Rank Prestige

socialposition - [Role - [ 7Status - 7 7Rank - [

Prestige -

Table3.2: OrderandSynonymity of role,status,rank,prestige

Rankandstatusarerelatedin the sensethat sincethe word statusis mostlyusedin conjunctionwith a rating like “high” or “low”, or that thecontext yieldsthisinformationthesetwo areusedsynonymical. Sometimeseventheword“pres-tige” is usedasa synonym. But all thesethreearebasedon the role a memberholdsin agivencommunity.

In thecontext of this booktheseabstractconceptswill beusedto describere-sultsof mathematicalevaluationwith respectto thecommunicationbehavior of amemberof thegroup.Thefollowing view is adopted:a membercanhave differ-entroles,for exampleanchoror lurker (seelater for a detailedexplanation).Therolehasaninherentrankor statusandat thesametimeaninherentprestige.Bothdependon the mathematicalcomputationdefinedon the communicationtrafficgeneratedby eachmember.

3.2.1 DegreebasedRank

In this subsectiontherankbasedon in- or out-degreewill bedescribed.Thusanintuitive introductionto the conceptof rank is given even thoughthe proposedmethodis not restrictedto degreecentrality. Degreein this context whendirectedweightedgraphsare observed is the numberof edges� that eitherdepartfroma vertex � (out-degree: 2�]� � � ) or are pointedtowardsthe vertex (in-degree:2�^� � � ). In the caseof weightedgraphswe alsodefinethe strengthfollowingBarratet al. [BBV04] as _<]� and _`^� asthenumberof messagesthateithergo outfrom vertex + or comeinto vertex + overall connectedvertices.

Centrality

Thereareseveraldefinitionsfor centrality a � � � . Centralityhereis definedfol-lowing WassermanandFaust[WF94] by thefact that in a non-directionalsettinga memberis saidto be central,if he is connectedto many othermembers.In adirectionalsettingcentralityis definedby thechoicesthemembermakes.This isreflectedin thenumberof differentothermembershecommunicateswith.

CHAPTER3 19

Degreecentralityasdefinedin WassermanandFaust [WF94, p.199]focusesonthechoicesthatarebeingmadeby themembersof anetwork, thusthecentral-ity in that caseis basedon the out-degree. A centralmemberis thusa memberthathasmany outgoingconnectionsto differentmemberswithin thenetwork. 2�]�of vertex + is definedasthenumberof verticesthat + is connectedto asgiveninEq.(3.1)

2 ]� @bc�edgfih ��� (3.1)

with h ��� @j- if �����LR@8. and h ��� @8. if ����� @8. . Thusthestandardizedcentralityinadirectedunweightedgraphcanbedefinedasin Eq.(3.2):

a � @ 2�]�0WMlkm-n4 with a � � � (3.2)

In theweightedcaseit is not possibleto standardizeanymore,sincethenum-berof possibleconnectionsis no longerknown. In thatcasejust the row sumistaken. This thenhasthe problemof comparabilitybetweennetworks. (Degreecentrality.)

Strengthis theconceptin theweightedcase.Thestandardizedform is givenin Eq.(3.3). It is givenastherow sumof theweightedadjacency matrix dividedby thetotal sumof matrix entries,with ����� � � and M thenumberof verticesormembersin thenetwork.

_ ]� @ o b �edgf �����o b ��dgf o b �pdgf ����� with _ ]� � � (3.3)

Thereareseveralderivedconceptsof centralitywhich will bepresentedwith-out further explanation. We will presenthereonly thosethat arerelevant in thecontext of directionalrelationships.It shouldalsobenotedthatall of thesecen-trality measurescan be generalizedto group centrality measures.This can beachievedfor exampleby defininghypernodesthatcontaintheverticesthatbelongto thatgroup.

Standardizedclosenesscentrality is definedfollowing again [WF94, p.200]for adirectionalandunweightedgraphasin Eq.(3.4):

aq, � @ Mlkm-o b�pdgf 2�Gr_`st0V+ � ,34 with aq, � � � (3.4)

with 2�Gr_`st0V+ � ,W4 denotingthe shortestdistancebetweennode + and , . Closenesscentralitygivesan indicationof how closea memberis to theothermembersofthegroup.Thiscanintuitivelybederivedfrom thedistancesbetweenvertex + andall othermembers.It shouldbe mentionedthoughthat if the actoris an isolate,

20 CHAPTER3

meaning,that he hasno connectionsat all, the sumof distancesbecomes. andthustheindex is undefined.(Closenesscentrality.)

Betweennesscentrality uva � � � putsthefocusonthefactthatavertex � with ahighbetweennesscentralityis likeabottleneck.All (or much)of thecommunica-tion within agivennetwork hasto go throughthis vertex. Betweennesscentralityasdefinedby Eq.(3.5) is thusbasedonthenumberof pathsthatgo throughacer-tain vertex whenconnectingtwo verticesin relationto all pathswithin a networkthatconnectthesetwo vertices.

uqa � @ ob �pdgfrw �tx5�vy ^���{z y ���0WMlk|-"4t0WM}k~E�4 with uva � � � (3.5)

with y ��� thenumberof pathsbetweenvertex + and , and y ^��� thenumberof pathsbetweenvertex + and , that containvertex G . This view requiresthe assump-tion that the choiceof communicationchannel/pathis equallydistributed. Thatmightnot bethecase,if preferentialbehavior is takeninto account.Betweennesscentrality hasbeenusedto detectclusterswithin networks [GN02], [NG03].(Betweennesscentrality.)

Informationcentralityis a strongerversionof betweennesscentrality, in thatit now takesinto accountthatpathsbetweentwo verticesmight not be takenforthe transportof informationon an equallydistributedbasis. Latora[LM04] ex-tendedthis index to incorporatesomenotionsfrom flows in non-directionalandweightedor unweightedgraphs,specifically the notion that information mighttake the shortestpossiblepath available betweentwo vertices. Thus the effi-ciency � ��� � � of two verticesregarding information transferis presumedtobe proportionalto

f� ^ b<��� ^������ �qw �e� . The efficiency of the network is then the aver-

age ��@�������q������� � �b � b 9 f3� . If now a graph w� is constructedwhereall edgesincidentto

vertex + have beenremoved, thenthe informationcentrality GVa � is definedasinEq.(3.6)

G�a � @ ��kQ� �� � � (3.6)

Informationcentralityis thustherelativedrop in thenetworkefficiencycausedbytheremoval of edgesincidentto theobservedvertex.(Informationcentrality.)K -centrality as definedby Bonacich[BL01] given in Eq.( 3.7) addsan ex-antestatus,givenasa vector � to thecentralitymeasureandweighsthe relativeimportanceof the grouped-derived rank in comparisonto the memberinherentstatusby acoefficient K .

@�K � � �� � (3.7)

with�

theunweighted,directedadjacency matrix.(K -centrality.)

CHAPTER3 21

In a graph-theoreticalsetting,on the otherhand,a centercanbe definedasin Jungnickel [Jun99,p.91]. A vertex � is central if it hasminimal excentricity.Excentricity � % beingdefinedas � %�� @¡  ��%�� 2�Gr_`st03+ � ,W4 . Thusavertex + is seenasat thecenterof agraphif Eq.(3.8) is calculated,

y a � @¢ lGVM � � %�� with y a � � � (3.8)

whichexpressesthefactthatthisvertex hastheminimallongestdistance2�Gr_`st0V+ � ,34of all verticesin thenetwork. Thismeasureis definedfor bothnon-directionalanddirectionalsettings.It is a generalizationof degreecentrality, in thatall distancesareused,notonly thoseto thenearestneighbors.(Graphcentrality.)

Prestige

Following [WF94, p.202]prestigein comparisonto centralityrepresentsthede-greeto which a vertex is chosenby otherverticeswithin a directedandpossiblyweightednetwork. It is calculatedasin Eq.(3.9).Thusa vertex hasa high pres-tige, if it hasa high in-degree,definedanalogousto theout-degreefor centralitywith 2�^� @ o b �pdgf h �p� thecolumnsumasthein-degreeof vertex + :

/ � @ 2�^�0WMlkm-n4 with / � � � (3.9)

Thestrengthin this caseis definedanalogousto thetheoutboundstrength.Itis thecolumnsumof matrix

�, asin Eq.(3.10).

_ ^ � @ o b ��dgf �����o b ��dgf o b �edgf ����� (3.10)

Proximity prestigeis the in-boundanalogonto closenesscentrality. It is de-finedasin WassermanandFaust [WF94,p.204]by Eq.(3.11):

/>/ � @ £ � z 0�Mlk|-"4o 2�Gr_`st03, � +¤4 z £ � with /�/ � � � (3.11)

with £ � representingthe numberof memberswho are in the influencerangeofvertex + , whichmeansmemberswhocanreachvertex + . Thusproximity prestigeasdefinedoverdirectedunweightedgraphs.(Proximity prestige.)

3.2.2 Rank Prestigeor Statusindex and eigenvector centrality

As was shown in section 3.2.1 neithercentrality nor prestigeby itself give acompletepicture of the relevanceof a vertex within a network. By finding anindex thatcombinesbothviews,thiscanbeachieved.Statusin thesensethatwill

22 CHAPTER3

beusedin thiswork will have themeaningthatavertex is centralandprestigiousat the sametime. Since [WF94] usestatusandrank prestigeassynonyms, thiswill alsobethecasefor therestof this work. Statusor rankthusin thiswork willbedefinedby thetraffic in emailvolumethatis exchangedandby thepatternthatbecomesvisible in theeigensystem.

Thestatusindex asdefinedby Katzusesfor thecalculationnotonly thedirectpathsbetweentwo neighbors,but alsoall pathsconnectingtwo verticessincethe+ -th power of the adjacency matrix

�hasasits entriesthe numbersof pathsof

length + betweentwo vertices.This leadsto theinterpretationthattherankof onememberdependson therankof thoseheis connectedto.

In generalthis is referredto aseigenvectorcentrality, sincethis is calculatedasfollows:

Theideaof therankprestigeindex is basedontheideathatnotonly thedirectneighborhoodof a vertex influencesthe rank of a vertex, but also the rank ofthe verticesthat areconnectedto vertex , by way of othervertices. This holdsalso in a directedgraph. This can be representedby the sum of an adjacencymatrix

�poweredto /Q@¥-§¦`¦<¦�¨ . That thepower of anadjacency matrix gives

the numberof + -lengthpathsbetweentwo verticesis known from graphtheory[Jun99, p.96]. Now the questionis, how to find the rank index given for eachmemberof a network basedon the sumof the poweredmatrices. This canbesolvedwith the help of two mathematicalproperties.For oneevery function ofa diagonalizablematrix leadsto a functionof its eigenvaluesandsecondthefactis used,thata powerseriesconvergesundercertainconditions.Thesetwo resultswill beintroducedin section 3.3.4.

Thecomponentsof thecorrespondingeigenvectorsaretheindicesthatrepre-senttherankof themember.

Statusasproposedby Katz [Kat53] is definedby the questionasked. In thecourseof this work statusis thushigh if onememberis identifiedasananchor.Thememberwith thehighestoverallstatuswill beidentifiedastheanchor corre-spondingto thehighestabsoluteeigenvalueof thesystem.

3.2.3 Communication Patterns

In additionto eigenvectorcentralitytheprojectorsderivedform theeigenvectorsyield an interpretation.As will be explainedin moredetail in section 3.3.5theorthogonalprojectorin Eq.( 3.58) givesa representationof structuralsimilarityfor memberswithin asubspaceandthepatternstructureof theoriginalspace.See[BL04] for adiscussionof thecasefor

©���������ª���with

� � � .

CHAPTER3 23

3.3 Linear Operators in Hilbert Space

In thissectionthemathematicalbackgroundis presented.As themethodproposedin this book is basedon the complex valuedrepresentationof communication,complex numberswill first be introducedin section 3.3.1. As a secondsteptheconstructof Hilbert spacewill bepresentedin section 3.3.2. Section 3.3.3willintroduceHermitianoperatorsin Hilbert space,followedby eigensystemanalysisof suchoperatorin section 3.3.4. In section 3.3.5orthogonalprojectorsarede-fined,which arecomputedfrom theeigenvectors,andthe interpretationof theseprojectorsin thecontext of communicationpatternanalysisis given. SinceHer-mitian operatorsbehave very predictableunderperturbation,which is of interestwhenanalyzingcommunicationpatterns,section 3.3.6will introducethosechar-acteristicsof Hermitianoperators.Finally section3.4will presenttheapplicationof the proposedmethodto two typesof graphs,namelythe star graphand thecompletegraph.Resultsin thebidirectionalsymmetriccasewill becomparedtothe undirectedcase,andresultsfrom Cvetcovic, Rowlinson andSimic [CS97]will beusedfor validation.Also eigensystemsof asymmetricbidirectionalgraphsandperturbedgraphsaregivenanddiscussed.

3.3.1 ComplexNumbersand Hermitian Matrices

Unlessotherwisestatedall numbersarecomplex ( � � ). Columnvectorswill bewritten in bold face , with components%5« � H�@¬-L¦<¦<¦­M . Thevectorspacewill bedenotedby

� @ � b . Capitalletterswill representmatrices�

with �U�ª� denotingthe entry in the k-th row andthe l-th column( + � ,®@¯-L¦<¦<¦­M ). Greekletterswillstandfor theeigenvalueswith

� � representingthek-th eigenvalue. �� will denotethe eigenvector correspondingto the + -th eigenvalue. The complex conjugatetransposeof a vector will be givenas 6° . The transposeof a vector will bedenotedas � .

Theequation%�±�� -�@Y. hasno solutionin�

. By introducingtheimaginaryunit G with G ± @ k�- this equationis solvable. The set of the real numbers

�augmentedby the imaginaryunit G andtheoperations� and ² definedsuchthat0 �� G�uq4 � 0Wa � GV2³4§@j0 �� a`4 � G�03u � 254 and 0 �� GVuq4´²60Wa � GV2³4§@j0 � a:k�uv254 � G�0 � 2 � uvat4with � � u � a � 2�� � is definedasthefield of thecomplex numbers

�.

As anintuitivealternative introductionto complex numbersconsidera vectorµ in a2-dimensionalplanein�

asin Fig.( 3.1).Whenwe now renamethe % -axis to “real” andthe ¶ -axis to “imaginary”, the

transitionto theGauss-planehasbeenachievedascanbeseenin Fig.( 3.2).It canthuseasilybeseenthatthecomplex number· canberepresentedlikea

vectorin algebraicform or equivalentlyin polarform:

24 CHAPTER3

x

y

x1

y1

p=x1+y1

Figure3.1: Two componentvectorin� ±

Re

Im

x1

y1 p = x1 + i * y1

Figure3.2: Two componentvectorin theGaussplane

·¸@ ��� GVuZ@ � · � � ^º¹ (3.12)

with therealpartof · beingdenotedas »��U03·�4L@ � , theimaginarypartas £  ¼03·�4½@u , with G theimaginaryunit ( G ± @¾k�- ). · denotesthecomplex conjugate·¸@ � k¿GVuof · . Theabsolutevalueandphasearedefinedby Eqs.( 3.13)and( 3.14).

� · � @ÁÀ � ± � u ± (3.13) @ÄÃ>ŪÆtÆ`Ç�È »��U0W·�4� · � . I¡ÂÉI|Ê (3.14)

Thefollowing fiveequationsarehelpfulwhendealingwith complex numbers:

CHAPTER3 25

· f · ± @ � · f<�Ë� · ± � � ^Ë�e¹"Ì F ¹tÍ � (3.15)·�@ · if andonly if ·�� � (3.16)·�·�@ � · � ± (3.17)· � ·�@8E>»��U0W·�4 (3.18)

· f · ± @¬03»��U0W· f 4λ��U0W· ± 4 � £  ¼03· f 4 £  ¼03· ± 4ª4� G�03»��U0W· ± 4 £  ¼03· f 4ikÏ»��U0W· f 4 £  ¼03· ± 4Î4 (3.19)

Eq.( 3.15)shows that themultiplicationof two complex numbersis achievedby multiplying the absolutevaluesand addingthe phasesof the two complexnumbers. Multiplication by a complex numberis thus equivalent to a rotationabout

 ± andscalingby � · ± � ascanbeseenin Fig.( 3.3).

Re

Im

p=|p| e^(iφ1)

q=|q|e^(iφ2)

r=|r|e^(iµ)=|p||q|e^(i(φ1+φ2))

Figure3.3: Multiplication of two complex numbersh and/Eq.(3.16)statesthatacomplex numberandits complex conjugate(or adjoint)

numberareequalif andonly if · is real. Eq.(3.17)expressesthatthemultiplica-tion of a complex numberwith its complex conjugategivesthethesquareof theabsolutevalueof thatnumber, which canalsobeexpressedas »��U03·�4 ± � £  ¼03·�4 ± .This againis a realnon-negative number. Eq.( 3.18)statesthefact, that thesumof a complex numberandits complex conjugateis two timestherealpartof thecomplex number. Eq.( 3.19)givesa representationof the productof a complexnumberandacomplex conjugatenumber. Thiswill berelevantwheninterpretingtheorthogonalprojectors.

In analogyto complex numberswe now introduceHermitianmatrices,sincescalarscanbeviewedas -�PÐ- matrices.

A matrix'

is calledHermitian,if andonly if

26 CHAPTER3

' ° @ ' (3.20)

with' ° representingthecomplex conjugatetransposeof H. Thecomplex conju-

gatetransposeof a matrix'

is found,whentheoriginal matrix'

is eitherfirsttransposedandtheneachentry is complex conjugated,or vice versa.Eq.( 3.20)meansthat thematrix entriescanbewritten as Ñ �p� @ Ñ �ª� . Theanalogonin

�is

thesymmetricmatrix. Hermitianmatricesarealsonormal(Eq. 3.21)asfollowsfrom Eq.(3.20):

'É' ° @ ' ° ' (3.21)

A matrix'

is unitarily similar( Ò ° ' Ò¯@�Ó , with Ò unitaryand Ó diagonal)to a diagonalmatrix if it is normal. This meansthat for normal matricestheeigenvectorsform anorthogonalsystem.

Any matrix Ô � � b canbewritten asin Eq.(3.22).With thefirst partgivingahermitianpartandtheseconda skew-hermitian( Ô ° @YkÕÔ ).

Ô @ -E 03Ô � Ô ° 4 � -E 0VÔ k Ô ° 4 (3.22)

Thecharacteristicsof Hermitianmatriceswhich areof specialinterestin thiswork are thoseof their eigensystem.Thesecharacteristicswill be discussedinsection 3.3.4.

3.3.2 Hilbert Spaceasa specialVector Space

Hilbert spaceis a

* complete

* normed

* innerproduct

vectorspace.In thefollowing thesecharacteristicswill beintroduced.For any given vectorspace

�on any field the following equationshave to

hold:

CHAPTER3 27

N�~! � � Ö � ! � � (closure)(3.23)

0 N�~! 4 � × @ N� 0 !}� × 4 Ö � ! � × � � (associativity)(3.24)N�~! @ !}�Ï Ö � ! � � (commutativity)(3.25)ؤ٠[ �� Ù @ Ö � � (neutralelement)(3.26)Ø -S[ - @ Ö � � (neutralelement)(3.27)Ø 0�k 4§[ �� 0­k 4§@ Ù Ö � � (additive inverse)(3.28)�� � � Ö � � �X� � �(3.29)

0 � uq4 @ � 03u 4 Ö � � uS� �Z� Ö � � (associativity)(3.30)� 0 N�~! 4L@ ��l� ��! Ö � � �Z�ÎÖ � ! � � (distributivity)(3.31)

0 �Õ� uq4 @ ��l� u Ö � � uÕ� �ZÚ­Ö � ! � � (distributivity)(3.32)

A norm / on agivenvectorspace�ÜÛ � b

is givenby

& &�Ý @¬0bc��dgf �n%��©�

Ý 4 f3Þ Ý (3.33)

Thenormsmostoftenusedarethe - -norm(& & f @ o b ��dgf �n%���� ), the E -norm

(& & ± @j0 o b��dgf �n%��©� ± 4

f3Þ ± ) andthe ¨ -norm(& &qß @¡  ��%����´%���� ).

In Hilbert spacethe norm is analogto the E -norm, but is basedon the innerproduct,asdefinedin Eq.(3.34). The inner productof and ! is a semilinear(conjugate-linearin thefirst partandlinearin thesecond)form on a givenvectorspace

�andis representedas

� �n!i# @ ° ! @bc��dgf %�� ¶ � (3.34)

Theinnerproductnormis consequentlydefinedby Eq.(3.35):à � ~�n1# @ & & (3.35)

28 CHAPTER3

This is the analogonto the Euclideanor E -norm& & ± @ à o b ��dgf �"%��¸� ±

which by Eq.( 3.17) is thesameasào b ��dgf %��q%�� , which following Eq.( 3.34) is

thesameasEq.(3.35).For agivenarbitrarynormthefollowing rulesmusthold:

& &Õá . (3.36)& & @�.�â @ Ù (3.37)& �� & @ �´�ã� & & � Ö � � � (3.38)& N�~! & I & & � & ! & (3.39)

A vectorspace�

is saidto becomplete,if everysequence� % b � converges.In

acompletenormedspacetheCauchyconditionäæåËç � ^ b � b w � �{è ß & �é k �ê & [ .is satisfied.

Thematrixnorm& � &

is aninducednormfor a givenvectornorm. Thusthematrixnorminducedby theinner-product-normis statedin Eq.(3.40).It explainsto whatextentavectoris stretchedby

�or shrunkby

� 9 f if � is nonsingular.

& � & @¡  ��%�ëíì>ë dgf & � & (3.40)

Therearetwo matrixnorms,whichareof interestin thiscontext:

* TheFrobeniusnormasdefinedin Eq.(3.41)

& � & ±î @ c �qw � �´������� ± @ms�ï � aq�U0 � ° � 4 (3.41)

* The 2-normasdefinedin Eq.( 3.42) following Eq.( 3.40) andEq.( 3.52).Thisnormwill betheoneusedasthestandardmatrixnorm,if nototherwisestated.

& � & @ � � �$ð�ñ � (3.42)

If�

is nonsingularthen& � 9 f & @ fò óvô¤õ÷ö>ò .

Eq.(3.42)definesat thesametime thespectralradius,which will beintro-ducedin moredetail in section 3.3.4.

In Hilbert spacethusthefollowing equationshold:

CHAPTER3 29

� ~�´6# á . with� ø�n1# @Ä. if andonly if @Ä. (3.43)� �� �n!$# @ � � �"!$# Ú � ~�´��!$# @ � � �n!$# � � � (3.44)� N�~!ù�"×5# @ � ~��×5#�� � !ù�´×5# (3.45)� �n!$# @ � !ù�n1# (3.46)

3.3.3 Hermitian Operators in Hilbert Space

A linearoperatoron a givenfield is definedasa functionthatsendsevery vector � ú into anothervector ! � ú by a linear transformation.Thusthe transfor-mationin

�f @bc^ dgf af ^ ! ^¦<¦<¦

b @bc^ dgf ab ^ ! ^

(3.47)

with a ��� �Ïû canalsobewritten as @jû ! with a ��� theentriesof thematrix û .û is thencalleda linearoperator.The adjoint space

� ° of a vectorspace�

is the setof all semilinearforms(e.g. inner productEq.( 3.44))on a vectorspace

�[Kat95, p.11] (Pleasenote

that Kato usesrow vectorsandthusthe inner productis definedaslinear in thefirst part andsemi-linearin the second). A selfadjoint linear operatoris calledHermitian.In Hilbert spacetheadjointspace

� ° canbeidentifiedwith thevectorspace

�itself. For a proof see [Kat95, p.252-253].Linearoperatorsasdefined

in Eq.(3.20)will beof interestfor therestof this book.Thefollowing (in)equalitieshold in anormedinnerproductspace:* Cauchy-Schwarzinequality:� �Vü �n1#"� I à �Vü � ü # à � ~�´6# (3.48)

with equalityif @ � ü � � � � .* Besselinequality: for any vectorü

andan orthonormalbasis³ý with HJ@- � ¦<¦<¦ � M of avectorspaceú bc «ªdgf � �3ü �n:ýÎ#´� ± I &Züù& ± (3.49)

30 CHAPTER3

If the ³ý formacompleteorthonormalsystemtheBesselinequalitybecomesParseval’sequality: bc «ªdgf � �3ü �":ýÎ#´� ± @ &þüù& ± (3.50)

* TriangleinequalityEq.(3.39)&(ü �~ & @ à �3ü �~ � ü �~1# I &Züù& � & & @ à �3ü � ü #6� à � �"1#(3.51)

with equalityif @ � ü asin Eq.(3.48)andin addition � � � .

In Hilbert spaceEq.( 3.48)becomesan equalityandEq.( 3.50)holdsdueto thefacttheconditionfor equalityhold.

ThusHilbert spaceis a veryconvenientvectorspacefor Hermitianoperators,sincefor thepurposesof this book,many featureshelp to leadto a quitenaturalinterpretationin thecontext of SNA.

3.3.4 Eigensystemanalysisof Hermitian Operators

In thissectionthemainpartof themethodologicalbasisof thiswork is introducedsincethe methodproposedis basedon the eigensystemanalysisof Hermitianmatrices.

The eigenvalue equationof a Hermitian matrix'

with eigenvector andeigenvalue

�is givenby Eq.(3.52):

' @ � (3.52)

Hermitianmatriceshave thefollowing characteristics:

* All eigenvaluesare real (Eq.( 3.53)) andcan thusbe orderedeither from� ��ð�ñ @ � f�ÿ ¦<¦<¦ ÿ � � ^ b @ � b , or asin thecontext of this bookby theirabsolutevalue.

� � � � Ö + with � � f � @¢  ��%��©� � �¸��ÿ ¦<¦<¦ ÿ�� � b � @¡ lG3M ��� � �¸�(3.53)

because�V' j� 1# @ � � ¬� 1# @ � � C�g1# and

� C� ' 1# @ � C� � 1# @� � ø�´1# andEq.(3.20)imply�V' ø�´1# @ � ù� ' 1# andthus

� @ � whichmeansby Eq.(3.16)

� � � (see [Mey00, p. 548], [Kat95,p. 53]).

* Thelargestabsoluteeigenvalueis thespectralradiusbecauseof Eq.(3.42).

CHAPTER3 31

* Eigenvectorsbelongingto differenteigenvaluesform anorthogonalsystem,whichcanbechosento beorthonormal

� � �n��Ë# @ � ��� (follows from (3.20)) (3.54)

with � �ª� @ � . if + R@Ä, �- if +O@Ä, (3.55)

Thisalsoholdstruefor arbitraryrotation:

� � ^º¹ � �¸� � ^º¹ � ��æ# @¡� 9U^º¹ � � ^º¹ � � ��¸�n��#@¡� ^Ë��¹ � 95¹ � � � ����n��æ#@¡� ^Ë��¹ � 95¹ � � � ��� (3.56)

* They canberepresentedbyacompletespectraldecompositionasin Eq.(3.57).Theeigenvaluescanbeinterpretedasweightsof thecorrespondingeigenspace.

' @bc��dgf � ����� (3.57)

with

�1� @ ��q °� @ � %f %f ¦<¦`¦ %f % b¦<¦<¦ ¦<¦`¦ ¦<¦<¦% b %f ¦<¦`¦ % b % b�� (3.58)

representingtheorthogonalprojectors.Note,that o b ��dgf ��� @ £ , ��°� @ ��� ,� ±� @ �1� . For moredetailseesection 3.3.5.

* Any function i0 ' 4 of a Hermitianmatrix'

canalsobeseenasa functionof its eigenvalues i0 � 4 . Now let i0 ' 4§@ o ßÝ d�� ' Ý bea power series,thenit follows that i0 � ^ 4X@ o ßÝ d�� � Ý ^ �­Ö Gþ@ - � ¦<¦<¦ÎM , since i0 ' 4X@� i0�Ò Ó�ÒQ9 f 4with Ò representingthematrixof all eigenvectorsand Ó adiagonalmatrixwith i0 � ^ 4 asentries.Thiscanbeseenwhenthefunctionis expanded:

32 CHAPTER3

i0 ' 4i@ ßc Ý d�� ' Ý@ßc Ý d�� 0 bc ��dgf � ����� 4 Ý by Eq.(3.57)

@¬0bc��dgf � ���1� 4 � � 0 bc ��dgf � ���1� 4 f � 0 bc ��dgf � ����� 4 ± � ¦`¦<¦

@¬0bc��dgf � ���1� 4 � � 0 bc ��dgf � ���1� 4 f � 0 bc ��dgf � ����� 41²�0 bc ��dgf � ����� 4 � ¦<¦`¦

@¬0bc��dgf � ���1� 4 � � 0 bc ��dgf � ���1� 4 f � 0ª0 � f���f ² � f��Lf 4� 0 � f���f ² � ± � ± 4 � ¦<¦<¦e4 � ¦<¦`¦

@¬0bc��dgf � ���1� 4 � � 0 bc ��dgf � ���1� 4 f � 0 bc ��dgf � ����� 4 ± � ¦`¦<¦ by �1���1� @ � �������

@ßc Ý d�� bc ��dgf � Ý � ���

@bc��dgf i0 � � 4 ���

(3.59)

It is awell known resultthatsuchpowerseriesconvergeundercertaincon-ditionswhichalsoholdfor everyadjacency matrix

' � � . [Mey00, p.618].Wenow useaspecialtypeof powerseries,thevonNeumannseriesgiveninEq.(3.60)andtheirconvergencecriteria,sincevonNeumannseriesexactlyreflecttherelationshipwe needto find therankindex introducedin section3.2.2.

£ � ' � ' ± � ' = � ¦<¦<¦5@ßc Ý dgf '

Ý

@C0 £ k ' 4 9f

â ,íGV  Ý è ß�' Ý @�.â � � ��ð­ñ ��� - (3.60)

CHAPTER3 33

which is equivalentto&®' & � - . Since

& ' & @ � � ��ð­ñ � (Eq.(3.42)). Thisis equivalentto the factestablishedabove thata functionof

'generatesa

functionof�. It followsthatthecorrespondingpowerseriesfor

�converges

if � � ��ð­ñ ��� - . This meansthat if the von Neumannseriesconvergesfor', this is connectedto theeigensystemof

'in thatthepower seriesfor

�convergesto 0­-5k � 4�9 f [ � � ��� - . Thustheeigensystemof

'approximates

theeigensystemof thepowerseriesandconvergesin thelimit.

* Eq.(3.57)togetherwith Eq.(3.50)yieldstheresult,thattheeigenvaluescanbeusedto explain thecontainedvarianceof asubspace.Thevariance� ± of is definedas � ± @ fb o � 0 %�� k��64 ± @ fb & k��� & ± with � representingthemeanand � theunit vector.

Proof:

&þ' kbc��dgf � ����� & ± @ �3' k bc ��dgf � ���1�¸� ' k bc ��dgf � �����<# Eq.(3.35)

@ �3' � ' # k �3' �bc��dgf � ���1�n# k � bc ��dgf � ���1�¸� ' #

k �V' �bc��dgf � ���1�n# Eq.(3.45)

@ &þ' & ± kSE &Z' & ± � �bc��dgf � �����¸� bc ��dgf � ���1�n# Eq.(3.35)

@Yk &(' & ± � c ^ w «� ^ � « � � ^ ���g«t# Eqs.(3.53), ( 3.57),( 3.58)

@Yk &(' & ± �bc^ dgf� ±^ @�.� &(')& ± @ bc

��dgf � ±�(3.61)

If onesumsnot over all M but only to an index   I M anddividesby thetotal sum of all

� ±� , the result shows the variancecoveredby the   firsteigenspaces.

* They arewell conditioned,which meansthata smalldisturbance�

of theHermitian

'matrixonly resultsin achangeof theeigensystemupto

& � &.

Seesection 3.3.6for details.

34 CHAPTER3

For matricesof theform discussedin chapters4 and 6, thefollowing charac-teristicshold:

Let'

be a matrix with Ñ �ª� @ . (no self reference)thenfollowing the factthat' @ �SÓ ��9 f the o b ��d�� � � @ s�ï � aq�U0 ' 4¼@ o b��d�� Ñ �ª� @ . . This leads

immediatelyto theexistenceof negativeeigenvalues.Of specialinterestis thematrix ! of order M �   with!Y@#" . b�$�b �� ° . �%$��'& (3.62)

with�

representinga MøPÏ  matrix. In general,this is the representationof abipartitegraph.

In thespecialcaseof n=1andm=k thismatrix representsadirected,weightedstargraphwith + � - vertices.Thespectrumof thatsystemis describedasfollows:�L0(!�4L@ � � � f � � � ± � ¦<¦<¦ � k � � F b 9 f � k � � F b � (3.63)

(see[Mey00, p. 555]) with � � fn� @ � � � F b � , � � ± � @ � � � F b 9 fn� , etc.The eigenvaluesareby definition of the sameabsolutevalue,but being the

squarerootsof thesingularvaluesof�

they have eithersign. For therespectiveeigenvectorsthis leadsto thefactthatthey areonly determinedup to multiplesofÊ

.As anexamplelet

'bea )�P*) Hermitianmatrixof astargraphwith Ñ �ª� @�.

and Ñ ��� @¬. for 03+:,W4�� � 03E+)�4 � 0,)�E�4 � . Theequationsfor theeigensystemarethengivenexplicitly following Eq.(3.52)by Eq.(3.64). Thephaseof theeigenvectorcomponentswill becalculatedby Eq.(3.65).

. %f�� Ñ f ± % ± � Ñ f = % = @ � %fÑ f ± %f�� . % ± � . % = @ � % ±Ñ f = %f�� . % ± � . % = @ � % =(3.64)

Thusit follows,thatthephasesfor %frw ± w = canbeexpressedas:

% = @ -� Ñ f =% ± @ -� Ñ f ±%f @ -� ± =c ��dgf � Ñ fW��� ± @Y-

(3.65)

CHAPTER3 35

The last equationholds, becausethe determinant2U�`st0 ' k � £ 4 @ k � = �� Ñ f = Ñ f = � � Ñ f ± Ñ f ± @ k � 0 � ± k o =��dgf � Ñ fW�m� ± 4}@ . . This holds for� = @ .

and� ± @ � o =��dgf � Ñ fW�Ï� ± (Eq. 3.34),which is the sameas

� frw ± @.- &�ü f &(Eq.3.35).

It canbeseen,that in theeigenvectorof anunperturbedstar, thephaseof theoriginalmatrixentryis keptbut complex conjugated.Thisis understandablewhenonelooksat theprojectors.

3.3.5 Orthogonal Projectors

The fact, that theentries/ ��� of theprojectors�g« in Eq.( 3.58)aregivenby %�� %��with + � ,�@Y- � ¦<¦<¦ � M theindex within theeigenvector�« , it followsfromEqs.(3.17)and( 3.19)that

/ ��� @ %�� %��@j03»��U0 %�� 4λ��U0 %�� 4 � £  ¼0 %�� 4 £  ¼0 %�� 4ª4 � G�0 £  ¼0 %�� 4λ��U0 %�� 4$k~»��U0 %�� 4 £  ¼0 %�� 4ª4@ � ����n��n#6� G �n�� P �$�(3.66)

with 0 �� P �� 4 thecrossproductof two vectors.Weregardeachcomplex number%�� and %�� asa two componentvector � � ± in this casewith components%��tw f @»��U0 %�� 4 and %��tw ± @ £  ¼0 %�� 4 . The crossproductof two vectorsis againa vector,that is orthogonalto the planespannedby the two original vectorsandhasthelengthidenticalto theareaof theparallelogramspannedby thetwo vectorswhichcanbecalculatedusingthedeterminantof thematrixbuilt by thetwo row vectors.This leadsto thefollowing interpretationof projector� « in Mat.( 3.67)( [Chi98],[BL04]):

* Eachentry gives in its real part a measureof the structuralsimilarity orclosenessbetweenthecommunicationpatternsof two memberswithin thissubspace,sincetherealpart is constructedby thescalarproductof thetwoeigenvectorcomponents.

* In its imaginaryparttheprojectorentrygivesameasureof thedivergenceofbehavior, sincetheimaginarypartis theabsolutevalueof thecrossproductbetweentheeigenvectorcomponents.

As anexampleusetheprojector� ó Ì in Eq.(3.67).� ó Ì @ � . fó Ñ f ± fó Ñ f =fó Ñ f ± . . .fó Ñ f = . . . � (3.67)

36 CHAPTER3

As will beexplainedin section4.3.2it alsoyieldsanadditionalinterpretationin thecontext of communicationpatternanalysis.

3.3.6 Perturbation Theory for Hermitian Operators

Thebehavior underperturbationiswell researchedfor Hermitianoperators[Kat95],[DS88] . And sincethis is amajoradvantageof theproposedmethodwhenthink-ing of futurework in this field, a very shortintroductionat leastfor perturbationof eigenvaluesis given.

Now assumean unperturbedHermitian matrix�

with eigenvalues K f � ¦<¦<¦ �K b , with K f á ¦<¦<¦ á K b anda Hermitianperturbationmatrix�

with eigenvalues� f � ¦<¦<¦ � � b , with � f á ¦<¦<¦ á � b . For theeigenvalues� f � ¦<¦<¦ � � b , with

� f á ¦`¦<¦ á� b of theresultingHermitianmatrix' @ � � � aWeyl typeperturbationbound

is givenby

K «i� � f á � « á K «i� � b (3.68)

See [HJ90,p.181-182], [Kat95,p.61]or [Mey00, p.551-552].The Bauer-Fike boundin Eq.( 3.69) for diagonalizablematricescanbe used

to giveanevenbetterboundfor Hermitianperturbations[HJ90, p.367-370]:

 lG3M0/ � � � � k~K �¸�& � & I21 0íÒ¼4 & � & (3.69)

with1 0�Ò¼4¸@ & Ò &�& ÒQ9 f & for a matrix Ò of eigenvectors,wherethe inverse

existswhich is thecasefor thematrix of eigenvectorsof a Hermitianmatrix. ForHermitianmatricesthe Bauer-Fike boundin Eq.( 3.69) even reducesto

& � &,

since1 0�Ò¼4S@ - . Sincethematrix Ò is orthogonal( Ò ° @ Ò 9 f ) and

& Ò & @ -due to orthonormality. Thus Hermitian matricesare well conditioned [HJ90,p.528]whichhasalreadybeenmentionedin section 3.3.4.

With theseboundsit would be possibleto checkif andhow a matrix'

isthe resultof a perturbation

�. This will beof someinterestin chapter 4, when

theconstructionandbehavior of complex Hermitianadjacency matricesandtheirrespectiveeigensystemswill bediscussed.SinceaHermitianmatrix

'canberep-

resentedby a Fouriersumof its subspaces,theaddends,orderedby their weights(the correspondingeigenvalues)canbe seenasan approximationof a perturba-tion. Theaddendwith a relative high weightcanbeseenastheunperturbedma-trix, with all otheraddendsasperturbationmatrices.This interpretationis highlyconvinient, whenthefirst oneor two subspacesalreadycover a largepartof thevariance.

CHAPTER3 37

3.4 Eigensystemof realadjacencymatrices for non-dir ectionalgraphs

For real valuedadjacency matricesextensive literatureaboutthe correspondingeigensystemsof graphscanbe found in Cvetkovic et al. [CS97]. As discussedthereintheeigenspectrumof a graphdoesnot identify thegraphuniquely, sincecospectrality, which means“having the sameeigenspectrum”,doesnot lead toisomorphism,whichmeans“having thesamegraphstructure”in thiscontext. Butif, in addition,theeigenspacesaretakeninto accounttheuniquenessof theeigen-systemcanbeestablished.As areferencefor laterusein chapter4.4,someresultsabouteigensystemsof certainkinds of graphsfrom Cvetkovic will be presentedwithoutproof.

Star graph

Let 3 be a stargraphas in Fig.(3.4)with 4 vertices � � � +¢@ - � E � ) � 4 . Furtherlet � f be thecenterof thatstar. Thenthenormedeigensystemof therealvaluedadjacency matrix

� @ ����� � �U�ª� @ - if �U0V+ � ,34�� � else ����� @j. hastheform asin Tab. 3.3.

1

2 3

4

Figure3.4: Non directionalstargraph

��k

� � ��tw f ��tw ± ��qw = �tw 51 1.73 0.71 0.41 0.41 0.412 -1.73 0.71 0.41 0.41 0.413 0 0 0.82 0.82 0.824 0 0 0.82 0.82 0.82

Table3.3: Eigensystemof astargraphin Fig.( 3.4)

38 CHAPTER3

As canbeseen,thereis onepair of non-zeroeigenvalueswith thesameabso-lutevaluebut differentsign(eigenvalue

� frw ± @6-�-�¦87�) (Eq.(3.63))).Theeigenvec-tors for the two nonzeroeigenvaluesshow, that thevertex � f hasa highervaluethantheothervertices,who haveequalabsolutevalues.This is intuitive in a star,thecenteris perceivedasthestrongestmember, while all theothersareperceivedto be lessrelevant in comparisonbut equallyrelevantamongstthemselves. Thisresultqualitatively correspondsto theresultfrom thedegreecentralitymeasureinEq.(3.2),whichyields a ] f @Y- � a ]± w = w 5 @Y- z ) .CompleteGraph

Let û bea completegraphasin Fig.(3.5)againwith 4 verticeslabeledasabove.Theeigensystemthenis givenin Tab. 3.4.

1

2 3

4

Figure3.5: Non directionalcompletegraph

k� ��tw f ��qw ± �tw = ��tw 5

1 3.00 0.50 0.50 0.50 0.502 -1.00 0 0.87 0.87 0.873 -1.00 0 0.87 0.87 0.874 -1.00 0 0.87 0.87 0.87

Table3.4: Eigensystemof acompletegraphin Fig.( 3.5)

This eigensystemsshows only onemaximaleigenvalue,while all the othershavethesameabsolutevalueandeventhesamesign.Theeigenvectorcomponentsof themaximaleigenvalueshow thatall verticesareof equalrelevance.

The star graphand the completegraphare two graphswhich representthetwo extremecommunicationpatternsfor group structures. A group can eitherbeextremelyconnected,ergo a completegraph,or stronglycentered,ergo a star.Thereexist all kindsof mixtures,andin additiontherecanalsobeisolatedvertices

CHAPTER3 39

within groups. But in the context of this book, all membershave at leastoneconnectionwithin thegroup. This is why isolatedverticesneednot bediscussedhere.

As wasshown in thischapter, theuseof complex Hermitianmatriceshasmanyadvantageswhencalculatingtheeigensystem.Theseare:

* Realeigenvalues

* Orthogonaleigenvectorsystem[ orthogonalprojectors

* Well conditionedmatrices

* Innerproductnorm

* Fouriersumrepresentation

Therelevanceof thesepropertiesfor thepurposeof interpretingthestructureof emailcommunicationwill beshown in thenext chapter.

40 CHAPTER3

Chapter 4

Analysis of GeneralBidir ectionalCommunication Patterns

This chapterdealsin section 4.1 with the codingof bidirectionalcommunica-tion patternsin a weightedadjacency matrix. In section 4.2 it is shown how acomplex Hermitianadjacency matrixcanbededucedfrom aweighteddirectionalgraphin a socialnetwork. Section 4.3 givesan interpretationof the resultsob-tainedin chapter3 in thecontext of bidirectionalcommunicationpatternanalysis.Section 4.4givessomeexamplesof designedsyntheticcommunicationmatricesandshows how theproposedmethodcanbeusedto analyzethecommunicationstructurehiddenin thesematrices.

4.1 Communication Behavior

In thefollowing we considerbidirectionalcommunicationpatternswhich canbemodeledasa directedandweightedgraph

@ ������������� asdescribedin section3.1.1.Let � � and � � bevertices(members)of thegroup( � � � � � � � ). Thecommu-nicationgoing from � � to � � andfrom � � to � � will be transformedinto theentry����� in thematrix

�.

Thusthefollowing possibilitiesfor communicationbetweentwo verticesmayarise:

* No self reference:����� @�. � Ö +* + and , communicate:������R@Ä.

– + communicateswith , on thesamelevel

– + writesmoreoftento , thenviceversa

– , writesmoreoftento + thenviceversa

41

42 CHAPTER4

Codingthecommunicationbetweenvertices+ and , is a problemin SNA, aslong as ����� � � , becauseonly one informationcanbe codedin a real numberinsteadof thenumberof messagessentfrom + to , andvice versa. It is usuallyachievedby makinga choiceof whatto code,e.g.only thedifferencein commu-nication,or thepredominantdirection,. . .andchoiceimpliesinformationloss.Asolutionto this problemis to regardthesetwo componentsasa two componentvectorin a2 dimensionalplane.Thusthisrealtwo componentvectorcanbetrans-formedinto a complex number, asexplainedin section3.3.1. As a consequence�U�ª� � � Ö + � , . Theasymmetryof communicationbehavior is thecharacteristic,whichhasto bedealtwith, whentrying to find feasiblesolutionswith eigensystemanalysis.

4.2 ComplexHermitian AdjacencyMatrix

The amount   of outboundcommunicationof a vertex + is translatedinto thereal part of the matrix entry,   @ »��U0 ����� 4 , whereasthe amount/ of inboundcommunicationof vertex + is denotedasthe imaginarypart, / @ £  ¼0 ����� 4 . Thusthe characteristicsof complex numbersasdescribedin section 3.3.1Eq.( 3.13)now have thefollowing interpretation:

* The absolutevaluecorrespondsto the traffic volumeexchangedbetweentwo members.

* Thephase 0 ����� 4 representstherelativedirectionof thecommunication.

Thecomplex adjacency matrix�

hasentriesasgivenin Tab.( 4.1):

communicationbehavior �U�ª� @¡  � Gæ/  0 ����� 4no self reference ����� @¡. arbitrary+�[ , ÿ ,[ +   ÿ / 9 . ��:5 9,�[ + ÿ +O[ ,   � / ; : 5 �+: ± ;+�[ ,@�,[ +   @ / : 5Table4.1: Communicationbehavior representation

The entriesof the complex adjacency matrix�

canbe completedby followingEq.4.1:

����� @ÄG ���÷� (4.1)

Theconstructionof thematrix�

is thusgivenby Eq.4.2

CHAPTER4 43

� @6! � G<! � � with u ��� � � (4.2)

with ! representingtheweightedrealvaluedadjacency matrix,thustheentriesof! reflectthenumberof messageswhichgo from vertex + to vertex , .In Fig.( 4.1) a samplecommunicationcanbe seen.The pointsrepresentthe

matrix entries �U�ª� of a samplematrix given in Eq.( 4.1). The diagonalline hasbeendrawn to demonstratetheunderlyingsymmetry.

Re = Outbound

Im = Inbound

mk

pk o

o

o

o

o

o

o

Figure4.1: Communicationbehavior beforerotation

It caneasilybe seenaswell from Eq.( 4.1) asfrom Fig.( 4.1) that�

canbetransformedinto a Hermitianmatrix

'by rotation.Rotationis achievedby mul-

tiplying�

following Eq.(3.15).Therotationfactoris � 9U^>= ? asshown in Eq.(4.3).

' @ �A@ � 9U^ = ? (4.3)

Proof:

1. Let theentryof�

beany complex number����� @¡ï>�`^º¹ with absolutevalue ïandphase

Â.

2. Thenby Eq.(4.1) �U�÷� @¡G �U�ª� @ÄGVï���9U^º¹3. �U�ª� @�.�@�Ñ ��� becauseof theexclusionof self references

4. After rotationaboutB : Ñ ��� @ ����� �`^8CN@¡ï��`^º¹´�`^8CN@¡ï��`^Ë�e¹ F C �5. Ñ �÷� @ ���÷� �`^8C�@ÄG3ï>��9U^�¹��`^�C�@�� ^>= Í ï>�`^Ë�DC³95¹ �6. ForHermitianmatricesÑ �ª� @ Ñ �÷� musthold(Eq.(3.20),therefore,ï��`^Ë��¹ F C � @ï>��9U^Ë�<= Í F C595¹ � .

44 CHAPTER4

7. Thisholds,if � B @Yk : ± kEB �  .

8. Solvingfor B leadsto E�B~@Ck : ± . Thus BÏ@Ck : 5 . qed

The alternative way given by Chino yields a matrix that is similar to'

inEq.(4.3).Theproof is thefollowing:

Let'GF @ f± 0(! � !��34 � G f± 0,!Ák�!��W4 by Eq.(3.22)and

'GH @¬0,! � G<!��34Î� 9U^I= ?by Eqs.(4.2, 4.3) then:

'GF @ -E 0,! � ! � 4 � G -E 0(!YkJ! � 4@ -E 0,! � GI! � ! � kÏG<! � 4@ -E 0,!l0�- � Gr4 � ! � 0­-XkÏGr4Î4@ À EE 0,!¿� ^>= ? � ! � � 9U^I= ? 4@ À EE 0,! � ! � � 9U^>= Í 4­� ^>= ? � � 9U^>= Í @Ck®G@ À EE 0,!¾kÐG<! � 4Î� ^>= ?@ À EE 'GH

(4.4)

with'GH

representingthecomplex conjugatedmatrixof'KH

.This leadsto aneigensystemthatis complex conjugatedandscaledby a con-

stantto theonepresented.As will beseenlaterin section 4.3.2theinterpretationof theorthogonalprojectorswill bemoreintuitive andrevealingwith thematrixgivenby Eq.(4.3) thanwith themethodproposedby Chino. In additionthecom-parisonwith theoriginal adjacency matrix is noteasyto achieve,sincemorethanabackrotationis neededin thecaseof Chino.

Tab.( 4.2) shows how the communicationbehavior patternsof Tab.( 4.1) re-mainvisible in therotatedmatrix

'with entriesÑ ��� aftertherotation:

1. No self reference:Ñ ��� @�.2. More outboundthaninboundtraffic ( + [ , ÿ ,Z[ + ) leadsto a negative

signof theimaginarypart /�L½@ £  ¼03Ñ �ª� 4 � .3. More inboundthanoutboundtraffic ( ,®[ + ÿ +Ð[ , ) leadsto a positive

sign /�L½@ £  ¼0VÑ ��� 4 ÿ .

CHAPTER4 45

Re

Im

mk

pk

o

o

o

o

o

o

o

-pk

Inbound

Outbound

Figure4.2: Communicationbehavior afterrotation

4. Outboundequalsinboundtraffic ( +©[ ,�@¡,�[ + ): /MLþ@¡.Fig.( 4.2)showshow thedatapointsin Fig.( 4.1)behaveunderrotation.

Note that the norm is invariantunderrotation. The absolutevalue- that is therepresentationof theamountof of communication- doesnotchange.

communicationbehavior ����� @¢  � G{/ Ñ �ª� @Ä NL � Gæ/�Lno self reference ����� @Ä. Ñ ��� @Ä.+O[ , ÿ ,�[ +   ÿ / /�L � .,�[ + ÿ +�[ ,   � / /�L ÿ .+O[ ,@�,�[ +   @Ï/ /�L½@�. ,  NL ÿ .

Table4.2: Communicationbehavior representationin thematrix'

4.3 Inter pretation of the Eigensystem

In thissectiontheinterpretationof thecomponentsof theeigensystemof thecom-plex Hermitianadjacency matricesintroducedin section4.2will bediscussed.In

46 CHAPTER4

the following the solutionof the eigensystem' @ � is considered.First, in

section 4.3.1theeigenspectrumwill be interpretedin the light of the input data,then,in subsection4.3.2,the interpretationof eigenvectorsandtheresultingor-thogonalprojectorswill beexplained.

4.3.1 Eigenspectrum

Sincethe input dataconsistsof inboundandoutboundmessages,theeigenspec-trumreflectsthe“energy” level, or traffic level within thedataset,since� � � @ &(' & @ & � &is the spectralradiusfollowing Eq.( 3.42). Thus

�changeswith the numberof

messagesexchanged.�

doesnotrevealif thenumberof in- or outboundmessageshaschanged.

The eigenvaluecanbe interpretedfollowing Eq.( 3.57) asthe weight of thecorrespondingeigenspace,thusit is alsopermissibleto seeit astheweight of apatternthat servesasa corrective perturbationor patternon themostprominenteigenspace.If now

� ± � . and� fþÿ . or viceversaandtheanchorcorresponding

to� f and

� ± is thesame,thenthecorrectionthat thepatterncorrespondingto� ±

bringsputsa strongerfocuson a characteristicwithin thecommunicationof thewholegroup.Thesignof theeigenvalueyieldsinformationif thecommunicationbehavior demonstratedby the subspaceis the patternwith the subgroup,or thecommunicationpatternthatthesubgroupmaintainsto therestof thegroup.For anexampleseesubsection4.4.3.This couldbeseenasa kind of filtering function.Whenthe secondsubspaceweightedby its eigenvalue is addedto the first, thestrongestpatternreceivesa modulation. The sign of the eigenvaluealsoyieldsinformation.

As wasshown in section 3.3.4,thespectrumexhibits a symmetryfor undis-turbedstargraphs.If thestaris beingperturbed,thesymmetrybreaksdown. Thismeansthat if, for example,a groupshows thecommunicationbehavior of a stargraph,additionalcrosscommunicationtakestherole of a perturbationaslong asit is smallcomparedto communicationin thestar. This is analogin thecaseof acompletegraph. In this casefor exampleonesidedcommunicationor very littlecommunicationbetweentwo nodestakestheform of aperturbation.

4.3.2 Eigenvectorsand orthogonal projectors

The absolutevalueof the eigenvectorcomponentsis the well known eigenvec-tor centrality index (seeBonacichandLloyd [BL01]). In this context this in-dex canbeinterpretedasa similarity in communicationpatternamongthegroupmembers.Thusthecomponentwith highestabsolutevaluecanbeinterpretedasthehighestrankedmemberwithin thatpattern.This hasbeendefinedin section

CHAPTER4 47

3.1.1as the anchorof the subgroup. The phaseinformationof the eigenvectorcomponentsgivesinformationaboutthedirectionalsimilarity comparedwith theanchorwithin thesubgroup.This meansthat thephaseshows,how thedirectionof communicationof the vertex is similar to the directionof the anchorin thatpattern. The anchorcanbe definedto have a phaseof

 @Ü. , the phaseof allothercomponentsis a relative information. If for exampletheeigenvectorshowsthedistributionof absolutevaluesthatrepresentastarlikepattern,thenthephasegivesanindicationof thedirectionof thepattern(inboundor outboundstar).

The orthogonalprojectorsare the bridge betweeneigenvectorsandoriginaladjacency matrix. As hasbeenexplainedalreadyin section 3.3.5,the real partof theprojectorconsistsof thescalarproductof theeigenvectorcomponents,thusgiving an index for similarity. While the imaginarypart consistsof theabsoluteamountof thecrossproductof theeigenvectorcomponents,thusgiving anindexof skewness.Sincetheprojectorsarepartsof theFouriersumandthusarepartoftheoriginaladjacency matrix, it followsthatthenumberof outboundmessagesinthe original matrix is connectedto the scalarproductof theeigenvectorcompo-nents,while thenumberof inboundmessagesis connectedto theabsolutevalueof thecrossproductof theeigenvectorcomponents.

4.4 Examplesof Eigensystems

After having thusdescribedtheconstructionof thecomplex Hermitianadjacencymatrix, this sectionwill presentsomeexamples,which are designedso as todemonstratehow themethodsworks.

As anintroductionto eigensystemsof complex Hermitianadjacency matricesthegraphpatternsusedin section.(3.4) will beused.All calculationshave beenperformedwith Mathematica[Wol03]. Thishasthefollowing consequences:

* Eigenvectorsarenotnormalized.

* In Mathematicatheeigenvectorshave beenrotatedto resultin a phaseof .for theeigenvectorcomponentwith the largestabsolutevalue. But this, asexplainedin section.3.3.4Eq.(3.56),doesnot disturbtheorthogonalityoftheeigenspaces.

* Eigenvalueshave beennormalizedby dividing eacheigenvalue with thematrixnorm.Thusthelargesteigenvalueis equalto � - � .

To show how the results,theenhancedmethodsprovidescanbe interpreted,theanalognon-directionalgraphswill beconsideredfirst,andthentheasymmetric

48 CHAPTER4

versionwill be discussed.The analogonto a non-directionalgraphis a bidirec-tionalbut equallyweightedgraph.For validationtheresultsof Cvetkovic [CS97]in thecaseof stargraphsandcompletegraphswill beused.

Threekindsof graphswill bediscussed:

* Thestargraph(subsection4.4.1)

* Thecompletegraph(subsection4.4.2)

* A graph representingtwo subgroupsconnectedby one link (subsection4.4.3)

4.4.1 Star Graph

The equivalentof a non-directionalstar graphin the complex representationisshown in Fig.( 4.3). Thecorrespondingcomplex adjacency matrix

�POis givenin

Eq.(4.5)andtherotatedmatrix� L ]V�� is givenin Eq.(4.6).

1

2 3

4

1

1 1

1

1 1

Figure4.3: Equivalentto non-directionalstargraphcorrespondingto Eq.(4.5)

�PO @ �QQ . - � G - � GA- � G- � G . . .- � G . . .- � G . . . �RR� (4.5)

� L ]V�O @ ' O @ �QQ . ->¦D4¤- -�¦S4¤- -�¦S4¤--�¦D4:- . . .-�¦D4:- . . .-�¦D4:- . . . �RR� (4.6)

The eigensystemof the rotatedmatrix� L ]V�O @ ' O is given in Tab.( 4.3). Since

therotatedmatrix hasentriesÑ ��� � � theresultingeigensystemfor thenon-zeroeigenvaluesis the same(modulo

Ê) asthat in Tab.( 3.3), eigenvalueshave been

CHAPTER4 49

normalized.Thiseigensystemshowsthefirst two eigenvalueswith sameabsolutevalue,but with differentsign.Thecorrespondingeigenvectorsarerealin thiscaseandshow aphasedifferenceof exactly

Ê(in thiscaseadifferentsign),apartfrom

the largestabsoluteeigenvectorcomponent.Characteristicfor a stargraphis thefact that two eigenvaluesof thesameabsolutevaluebut with differentsignexist,andthat the distribution of the absolutevalueof the eigenvectorcomponentsissuchthatthecenterof thestarhasthehighestvalue,while all othermembershavea lowerbut equallydistributedvalue.

+ � � ��tw �,@C- ,�@8E ,�@6) ,@T4

1 -1.0 0.71 -0.41 -0.41 -0.412 1.0 0.71 0.41 0.41 0.413 0 0 -0.58 -0.21 0.794 0 0 -0.58 0.79 -0.21

Table4.3: Eigensystemfor' O

in Eq.(4.6)

Theeigensystemin Tab.( 4.3)corresponds(moduloÊ

) for thenon-zeroeigen-valueswith theresultsof Cvetkovic et al. [CS97,p.235].

If the star is perturbedwithin the star, which meansthat additionalcommu-nicationtakesplacebetweenthealreadyconnectedmembersof thenetwork, theonly effect is achangein theabsolutevalueof theeigenvalues,sinceby Eq.(3.42)only

&S' &becomeslarger, but no additionalmembersgetconnected.If thestar

is subjectedto asymmetricperturbationasin Fig.( 4.4)betweenverticesE and ) ,which meansthatthematrix entries� ± = and � = ± aredifferentfrom . asmatrix

� Ýin Eq.(4.7)shows,

� Ý @ �QQ . - � GA- � G - � G- � G . - � G .- � GA- � G . .- � G . . . �RR� (4.7)

thefollowing effectscanbeobservedin theeigensystemgivenin Tab.( 4.4):

* The symmetryfor the largest two eigenvaluesbreaksdown accordingtoEq.(3.69):

ò ó Í­9 / Í òëVU�ë @ ò 9 �XW Y>Z F f>W � òf>W 5�f @¾.³¦�E�E I - , with�

beingtheperturbationmatrixwith only theentries� ± = � � = ± R@�. .* Theeigenvectorsloosetheir symmetry, which meansthat theeigenvectorsof thetwo largesteigenvaluesdonothavethesameweightsanymoreaswellasthefact that theeigenvectorcomponentfor vertex + in eacheigenvectordoesnotonly changethesign,whichwouldmeanaphaseshift of

Ê(which

50 CHAPTER4

is equivalentto arotationby -\[>.^] counterclockwiseor theratiobetweenthecircumferenceû of acircleandits diameter2 giving

F � @T):¦{-�4X¦<¦<¦ .* The distribution of the eigenvector componentswithin eacheigenvector

loosesits starlikecharacteristic.Theanchoris still visible,but thebehaviorof theotherverticeschanges.

1

2 3

4

1

1 1

1

1 1

1

1

Figure4.4: Symmetricalperturbationof non-directionalstargraphrepresentedinEq.(4.7)

+ � � ��qw �,�@Y- ,�@ÄE ,�@T) ,�@24

1 1.0 0.61 0.52 0.52 0.282 -0.68 0.75 -0.3 -0.3 -0.513 -0.46 0 -0.71 0.71 04 0.14 0.25 -0.37 -0.37 0.82

Table 4.4: Eigensystemfor the perturbedstar in Eq.( 4.7) after rotation inFig.( 4.4)

In addition it canbe said, that if one reversesthe directionof the edgebe-tweentwo vertices,the eigenvaluesarekept, but the eigenvectorsbecomecom-plex conjugated.This canbeseenif thestargraphis subjectedto anasymmetricperturbationasin Fig.( 4.5)

CHAPTER4 51

1

2 3

4

1

1 1

1

11

1

Figure4.5: Asymmetricperturbationof anon-directionalstar

Theeigensystemof thegraphin Fig.(4.5)representedbymatrix�`_

in Eq.(4.8)is given in Tab.( 4.5). In the caseof the perturbationin the otherdirection,theeigensystembehavesaspredicted(seeTab.( 4.6)).

�`_ @ �QQ . - � G - � G - � G- � G . - .- � G G . .- � G . . . �RR� (4.8)

52 CHAPTER4

a bXcd

ef bgVh b cig jk h b cilgVh bXcmg jk h b cmlgVh b cng jk h bXcnlgVh b cog jk h b col

11.

00.

650

0.48

-0.2

0.48

0.2

0.33

02

-0.8

40.

710

0.40

-2.7

0.40

2.7

0.43

3.1

3-0

.30

0.20

1.4

0.65

2.8

0.65

00.

34-1

.74

0.13

0.20

00.

432.

60.

43-2

.60.

770

Tabl

e4.

5:E

igen

syst

emfo

rth

epe

rtur

beds

tarw

ithas

ymm

etric

com

mun

icat

iong

iven

inF

ig.(

4.5)

a b cd

ef bg h b cig jk h bXcilg h b cmg jk h b cmlgVh b cng jk h b cnlgVh b cog jk h b col

11.

00.

650

0.48

0.2

0.48

-0.2

0.33

02

-0.8

40.

710

0.40

2.7

0.40

-2.7

0.43

-3.1

3-0

.30

0.20

-1.4

0.65

-2.8

0.65

00.

341.

74

0.13

0.20

00.

43-2

.60.

432.

60.

770

Tabl

e4.

6:E

igen

syst

emfo

rth

epe

rtur

beds

tara

sin

Fig

.(4.

5)in

reve

rsed

dire

ctio

n

CHAPTER4 53

Theeigensystemalsoindicatesthelocationof theperturbingevent.Thephaseof theeigenvectorcomponents

 0 %frw ± 4 and 0 % ± w ± 4 aswell as

 0 %frw = 4 and 0 % ± w = 4

nolongershow aphaseshift ofÊ

ascanbeseenin Tabs.(4.5, 4.6).Thisis exactlywheretheperturbationtookplace.

Theeigenvaluesleadto cumulative coveredvarianceasin Tab.( 4.7). As canbeseenthefirst two subspacescoveralreadyover D�.qp of thevariance.Theeigen-projectorsderived from the eigensystemin Tab.( 4.5) are given by Eqs.(4.9 –4.12). The projectors��� � +¢@ - � E � ) � 4 are thosecorrespondingto eigenvalues� � � +O@Y- � E � ) � 4 . � f � ± � = � 5

coveredvariancepereigenvalue .³¦�B�B .:¦D)�D .:¦p.�B .:¦e.³-cumulativecoveredvariance .³¦�B�B .:¦pD+4 .:¦pD�D -�¦e.>.

Table4.7: Cumulativecoveredvarianceof eigenvaluesfrom Tab.( 4.6)

Thefirst two projectors�Lf and � ± have thefollowing interpretation:

* Theanchoris identifiedby thehighestabsolutevalueanidentifiedasvertex- . / � f3�frf @j.:¦S4UE and / � ± �frf @j.:¦eB>. , where/ � ���«Î� representstheelementin row +andcolumn , of matrix �1� .*r�Lf correspondsto thehighestabsolutevaluedeigenvalue,thusmoretraffic

is beingexchangedin thatpatternand � ± servesasthecorrectivepattern.Ifonetakesinto accountthat

� ± � . , it canbe seenthat thesecondpattern,whenaddedasdescribedby theFouriersum,hastheeffect thatthediagonalentriesof thesumwill go towards. or below, andthattheentriesin thefirstrow andfirst columnwill beaugmented,while theentriesin thesubmatrix/ � f3� F � ± �±r± ¦<¦<¦3/ � f3� F � ± �5>5 will tendtowards . . This thenalreadycomescloseto astarlikeadjacency matrix.

* Thepatternin �Lf representsthefact thatall membersareconnectedwith aslight tendency towardsa star. Only � ± showsastrongerstararoundvertex- , sinceall othermatrix entriesare lower by half at least. This suggeststhat �Lf representsthe communicationwithin the whole group, while � ±representsthecommunicationwithin thestar.

�Lf @ �QQ .:¦S4UE .:¦D):-Zk~.:¦p.^s>G�.:¦D):- � .:¦p.^s>G .³¦�E³-.:¦�)³- � .:¦p.^s>G .:¦eE+) .:¦eE�E � .:¦p.�D>G .:¦{-\s � .³¦e.^)�G.:¦D):-Zk~.:¦p.^s>G�.:¦eE�EÕk~.:¦p.�D>G .:¦eE+) .:¦{-\sSk~.:¦p.^)>G.:¦eE³- .:¦{-\sSk~.:¦p.^)>G�.:¦{-\s � .:¦p.^)>G .³¦æ-�- RR� (4.9)

54 CHAPTER4

� ± @ �QQ .:¦eB>. kÕ.³¦�E+s�kÏ.³¦æ-�-`G k .:¦eE+s � .:¦{-�-<G k .:¦D)�.k .:¦eE+s � .:¦{-�-<G .:¦{-\s .:¦{-�-XkÏ.:¦{-"E�G .:¦{-\sSk~.:¦p.q7�Gk .:¦eE+s�k~.:¦{-�-<G .:¦{-�- � .:¦{-"E�G .:¦{-\s .:¦{-\s � .³¦e.q7´Gk .:¦D)�. .:¦{-\s � .:¦p.q7�G .:¦{-\s�kÏ.:¦p.q7�G .³¦æ-\[ �RR� (4.10)

� = hasvertex ) asanchor, but vertex E hasthe sameabsolutevalue,whichcanbeinterpretedasa structuralsimilarity betweenthetwo. Only thedifferencein phaseshows thatcomparedto vertex ) thatvertex E hasa slight differenceindirectionof communication.

� = @ �QQ .:¦e.�4 .:¦e.�E � .:¦{-\)>G .:¦p.�EÕk~.:¦{-\)>G kÕ.³¦e.q7.³¦e.�EÕk~.³¦æ-\)�G .:¦S4UE k .:¦D4�.�k~.³¦æ-�4>G k .:¦p.+4 � .:¦eE�E�G.³¦e.�E � .:¦æ-�)>G k .:¦S4�. � .:¦{-�4�G .³¦D4UE kÕ.:¦p.+4�k~.:¦eE�E�Gk .:¦p.q7 kÕ.:¦p.+4�k~.:¦eE�E�G kÕ.³¦e.+4 � .:¦�E>E�G .:¦{-"E �RR� (4.11)

In �t5 vertex 4 is theanchorandverticesE and ) againhave thesameabsolutevaluebut differentphase.

�u5 @ �QQ .:¦p.+4¤- kÕ.:¦p.q7 � .³¦e.�B´G k .:¦p.q7Sk~.:¦p.�B�G .³¦æ-\sk .:¦e.^7�k~.³¦e.�B´G .:¦{-\[ .³¦e.q7 � .:¦{-v7�G kÕ.³¦�E+[�kÏ.³¦æ-\[�GkÕ.³¦e.q7 � .:¦e.�B�G .:¦p.q7SkÏ.:¦{-v7�G .:¦{-\[ kÕ.³¦�E+[ � .:¦æ-�[>G.³¦æ-\s kÕ.:¦eE+[ � .³¦æ-\[�G k .:¦eE+[�k~.:¦{-\[>G .³¦�B>D RR�(4.12)

Now consideradirectedandweightedstargraphof anasymmetriccommuni-cationbetween5 membersor verticesasin Fig.( 4.6).

1

2 3

45

2

1 2

3

2

1

1

1

Figure4.6: Asymmetricstargraph

The original complex adjacency matrix�wOyx

correspondingto the graphinFig.( 4.6) is givenin Eq.(4.13).

CHAPTER4 55

�POvx @ �QQQQ. - � E�G E � )>G E � G - � GE � G . . . .) � E�G . . . .- � E�G . . . .- � G . . . .

�RRRR� (4.13)

After rotation�wOyx

becomestheHermitianmatrix' Oyx

:

� L ]V�Oyx @ ' Oyx @ �QQQQ. E³¦{- � .³¦87�Gz):¦eB � .:¦�7�G E³¦{-XkÏ.:¦�7�GA-�¦D4E³¦{-®k~.³¦87�G . . . .):¦eB�k~.³¦87�G . . . .E³¦æ- � .:¦87´G . . . .-�¦S4 . . . .

RRRR� (4.14)

The correspondingeigensystemis given in two different representationstomakecertainaspectsmorevisible. To givea betterview of theeffect of perturba-tion on theeigenvaluesno normalizationhasbeenperformed.As canbeseeninbothTab.( 4.8)andTab.( 4.9) therearetwo non-zeroeigenvaluesof thesameab-solutevalue( � � fn� @ � � ± � @¾B³¦e. ) but with differentsign(

� f{� . � � ± ÿ . ). As wasshown in Eq.( 3.63)this is indeedthecharacteristicof theadjacency matrix of astargraph.As canalsobeseentheeigenvectorsbelongingto thetwo eigenvaluesarethe samein absolutevaluesbut differ by

Êin phaseasshown in Eq.( 3.65).

Note that the entry in the first row (vertex - ) is the centerof the stargraphandis indicatedassuchby theeigenvectorcomponentof thehighestabsolutevalue.This membercanthusbe describedasthe mostcentralandprestigiousmemberof thegroup,ergo theanchor. It canalsobeseenthattheentryin row 3 (memberwith ID 3) hasthesecondhighestabsoluteeigenvectorcomponent,which meansthatthis vertex hasastrongsimilarity in behavior comparedto vertex - .

� � � f � ± � = � 5 � x-5.0 5.0 0 0 0�tw f 0.71 0.71 0 0 0�tw ± -0.3+0.1i 0.30-0.10i -0.27+0.09i -0.36+0.27i 0.72�tw = -0.5+0.1i 0.50-0.1i -0.14+0.03i -0.19+0.11i -0.64-0.08i�tw 5 -0.3-0.1i 0.30+0.10i -0.08-0.03i 0.86 0.18+0.13i�tw x -0.2 0.20 0.94 -0.08+0.03i 0.13+0.04i

Table4.8: Eigensystemfor' Oyx

in Eq.(4.14)with ·¿@ �Õ� G�u

56 CHAPTER4

� � � f � ± � = � 5 � x-5.0 5.0 7Á. 0 0� · �  03·�4 � · �  0W·�4 � · �  0W·�4 � · �  0W·�4 � · �  03·�4��tw f 0.71 0 0.71 0 0 undef 0 undef 0 undef��tw ± 0.32 2.82 0.32 -0.32 0.28 2.82 0.45 2.50 0.72 0��tw = 0.51 2.94 0.51 -0.20 0.14 2.94 0.22 2.62 0.64 -3.02��tw 5 0.32 -2.82 0.32 0.32 0.087 -2.82 0.86 0 0.22 0.64��tw x 0.20Ê

0.20 0 0.94 0 0.09 2.82 0.14 0.32

Table4.9: Eigensystemfor' Ovx

in Eq.(4.14)with ·¿@ � · � � ^�¹"��| �

1

2 3

45

2

1 2

3

2

1

1

1

2

2

Figure4.7: Perturbedstargraphcorrespondingto Eq.(4.15)

Now considera symmetricperturbationof the form given in Fig.( 4.7) with therespectivematrix

�POvx Ýin Eq.(4.15).

�POvx Ý @ �QQQQ. - � E�G E � )>G E � GA- � GE � G . E � E�G . .) � E´G¯E � E�G . . .- � E´G . . . .- � G . . . .

RRRR� (4.15)

whichafterrotationbecomestheHermitianmatrix' Oyx Ý

in Eq.(4.16):

� L ]V�Oyx Ý @ ' Ovx Ý @ �QQQQ. E³¦æ- � .:¦87´G})³¦�B � .³¦87�G E³¦{-XkÏ.:¦�7�GA-�¦S4E³¦{-XkÏ.:¦�7�G . E³¦�[ . .):¦eBSkÏ.:¦�7�G E³¦D[ . . .E³¦{- � .:¦�7�G . . . .-�¦S4 . . . .

RRRR�(4.16)

CHAPTER4 57

� � � f � ± � = � 5 � x6.2 -4.5 -2.5 0.79 0��tw f 0.61 0.70 -0.26-0.01i 0.23-0.08i 0��tw ± 0.46-0.13i 0.03+0.07i 0.78 -0.34+0.21i 0��tw = 0.56-0.13i -0.57+0.07i -0.50-0.06i -0.25+0.17i 0��tw 5 0.21+0.07i -0.33-0.11i 0.22+0.08i 0.70 -0.51-0.17i��tw x 0.14 -0.22 0.15 0.42-0.14i 0.85

Table4.10:Eigensystemfor' Ovx Ý

in Eq.(4.16)with ·¿@ �S� GVu� � � f � ± � = � 5 � x6.2 -4.5 -2.5 0.79 0� · � Â 0W·�4 � · � Â 0W·�4 � · � Â 03·�4 � · � Â 03·�4 � · � Â 03·�4

0.61 0 0.70 0 0.27 -3.10 0.25 -0.32 0 undef0.48 -0.27 0.073 1.19 0.78 0 0.40 2.60 0 undef0.57 -0.22 0.58 3.02 0.50 -3.03 0.31 2.54 0 undef0.22 0.32 0.35 -2.82 0.23 0.37 0.70 0 0.53 -2.820.14 0 0.22

Ê0.15 0.04 0.44 -0.32 0.85 0

Table4.11:Eigensystemfor' Ovx Ý

in Eq.(4.16)with ·¿@ � · � �`^º¹"��| �Theeigensystembelongingto matrix

' Ovx Ýhastheform shown in Tabs.(4.10,

4.11).As canbeseenwhencomparingTab.( 4.9)andTab.( 4.11)theanchorsof the

first two eigenspacesdid notchange,while in therestof thesubspacestheanchorschanged.Theperturbationthushadeffectsnot in theoverall communicationpat-tern,but in thecorrective patterns.In Tab.( 4.12)thecoveredvarianceis shown.As canbeseenthefirst two eigenvaluesexplain already7¬D�.^p of thetotal vari-ance. � f � ± � = � 5 � x

coveredvariancepereigenvalue .:¦eB>D .:¦D):- .:¦æ- .:¦p.:- .cumulativecoveredvariance .:¦eB>D .:¦D[�D .:¦eD>D -�¦p.�. ->¦e.�.

Table4.12:Cumulativecoveredvarianceof eigenvaluesfrom Tab.( 4.6)

Theprojectorsderivedfromtheeigensystemin Tab.( 4.10)aregivenin Eqs.(4.17)- ( 4.20).

58 CHAPTER4

~� � � i�� �������u����������� ������������ ������������������� ������

����� ����������������� ������������������������ �������

����� ������������ ����������������� �������������������

����� ������ ������ ������������ ����������������� �������

���������� ������������ �������������������������

� �������u�(4

.17)

~� � � m�� �������u������� ��� �� ���� ������ �� ���� ������ �������������

����� �������������� �� ��� � ������ ������� ������ �������

������ ������� ������� ������ ��� �������� �������� ��� �������

������ ������� ������� ������������� �� �������������� �� ����

������������ ������������ �� ��������� �������� ���

� �������u�(4

.18)

~� � � n�� �������u������������ �� ��������� ������� ������ ������� ������

������ �������� ��������� ������ �� ��� �� ��� ������

����� ������� ������ �� ��� ������������ �������������

������ ������������ �� ��� � ������ ����������������� �������

������� ���������� ������ ���������

� �������u�

(4.1

9)

~� � � o�� ������������������� ������� ������ �������� ��� �� ����� ���

����� �������� �������� ������� ������ ������ � �� ��� ������ �

������ ������������ ����������������� ������� �� ��� ������ �

� ��� ������� ������ �� ��� � ������ ����������������� �� ����

����������� �� ��� � ������ ������ �� ��� �� ����� ���

� ���������

(4.2

0)

CHAPTER4 59� Ovx Ýx doesnot contributeandhasbeenomitted.The interpretationof theprojectorsis similar to the interpretationof thepro-

jectorsfor the non-directionalstar. The starcanstill be seen,but the effectsofthe asymmetricbilateralcommunicationandthe perturbationaresuperimposed.Again it canbeseen,that � Oyx ݱ is therepresentative of thestar, while in � Oyx Ýf thecommunicationpatternseemsto point towardsa moreequallydistributedform.This againhasthe effect thatwhenboth subspacesareweightedandadded,thesecondsubspacewill enhancethestarlikepattern.

As predictedby Eq.(3.68)theeigenvaluesof matrix' Oyx

is now disturbed.Inthis intuitive exampletheconsequenceof theperturbationcanbe seenwhenweview

' Ovx Ýasthesumof theunperturbedmatrix

' OvxandtheHermitianperturba-

tion matrix � Oyx of theform

� Oyx @ �QQQQ. . . . .. . E � E�G�. .. E � E´G . . .. . . . .. . . . .

�RRRR� (4.21)

Theeigenvaluesof matrix � Ovx afterrotationare� � îv��� �f @¬E³¦D[ � � � îv��� �± @ kÕE³¦D[ .

Thusthe largesteigenvalueof� H �����f @�s:¦�E of theperturbedmatrix

' Ovx Ýshould

be insidethe Weyl-type boundsfollowing Eq.( 3.68)given by� H � �f � � � îv��� �f @B5¦e. � E³¦�[Ð@¡75¦�[ á s:¦eE á E³¦eEÐ@AB³¦p.�kÄE³¦D[Ï@ � H � �f � � � îv��� �± andthe smallest

eigenvalue� H �����± @¬k{4¤¦�B shouldbeinsidetheboundgivenby

� H ���± �ù� � � îv��� �f @kÕB � E³¦D[l@ kSE³¦eE á k{4¤¦eB á kÕB¿kmE5¦�[l@ kw7³¦D[}@ � H ���± �¡� � � îv��� �± . Tab.( 4.10)confirmsthis.

As canbe seen,the eigenvaluesrise in absoluteamount,dueto the fact thatmoretraffic volumehasoccurredwithin thegroup. This correspondsto the factthat the norm of the matrix

' Oyx Ýis greaterthan the norm of

' Ovx. Sincethe

norm of a Hermitianmatrix by Eq.( 3.42)definesthe spectralradiusthis meansanincreasein thelargesteigenvalueof

' Ovx Ý.

Note,thatdueto theperturbationthesymmetryof theeigenvaluesin Tab.( 4.8)is destroyed. In additiontheperturbationleadsto a breakin symmetrybetweenthe eigenvectors. The eigenvector componentsaffectedby the perturbationdono longershow the regular phaseshift of

Êanymore. For memberswith ID 2

and3 this is visible in Tab.( 4.11). However, a perturbedstargraphcanstill berecognizedby thefactthattherestill existsananchor, which canbeidentifiedbythehighestabsolutevalueeigenvectorcomponentin two eigenvectorsat thesameID position.

Matrix� ó Ì in Eq.(4.22)representstheproduct

� f�� Ovx Ýf . By adding,accordingto the Fourier sumthe product

� ± � Ovx ݱ we get matrix�� Í� � Ì ó ��¢"� in Eq.(( 4.23).

60 CHAPTER4

Theremainingpartialsum û ��� �v£ ó � ¢ � only coversabout -\p of thevariance.

� ó Ì @�QQQQ

E³¦�)³- -�¦�7�4 � .:¦D4�D>GÜE³¦{-"E � .:¦D4�D>G¯.³¦87>DSk~.³¦�E+s�G .:¦�B�)->¦87�4�kÏ.³¦D4�D�G -�¦D4�E ->¦87>.�kÏ.³¦e.^[�G .³¦�B�4�k~.³¦�)q7´G .³¦D4�.Sk~.³¦æ-�-`GE5¦æ-"ESkÏ.³¦D4�D�G ->¦87>.�kÏ.³¦e.^[�G E³¦e.�B .³¦�sq7Õk~.³¦D4¤-`G .³¦D4�DSk~.³¦æ-�-`G.:¦�7>D � .³¦�E+s�G¯.:¦eB�4 � .:¦�)^7�G .:¦Dsq7 � .:¦D4:-<G .:¦�)>. .:¦{-\[ � .:¦e.+s>G.:¦�B�) .:¦S4�. � .:¦æ->-<G .:¦S4�D � .:¦æ->-<G¯.³¦æ-\[Sk~.³¦e.^s�G .:¦æ-nE RRRR�

(4.22)

�� Í� � Ì ó � @

�QQQQ.:¦{- -�¦87 � .:¦87´G¤):¦pD � .:¦�7�G -�¦D[�k~.³¦�s>G -�¦�E-�¦�7Sk~.:¦�7�G -�¦S4 -�¦�7SkÏ.:¦{-<G�.:¦Ds�k~.³¦�)>G .:¦D4):¦pD�k~.:¦�7�GÜ-�¦�7�k~.³¦æ-<G .:¦Ds kÕ.:¦{- k .:¦{--�¦D[ � .:¦Ds>G .:¦�s � .:¦�)�G k .:¦æ- kÕ.:¦eE k .:¦{--�¦eE .:¦S4 k .:¦æ- kÕ.:¦{- k .:¦{-

RRRR� (4.23)

If we now rotatethis matrix backby � ^>= ? , thematrix�w¥ L ]V�� Í� � Ì ó � ¢ � in Eq.(4.24)

alreadylooksquitelike theoriginalmatrix� ± apartfrom theentries� ±r± � � ± 5 � �^5 ± .

This matrix could also be written as the original matrix�POvx

plus the originalperturbationmatrix � Oyx plus a matrix û in Eq.( 4.25) with the remainingnon-zeroentries.This û is thebackrotatedpartialsum û �

�� �v£ ó ��¢"� .

� ¥ L ]V�� Í� � Ì ó ��¢"� @ �QQQQ. - � E�G E � )>G E � -<GA- � -`GE � -<GA- � -<GA- � -<G - .) � E�GA- � -<G . . .- � E�G�. � -<G . . .- � -<G . . . .

�RRRR� (4.24)

û¾@ �QQQQ. . . . .. k�-Xkm-<GA- � -<GAk�- .. - � -<G . . .. k�-<G . . .. . . . .

�RRRR� (4.25)

As aconclusionit canbestatedthatastargraphlikestructurecanbeidentifiedif theeigensystemhasthefollowing characteristics:

* Two eigenvalueswith differentsign,but relatively equalor equalabsolutevalue (in comparisonto the other eigenvalues): � � �¾�§¦l� � � � � + � ,}@- � ¦<¦`¦ � M and + R@Ä, .

CHAPTER4 61

* Theeigenvectorsto theabove mentionedeigenvaluesfeaturethesamean-chor: � ï y   ��%5«��n%��tw «�� @ � ï y   ��% � �n%���w � � with H¸@Ä  .* The distribution of absolutevaluesof eigenvectorcomponentsshows onehigh absolutevalueandthe rest is relatively lower. In the caseof a clearstar, all otherabsolutevaluesarethesame.

If thestaris clearlyidentifiable,it canevenbeventuredto constructanunper-turbedmatrix andcheckfor theperturbation,otherwisethealternativeway usingtheFouriersumasafilter tool shouldbeinvestigated.

4.4.2 CompleteGraph

The spectralbehavior of a non-directionalcompletegraphas in Fig.( 4.8) andrepresentedby matrix Ô F 5 in Eq.(4.26)andits correspondingHermitianmatrixÔ L ]V�F 5 @ 'GF 5 in Eq.( 4.27) is characterizedby the fact that only oneeigenvalue  ��%��m� � �|� @¨- � f exists, while all otherswill be of the sameabsolutevalue� �J� � f � + R@ - . Sincethe sumof all eigenvalueshasto be . asexplainedinsection 3.3.4they alsohave theoppositesignof thestrongesteigenvalue. In ad-dition all eigenvectorcomponentsof thefirst eigenvectorwill haveequalabsolutevalue.This holdsalsowhenusingcomplex Hermitianmatrices(seeeigensystemin Tab.( 4.13).

1

2 3

4

1 1

1

1

1

1

1

1 1

1

1

1

Figure4.8: Completesymmetricbidirectionalgraphcorrespondingto Eq.(4.26)

Ô F 5 @ �QQ . - � -`G - � -<GA- � -<G- � -<G . - � -<GA- � -<G- � -<GA- � -`G . - � -<G- � -<GA- � -`G - � -<G . �RR� (4.26)

Ô L ]V�F 5 @ 'KF 5 @ �QQ . -�¦D4:- ->¦D4¤- -�¦S4¤--�¦S4¤- . ->¦D4¤- -�¦S4¤--�¦S4¤- -�¦D4:- . -�¦S4¤--�¦S4¤- -�¦D4:- ->¦D4¤- . RR� (4.27)

62 CHAPTER4

k� � ��tw �

,�@C- ,�@8E ,�@6) ,@241 1. 0.5 0.5 0.5 0.52 -0.33 -0.09 -0.53 -0.20 0.823 -0.33 0.04 -0.54 0.79 -0.294 -0.33 0.87 -0.29 -0.29 -0.29

Table4.13:Eigensystemof'GF 5 in Eq.(4.27)

Thiseigensystemfor all eigenvaluesandtheeigenvectorcorrespondingto thelargesteigenvaluecorrespondto Cvetkovic et al. [CS97, p.235].

Now consideran asymmetriccompletegraphasin Fig.( 4.9). The complexadjacency matrix is givenin Eq.(4.28)andtherotatedHermitianmatrix Ô L ]V�F 5 ð @'KF 5 ð in Eq.(4.29).

1

2 3

4

1 2

4

2

3

3

0

0 5

5

1

1

Figure4.9: Completedirectionalcomplex graph

Ô F 5 ð @ �QQ . E � )>G ) B�G) � E�G . E � -<G©4 � B�G)>G - � E�G . - � -<GB B � 4�GA- � -<G . �RR� (4.28)

Ô L ]V�F 5 ð @ 'GF 5 ð @ �QQ . ):¦eB � .:¦�7�G E5¦æ-Xk E³¦{-<G ):¦�B � ):¦�B´G):¦�BSkÏ.³¦87�G . E³¦æ-Zk~.³¦87�G3Gzs:¦�) � .:¦87´GE5¦æ- � E5¦æ-<G¯E³¦{- � .:¦�7�G . -�¦S4):¦�BSk�)³¦�B�Gzs:¦D)�k~.:¦�7�G ->¦D4 . �RR� (4.29)

Theeigensystemof this graphis givenin Tabs.(4.14)and( 4.15)Again the effect of bidirectionalasymmetriccommunicationis visible. The

eigenvaluesloosetheir strict behavior asdo theeigenvectors.As a conclusionit canbestatedthata completestructurecanbe identifiedif

theeigensystemhasthefollowing characteristics:

CHAPTER4 63

k� � ��tw �

,�@¾- ,�@ÄE ,�@T) ,�@241 1.0 0.44+0.26i 0.55+0.12i 0.21+0.19i 0.582 -0.67 -0.12-0.39i -0.53+0.05i -0.06+0.19i 0.723 -0.37 0.64 -0.36+0.47i -0.04-0.48i -0.06-0.08i4 0.05 -0.06-0.40i 0.04+0.22i 0.80 -0.15+0.34i

Table4.14:Eigensystemof matrix'GF 5 ð in Eq.(4.29)� � � f � ± � = � 5

1.0 -0.67 -0.37 0.05� · �  0W·�4 � · �  03·�4 � · �  03·�4 � · �  0W·�4��tw f 0.51 0.53 0.41 -1.9 0.64 0 0.40 -1.7��tw ± 0.56 0.21 0.53 3.1 0.59 2.2 0.23 1.4��tw = 0.29 0.75 0.20 1.9 0.48 -1.7 0.80 0��tw 5 0.58 0 0.72 0 0.1 -2.3 0.37 2.0

Table4.15:Eigensystemfor' Ovx Ý

in Eq.(4.16)with ·¿@ � · � � ^º¹"��| �* Thereis only oneeigenvalueof relative high absolutevalue,while all oth-

ershave the sameor almostthe sameabsolutevalueandaresmallerthanthe largesteigenvalue:

Ø � � �$ð�ñ �eÿ�ÿ�� � �~� Ö + R@   ��% and � � ��ð­ñ � @o b�$ð�ñ�ªd¤��« ��dgf � � �U�* Theeigenvectorcorrespondingto thelargesteigenvalueshowsadistributionof absolutevaluesof eigenvectorcomponentsthat is equallydistributedorcloseto that. All membersareof relative equalimportance,which is intu-itive.

4.4.3 Two subgroupsconnectedby onelink

An interestingexamplethatgivesmoreinsightinto theinterpretationof thesignoftheeigenvaluescanbeconstructedasfollows. Considera network with two sub-groupssuchthattheonesubgroupis astarstructurewhile theotheris acompletegraph.Thesetwo subgroupssharea link. (seeFig.( 4.10)).Thematrix ¬ for this graphis givenin Eq.( 4.30). Thestructureof thetwo sub-groupsis evidentin thematrix ¬ . Theeigensystemof therotatedmatrix is givenin Tab.( 4.16). The fact that the secondandthird eigenvaluesharethe samean-chorpointsto thefact thata secondinterpretablecharacteristiccanbeseen.Theeigenvectorsgive this interpretation. The subspacecorrespondingto the nega-tive eigenvalue

� ± @ k .:¦Ds^[ givesthepatternfor theconnectionsin thecomplete

64 CHAPTER4

6

8 7

9

1

5 3

42

Figure4.10:A network consistingof two subgroupsandsymmetricbilateralcom-munication

group,while the subspacecorrespondingto the positive eigenvalue� = @ .:¦Ds+4

showsthatthetwo subgroupsbreakapartif vertex s is removed.

¬¢@�QQQQQQQQQQQQ

. - � -<GA- � -`G - � -<GA- � -<G . . . .- � -<G . . . . . . . .- � -<G . . . . . . . .- � -<G . . . . . . . .- � -<G . . . . - � -<G . . .. . . . - � -<G . - � -<GA- � -`G - � -<G. . . . . - � -<G . - � -`G - � -<G. . . . . - � -<GA- � -<G . - � -<G. . . . . - � -<GA- � -<GA- � -`G .

�RRRRRRRRRRRR�(4.30)

This behavior alsoholdsfor arbitrarymatricesof this kind. This follows theobservationsof Barnettet al. [BR85].

4.5 Conclusion

It canthusbeseenfrom theseexamples,that theeigensystemof a complex Her-mitian adjacency matrix holdsall the information,that therealadjacency matrix

CHAPTER4 65

k� � �tw �

,�@Y- ,�@�E ,@T) ,�@­4 ,@�B ,�@6s ,�@®7 ,�@6[ ,@ÄD1 1 0.09 0.03 0.03 0.03 0.2 0.52 0.47 0.47 0.472 -0.68 0.64 -0.3 -0.3 -0.3 -0.45 0.32 -0.08 -0.08 -0.083 0.64 0.69 0.35 0.35 0.35 0.35 0 -0.12 -0.12 -0.124 -0.43 -0.32 0.24 0.24 0.24 -0.3 0.71 -0.21 -0.21 -0.215 -0.32 0 0 0 0 0 0 0.76 -0.12 -0.646 -0.32 0 0 0 0 0 0 -0.34 0.81 -0.487 0.11 -0.09 -0.26 -0.26 -0.26 0.74 0.34 -0.21 -0.21 -0.218 0 0 0.72 -0.04 -0.69 0 0 0 0 09 0 0 -0.51 0.81 -0.29 0 0 0 0 0

Table4.16:Eigensystemof therotatedmatrix ¬ in Eq.(4.30)

carriedandin additiondoesalsoprovide informationof thedirectionof commu-nication.Informationsthatcanbegainedfrom this analysisare:

* The patternstructureof bidirectionalasymmetriccommunication,whichcanberepresentedasaFouriersum.

* Structuralsimilarity thatis representedin theprojectorsasin realadjacencymatricesanddirectionalsimilarity from thephaseinformationavailableincomplex numbers.

* Patternhomogeneitythat is expressedin theeigenvectorscorrespondingtopositiveandnegativeeigenvalues.

* Perturbationtheorycanbeusedto explaineigensystembehavior.

The methodof eigensystemanalysisof complex Hermitianadjacency matri-cescanthusbroadenthesetof informationthatcanbegained.Thefact that thisanalysisis performedin Hilbert spaceis a moregeneralapproachthenthe onesavailableat themoment.Theuseof complex numbersallow for theuseof bidi-rectionaldatawithout informationloss.

66 CHAPTER4

Chapter 5

Data sets

In this chapterthedatasetsusedin this work will beshortlydescribedandtheircharacteristicswill beintroduced.Two well known datasetswill beused.

1. TheEIESfull datasetasdescribedin Freeman[FF79]

2. TheEIESsubsetasdescribedin Freeman[Fre97]

Thesedatasetsarebeingused,sothatthedifferencesandsimilaritiesbetweenthemethodproposedin chapter 4 andthestandardmethodsbecomeclearlyvis-ible. The completeEIES dataset is usedto show how communicationpatternsaredetected,the subsetis usedto show how the methodyields resultsthat aredifferentfrom thosereturnedby thestandardmethodsbut not implausible.

5.1 EIES data set

This dataset is basedon researchdoneby L. andS. Freemanin the late 1970s[FF79]. It is thoroughlydiscussedin [WF94]. It describesthe communicationbehavior of a set of researcherswho usedan electronicinformation exchangesystem(EIES).This researchfocusedon theway CMC influencedthecommuni-cationbehavior within a groupof researcherswho aregeographicallyapartandsharethesameinterestinto anewly developingresearchfield.

This datasetgivesthe opportunityto seethat differentcommunicationpat-ternsexist in onegroupwith respectto direction,even if the groupis not huge.Tab.( 5.1) representsthe weightedadjacency matrix. In this dataset messagesfrom onememberto oneothermemberaswell asfrom onememberto all othermembershavebeenincluded[Fre04]. For thepurposesof thisbooktheentriesonthediagonalweresetto y ��� @Ä. .

67

68 CHAPTER5

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34

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910

1112

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34682

52177

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5610

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355

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1070

Table5.1:F

ullEIE

Sdataset

CHAPTER5 69

Theplotsof in- andout-degreeasdefinedin section3.2.1canbeseenin Fig.(5.1)and Fig.( 5.3). The correspondingplots of strengthas given in section 3.2.1is shown in Fig.( 5.2) andFig.( 5.4). Thesefiguresshow that while the degreedistributiondoesnotdropdramaticallyafterthehighestdegreevalue,thestrengthdropssharply. This canbeexplainedby thefact thatthememberwith ID - , whoactuallyis Linton Freeman,wasthesystemadministratorin thisexperiment.Thushehadto write many messagesto all othermembers[Fre04].

1 5 10 15 20 25 30k

10

15

20

25

dki

Figure5.1: EIES:In-degreesortedby amount

1 5 10 15 20 25 30k

0

500

1000

1500

2000

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ski

Figure5.2: EIES:Inboundnormalizedstrengthsortedby amount

70 CHAPTER5

1 5 10 15 20 25 30k

0

5

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15

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25

30

dko

Figure5.3: EIES:Out-degreesortedby amount

1 5 10 15 20 25 30k

0

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sko

Figure5.4: EIES:Outboundnormalizedstrengthsortedby amount

CHAPTER5 71

Thestandardindicesfor this datasetare:

* Numberof authors:)�E* Numberof messages:-"B>.^)�4* Inboundstrengthdistribution : As canbeseenin Fig.( 5.2)oneauthor(ID

1) clearly hasthe highestinboundstrengthasdefinedin Eq.( 3.10). Mostof theothermembershave a strengthof below B>.�. . Only 7 authorshave astrengthin therangeof B>.�. to -"B>.>. .* Outboundstrengthdistribution: As canbeseenin Fig.( 5.4)theauthorwith

ID 1 againclearlystandsapart. Most of theothermembershave a degreeof below B>.�. . 5 authorshave a strengthin therangeof B>.�. to -"E>.�. . and2authorsarein therangeof -"B>.>. to E�B>.�. .* Outboundstrength/inboundstrength:Tab.( 5.2)givestheoutboundstrength

andinboundstrengthindicesnormalizedwith thetotalamountof messages.

* Centrality: Following Eq.( 3.2) is given in Fig.( 5.3). In Tab.( 5.2) this isalsogivenin comparisonto thestrength.

* Prestige:Following Eq.(3.9) is givenin Fig.( 5.1). In Tab.( 5.2) this is alsogivenin comparisonto thestrength.

* Eigenvectorcentralitycalculatedasin Freeman[FF79]seeTabs.(7.6),( 7.7),( 7.8) and( 7.9) in theappendix:As canbeseen,theeigenvaluescomeinpairs,thusthetwo correspondingeigenspacesareusedto evaluatethehier-archicalstructure.

72 CHAPTER5

AuthorID Outboundstrength Inboundstrength Centrality Prestige1 0.21 0.17 0.032 0.0302 0.075 0.081 0.029 0.0253 0.00093 0.0067 0.003 0.0114 0.022 0.021 0.019 0.0195 0.010 0.0059 0.029 0.0086 0.016 0.014 0.016 0.0137 0.0013 0.0082 0.001 0.0078 0.11 0.092 0.026 0.0219 0.013 0.025 0.011 0.01110 0.024 0.025 0.021 0.01911 0.056 0.049 0.021 0.01912 0.0055 0.013 0.006 0.01213 0.00100 0.0098 0.003 0.00914 0.0054 0.013 0.006 0.01015 0.015 0.015 0.006 0.00816 0.013 0.022 0.012 0.01517 0.015 0.019 0.018 0.01718 0.013 0.015 0.011 0.01119 0.0043 0.011 0.007 0.01120 0.00053 0.0065 0.002 0.00821 0.0015 0.0074 0.004 0.01022 0.0047 0.019 0.004 0.01023 0.0039 0.0080 0.007 0.00924 0.043 0.042 0.032 0.01325 0.0073 0.0073 0.007 0.00626 0.0045 0.011 0.005 0.00827 0.020 0.027 0.017 0.01828 0.0016 0.0052 0.004 0.00829 0.15 0.091 0.029 0.02630 0.019 0.027 0.014 0.01431 0.069 0.076 0.032 0.02732 0.071 0.061 0.018 0.018

Table5.2: Outbound,inboundstrength,centralityandprestigeof theEIEScom-pletedataset

CHAPTER5 73

5.2 Subsetof EIES data set

The EIES subsetis a seven membersubsetof the total EIES dataset(Tab. 5.3,[Fre97]). Thesesevenmembersarethemostprominentin thegroup.Theentriesnow representthe numberof messagesthatwent from onememberto the othermember [Fre04]. This subset,sinceit is smallerandthusmoreopento intuitiveinvestigation,is usedto giveacomparisonbetweenthecalculationdoneby Free-manandtheresultsobtainedwith themethodproposedin chapter4 [HGS04].

Author ID Freeman White Alba Bernard Doreian Mullins Wellman1 2 4 8 11 24 29

Freeman 1 0 115 17 93 53 33 84White 2 84 0 4 5 5 0 15Alba 4 16 10 0 15 3 3 4

Bernard 8 127 22 17 0 57 12 34Doreian 11 57 9 4 57 0 8 10Mullins 24 23 4 3 9 8 0 33Wellman 29 118 24 5 35 15 45 0

Table5.3: Numberof messagesexchangedbetweenmembersin theEIESsubset

Thestandardindicesfor this datasetare:

* Numberof authors:7* Numberof messages:-"E>D�B* Outboundandinboundstrength,centralityandprestige:SeeTab.( 5.4).

* Themostcentralandprestigiousmemberis Freeman,followedby Bernard(seeTab.( 5.4)).

* Eigenvectorcentrality:Calculatedwith thedichotomizationandsubsequentsymmetrizationseeTab.( 5.5). Themostcentralmemberdueto theeigen-systemis Wellman.This is only understandablewhenthepreprocessingistakeninto account.

In Fig.( 5.5) thein-degreeis givenby authorID. It canbeseenthat thegraphis almostcompletelyconnected.This is supportedby theout-degreedistributionin Fig.( 5.7).

74 CHAPTER5

1 4 2 7 2 6 3k

1

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dki

Figure5.5: EIESsubset:In-degree;k: authorID

1 4 2 7 5 6 3k

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ski

Figure5.6: EIESsubset:Inboundnormalizedstrengthsortedby amount;k: au-thor ID

CHAPTER5 75

Thestrengthof inboundandoutboundtraffic in thesubsetis givenin Fig.(5.6)andFig.( 5.8).Clearlynow theauthorwith ID 1 (Freeman)is themostactiveandauthorwith ID 4 (Bernard)comessecond,but verycloseto all others.

76 CHAPTER5

1 4 7 5 2 6 3k

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dko

Figure5.7: EIESsubset:Out-degreesortedby amount

1 4 7 5 2 6 3k

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sko

Figure5.8: EIESsubset:Outboundnormalizedstrengthsortedby amount

CHAPTER5 77

Author IDOutbound Inbound

Centrality Prestigestrength strength

Freeman 1 0.31 0.33 0.24 0.24White 2 0.087 0.14 0.20 0.24Alba 4 0.039 0.039 0.24 0.24

Bernard 8 0.21 0.17 0.24 0.24Doreian 11 0.11 0.11 0.24 0.24Mullins 24 0.062 0.078 0.24 0.20Wellman 29 0.19 0.14 0.24 0.24

Table5.4: Outboundandinboundstrengthfor theEIESsubsetnormalizedby thetotal numberof messages,aswell ascentralityandprestige

Thesetwo datasetswill be analyzedwith the proposedmethodin chapter6, and the resultsobtainedwill be comparedwith the standardresultsand aninterpretationwill begiven.

Author/Eigenvalue 1.0 0.13 0.071 0.071 0.011 0 0Freeman 0.40 0.25 0.20 0.61 0.47 0 0White 0 0 0 0 0 0 1.0Alba 0.16 0.48 -0.47 -0.38 -0.15 0.71 0

Bernard 0.51 0.064 0.47 0.38 -0.71 0 0Doreian 0.40 0.25 0.27 -0.22 0.47 0 0Mullins 0.16 0.48 -0.47 -0.38 -0.15 -0.71 0Wellman 0.60 -0.65 -0.47 -0.38 0.061 0 0

Table5.5: Eigensystemof theEIESsubsetascalculatedwith themethodusedbyFreeman

78 CHAPTER5

Chapter 6

Resultsand Discussion

Thischapterwill presentresultsobtainedwith theproposedmethod.To show thedifferencesbetweenthestandardmethodsandtheproposedmethodtheverywelldescribedEIESdatasetof Freeman(seesection 5.1) andthesubsetof thatdatasetdescribedin section 5.2will beused.

For eachdatasetthefollowing resultswill bepresentedanddiscussed:* Eigensystem

– Full dataset: 7.2Tabs.(7.2),(7.3),( 7.4)and( 7.5)

– Subset:seesection6.2.1* Eigenvaluedistribution

– Full dataset:seeFig.( 6.2)

– Subset:seeFig.( 6.29)* Cumulativevariancecoveredby eigenvalues

– Full dataset:seeFig.( 6.3)

– Subset:seeFig.( 6.30)* Distributionof theabsolutevalueandphaseof eigenvectorcomponents

– Full dataset:seeFigs.(6.4)-(6.14)andFigs.(6.5)-(6.15)

– Subset:seeFigs.(6.31)-(6.33)andFigs.(6.32)-(6.34)* Eigenprojectordensityplotsfor relevantsubspaces(realandimaginarypartseparately)

– Full dataset:Figs.(6.16)-(6.26)andFigs.(6.17)-(6.27)

– Subset:Figs.(6.35)-(6.37)andFigs.(6.36)-(6.38)

79

80 CHAPTER6

6.1 EIES Data Set

6.1.1 Eigensystemof the EIES Data Set

Theeigensystemof thecompleteEIESdataset(seeTab.( 5.1))obtainedwith themethoddescribedin chapter4 is givenin theappendixin Tabs.(7.2),(7.3),( 7.4)and( 7.5). Fig.(6.1)showstheeigenspectrumwhensortedby value.For aclearerview of a possiblesymmetryFig.( 6.2)shows thedistribution of theeigenvalues,wherethe positive eigenvalueswith index H and the negative eigenvalueswithindex aregiven.It suggeststhebrokensymmetryof astargraphasdiscussedinchapter3.

5 10 15 20 25 30j

-1

-0.5

0

0.5

1

PSfragreplacements

� «

Figure6.1: Eigenspectrumof thecompleteEIESdatasetsortedby value

In addition the first and thus largesteigenvalue covers already s^sqp of thevarianceof the data(seeTab.( 6.1)). Togetherwith the secondeigenvaluetheyexplain [^sqp of the variance. The first six eigenvaluescover about D+[qp . For aclearerrepresentationof thecumulativevarianceseeFig.( 6.3).

Thus, to understandthe patternstructurewith the most messagevolume itwould be sufficient to look at the first two eigenspaces.For a broaderview thefirst six eigenspacesaregivenandinterpretedhere.If moredetailis necessary, theothersubspacescanbetakeninto account.

As thedistributionof theeigenvectorcomponentsgivesahint aboutthestruc-turewithin thepattern,first thedistribution of theabsolutevalueof componentsof theeigenvectorscanbeseenin Figs.(6.4),( 6.6),( 6.8),( 6.12),( 6.10),( 6.14)with their respective phasedistribution in Figs.(6.5),(6.7), (6.9), (6.13), (6.11),(6.15). The % -axis is denotedby , which is the runningindex within the eigen-vector. This index is not equivalentto theauthorID, sincethevalueshave been

CHAPTER6 81

2 4 6 8 10 12 14 16

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PSfragreplacements

� F« � � 9¯j,f

Figure6.2: Eigenspectrumof thecompleteEIESdatasetsortedby absolutevalue

5 10 15 20 25 30 35k

0.2

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0.6

0.8

1

PSfragreplacements

� ±° w �

Figure6.3: Cumulativecoveredvarianceby eigenvalues� �

82 CHAPTER6

+ Coveredvarianceper� � Cumulativecoveredvariance

1 0.66 0.662 0.20 0.863 0.06 0.924 0.03 0.965 0.02 0.986 0.01 0.98

7-10 � .³¦e.:- 0.9911-32 78. 1.0

Table6.1: Cumulativecoveredvarianceof eigenvaluesfrom Tabs.(7.2)-(7.5)

sortedby eitherabsolutevalueor amount.

First suspaceof EIES data set

Let usstartwith analyzingthe subspacecorrespondingto� f in moredetail. As

canbeseenin Fig.(6.4)thedistributionstartsof ratherlow andalreadythesecondlargestcomponentonly hasan absolutevalueof about .:¦D4 . From +Q@¯-n.Z¦<¦<¦ªE�Eandagainfor the rest thereis almostan equaldistribution visible. This pointstowardsa connectedgraph. Thephasedistribution, asgiven in Fig.( 6.5), variesbetween. and

: 5 , whichsuggests,thatthecommunicationis mostlybalanced.

1 5 10 15 20 25 30l

0.2

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0.8

1

PSfragreplacements

± ²�³(´�±

Figure6.4: Absolutevaluedistribution for theeigenvectorcorrespondingto µ ³As a representationof theprojectorsfollowing Eq.( 3.66)therealpartof the

projectors,thepatternsimilarity of eigenvectorcomponentsof eacheigenvector,andtheimaginarypart,theabsolutevalueof thecrossproductof theeigenvector

CHAPTER6 83

5 10 15 20 25 30l

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-1.57

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PSfragreplacements

¶¸· ²�³(´º¹

Figure6.5: Phasedistributionof theeigenvectorcomponentscorrespondingto µ ³componentscanbevisualizedin densityplots,wherethegray level indicatestherelative absolutevalue. The densityplot givesthe completematrix of the real,respectively the imaginarypartof theprojector. Thenumbersat thebottomandat the left handsideof theplot indicatetheauthorasin theoriginal matrix. Thisrepresentationis now the link backto theoriginal complex Hermitianadjacencymatrix, andthusby back-rotationto theoriginal complex adjacency matrix. Thematrix is symmetricfor therealpartandskew-symmetricfor theimaginarypart.

Fig.( 6.16)shows, that theauthorswith ID »½¼�¾M¼�¿^¿^¼À»+ÁM¼ÀÂM¿ and Âq» have a highpatternsimilarity to authorwith ID ¿ , who is the anchorin this patternascanbeverifiedby checkingagainstthecompleteeigensystem,andalsoamongsteachother. This is representedby thebrightsquareswhich lie in thebottomrow. Sincethis plot is symmetric,this also holds for the first column. A high brightnessstandsfor a high similarity. Fig.( 6.17)in additionshows, thatauthorwith ID »�Áhasa high (first row) skewnesscomparedto authorwith ID ¿ , which meanshecould be viewed aswriting moremails to authorwith ID 1, thenhe getsback.Whereasauthorswith IDs » and ÂM¿ have a low skewness,andarethusmorein-boundoriented.Sincethis plot is skew-symmetric,a bright squarein onehalf ofthematrix correspondsto a darksquarein theother.

Theprojector ÃÅÄ�Æ of this first subspaceweightedby thecorrespondingeigen-valueandrotatedbackwardsby Ç�È�É Ê is partly given in Tabs.(6.3)and( 6.4). Forreasonof betterclarity only the ten highestranked membersin that eigenvectorhavebeentaken.Theindiceson topandon theleft handsidegive theauthorID.

If thisresultis now comparedto asubmatrixof thecomplex adjacency matrix,whichhasbeenextractedfrom thecomplex adjacency matrixby usingtheauthorsbasedon thestatusindex within thesubspace,this givesa clearerview what the

84 CHAPTER6

subpatternis. Thesubmatrixbasedon theeigenvectorcentralitygivenin thefirsteigenvector is given in Tab.( 6.5). As canbe seen,the membersareconnectedandthe subgraphis complete.This hasalreadybeensuggestedby theprojectorandthedistribution of theabsolutevaluesof theeigenvector. Also thedirectioninformationgiven by the projectorcanbe found in this submatrix. The authorwith ID 1 hasmoreinboundthanoutboundtraffic, while authorswith IDs 29 and8 aremoreoutboundthaninboundoriented.

Secondsubspaceof EIES data set

Thesecondsubspacealreadyservesasa correctionfor thefirst (Fouriersum),inthatit almostcancelsthediagonalentries,whichwerepresentin thefirst projector,andat the sametime shows the secondstrongestpattern(variancecoverageofabout»+ËqÌ ) andin thiscasealsodescribesthebehavior of thesubgroupconsistingof the strongestmembersin this pattern. Comparedto the first subspace,givenin Fig.( 6.4), Fig.( 6.6) shows, that thehighestvaluedcomponentranksat aboutËMÍ8Î andthenext followsonly afteragapof about ËMÍ�» . Againafteradropof aboutËMÍ8» onemoreauthorsstandsout, while the restthendropsbelow ËMÍÏ¿ andout ofrelevance.Thissuggestsastarpatternwith about majorcentersandaminoroneof stars, ÐrÑ¡¿ÒÍ�Í�ÍXÓ . The phasedistribution in Fig.( 6.7) shows, that the phaserangesfrom Ô{ÕKÍ�Í�Í�Õ , which is partly due to correctionsfor the first subspace.This is probablythecasefor

¶ ÖØ× Õ .

1 5 10 15 20 25 30l

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± ²ÚÙ>´�±

Figure6.6: Distributionof± Û�Ù>´>±

The real part of the projectorgiven in Fig.( 6.18) suggests,that the authorswith IDs » and »+Á againhave the strongeststructuresimilarity comparedwiththe anchor, who is the authorwith ID ¿ . The patternin questionis a star like

CHAPTER6 85

5 10 15 20 25 30l

-3.14

-1.57

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3.14

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¶¸· ²ÚÙ>´º¹

Figure6.7: Distributionof¶Å· Û�Ù>´Ï¹

behavior. Fig.( 6.19)shows, thattheauthorwith ID ¾ hasa low skewness,whichcorrespondsto ahigheroutboundbehavior towardsauthor ¿ .

Again thebackrotatedweightedprojector ÃÅÄ Ü is givenpartly for thesecondsubspacein Tabs.(6.6) and ( 6.7). As can be seen,this projectorcorrectsthediagonalentriesof ÃtÄ�Æ almostcompletely. In additionit strengthensthepositionof theanchor(authorwith ID ¿ ) andtheauthorwith ID »+Á .

Thesubmatrixcomposedalongthesecondeigenvectoris givenin Tab.( 6.8).This matrix supportsthe interpretationgiven alreadyfor the correspondingpro-jector ÃtÄ�Ü . A starpatternis clearlyvisible. Thestrongeststaris aroundtheauthorwith ID ¿ . Theauthorwith ID »+Á is alsothecenterof astrongstar.

The first two subspacesthusgive informationaboutthe fact, that while therelevant membersarestronglyconnectedamongstthemselvesand looselycon-nectedwith the restof the group,therearedistinct starlike patternsfor eachoftherelevantmemberswith theauthorwith ID ¿ asthestrongestof them.

Third subspaceof EIES data set

Thedistributionof theabsolutevaluesof thethird eigenvectorisgivenin Fig.(6.8).Theanchorhaschanged,which pointstowardthefact,thata differentgroupandpatternis described.This time, the authorwith ID ¾ is the anchor. In additionastheanchorof thefifth eigenvectorsuggestsin Tab.(7.2) , thereis a starpatternagaincenteredon the authorwith ID ¾ . The distribution supportstheview, thatwhile thereare4 relevantmembers,therestareequallylessrelevant. Thephasedistribution in Fig.( 6.9)suggests,thatthereis symmetry, which pointstowardsaÝ,Þ�ß�àvá�Þ0âäã®àváÚå�ß�àváÚÞ�â

patternwith astarpattern,asin subspaces¿ and  .

86 CHAPTER6

1 5 10 15 20 25 30l

0.2

0.4

0.6

0.8

1

PSfragreplacements

± ²Úæ>´�±

Figure6.8: Eigenvectorcomponentdistribution for the3rdeigenvector

5 10 15 20 25 30l

-3.14

-1.57

0

1.57

3.14

PSfragreplacements

¶¸· ²Úæ>´ç¹

Figure6.9: Eigenvectorcomponentdistribution for the3rdeigenvector

CHAPTER6 87

Therealpartof theprojectorgivenin Fig.( 6.20)showsthehighpatternsimi-larity betweentheauthorwith ID ¾ andtheauthorswith IDs ¿^¿^¼�Â^Ë and Âq» . WhileFig.( 6.21)showsamoreoutboundcommunicationfor theauthorswith IDs » and»�Á comparedto theauthorwith ID ¾ anda moreinboundcommunicationfor theauthorwith ID Â^Ë and ¿ comparedto theauthorwith ID ¾ .

The submatrixdeducedfrom the third eigenvector is given in Tab.( 6.9). Itconfirms, that there is a strongly connectedgroup aroundthe anchorwith theauthorwith ID ¾ . For acomparisonseetheprojectorÃÅÄ�è givenin Tabs.(6.10)and( 6.11).

Fifth subspaceof the EIES data set

Thefifth eigenvectoragainhasa similar behavior asthesecond,suggesting,thatthis patternis thecompletepatterncorrespondingto thestarpatternfoundin thethird subspace.Thedistribution is givenin Fig.( 6.10). Thephasedistribution inFig.(6.11)supportsthatview, in thatit is similar to theoneof thefirst eigenvector,if onetakesinto account,thata phasebetweenËêé ¶ éìëí hasa similar interpre-tationasaphasebetweenÔ æí\Õîé ¶ éïÔ{Õ . In thiscasethestarpatternof thethirdeigenspaceis strongerthen the completegraphpatternof the fifth eigenspace,which is reflectedin the absolutevalueof the respective eigenvalues. The fact,that therelies anothereigenvalueandthusanotherpatternbetweenthe two canbeexplainedby theinterlacingeffect of eigenvalueswhich occursif perturbationtakesplace.

1 5 10 15 20 25 30l

0.2

0.4

0.6

0.8

1

PSfragreplacements

± ²Úð>´>±

Figure6.10:Eigenvectorcomponentdistribution for the5th eigenvector

Theprojectorin Fig.( 6.22)shows thestructuralsimilarity within thepattern.Clearly, the authorswith IDs Â^ËM¼ÀÂ^»½¼ÀÁ and ¿^¿ have a high structuralsimilarity

88 CHAPTER6

5 10 15 20 25 30l

-3.14

-1.57

0

1.57

3.14

PSfragreplacements

¶¸· ²Úð>´ç¹

Figure6.11:Phasedistribution for the5theigenvector

with author ¾ . Fig.( 6.23)reveals,thatauthorsÂ^ËM¼�Á and » have a low skewness,translatinginto aninboundpatterncomparedto author¾ , while authors¿�Ó�¼À»^» and»�Ó aremoreoutboundin comparison.

The fact, that therearetwo disturbedstarpatternsaroundtheauthorwith ID¿ andthe authorwith ID ¾ , combinedwith the result that the authorwith ID ¾alsoplaysa relevantrole in thegroupof theauthorwith ID ¿ suggests,thatthesetwo starpatternsarecloselyconnected.While the first group is morestronglyconnectedwith everybodyelse,while still keepingastarpattern,thesecondgrouparoundthe authorwith ID ¾ hasa strongerfocuson the starpatternthanon thecompletepatternwhich might allow theconclusionthat thegroupof theanchorwith ID ¿ is more“open to the public”, while thegrouparoundtheanchorwithID ¾ is moreclosed.

Fourth subspaceof the EIES data set

Thedistribution in thefourth eigenvectoris givenin Fig.( 6.12). Now theauthorwith ID »+Á is theanchor. Theotherrelevantmembersarethosewith IDs » and »�Ó .It is probable,thatagainthis is anew subpatternandalsoanew group.Thephasedistribution in Fig.( 6.13)againsuggestsastronglyconnectedpattern.

Thedensityplotssupporttheabove interpretation.Fig.( 6.24)shows, thattheauthorswith IDs »½¼�»�Ó and ÂM¿ seemto behighly patterncorrelatedwith theauthorwith ID »+Á . In additionFig.(6.25)shows,thatwhile theauthorswith IDs » and »�Óhaveamoreinboundbehavior, theauthorwith ID ÂM¿ hasamoreoutboundpattern.

CHAPTER6 89

1 5 10 15 20 25 30l

0.2

0.4

0.6

0.8

1

PSfragreplacements

± ² í ´>±

Figure6.12:Eigenvectorcomponentdistribution for the4th eigenvector

5 10 15 20 25 30l

-3.14

-1.57

0

1.57

3.14

PSfragreplacements

¶¸· ² í ´º¹

Figure6.13:Phasedistribution for the4theigenvector

90 CHAPTER6

Sixth subspaceof the EIES data set

Fig.( 6.14)givesthedistributionof theabsolutevaluesof theeigenvectorcompo-nentsof thesixtheigenvector, while Fig.(6.15)againshowsthephasedistribution.Theanchorof thatsubspaceis theauthorwith ID » .Theauthorwith ID »�Ó is alsoclose,but overall the relevanceof both authorsis alreadynot very high with acentralityindex of approximatelyËMÍSÓ .

1 5 10 15 20 25 30l

0.2

0.4

0.6

0.8

1

PSfragreplacements

± ²Úñ>´�±

Figure6.14:Eigenvectorcomponentdistribution for the6theigenvector

5 10 15 20 25 30l

-3.14

-1.57

0

1.57

3.14

PSfragreplacements

¶¸· ²Úñ>´ç¹

Figure6.15:Phasedistribution for the6theigenvector

Theprojector ÃtÄ ò is againgivenin his realpart in Fig.( 6.26)andin its imag-inary part in Fig.( 6.27). The authorwith ID » shows a relative high visibility.

CHAPTER6 91

As canbe seen,he seemsto be moreinboundoriented,asin Fig.( 6.26)all hisrelevantpartnersshow a low patternsimilarity.

If onenow takesa look at thetwo partialsumsof weightedprojectors,namelyó ³>ô Ù Ñ ·<õ ÙöÀ÷ ³ µ ö à ö ¹ Ç È>ø Ê given in Tabs.(7.10), ( 7.11), ( 7.12) and ( 7.13) intheAppendixand

ó æXô ñ Ñ · õ ñöÀ÷ æ µ ö à ö ¹ Ç�È ø Ê givenin Tabs.(7.14),( 7.15),( 7.16)and( 7.17)intheAppendix,thenit canbeseenthat

ó ³>ô Ùis alreadya very good

approximationof theoriginal matrix whenbackrotated.ó æXô ñ

on theotherhandshows that oncethe main bulk of communicationhasbeenfiltered out, that theauthorwith ID ¾ , which is Bernard,is actuallythecentralmemberof thegroup.

6.1.2 Comparisonwith dichotomizedandsymmetrisizedeigen-system

If theeigensystemof thefull datasetis calculatedwith themethodusedby Free-man(seeTabs.(7.6) -( 7.9) in the Appendix),a comparisongiven in Tab.( 6.2)showsthattheanchorsof thesubpatternsdonotmatch,exceptfor thefirsteigenspace.ù · µ ö ¹ and ú · µ ö ¹ representtheFreemanmethodandtheHermitianmethod.Thenumbersin bracketscorrespondto theeigenvaluessortedby absolutevalue.TheauthorIDs aregivenfrom left to right for eacheigenvector, goingfrom theanchorto the10thrankedmemberin thateigenvector. Theeffect seenis mostprobablyaresultof thedichotomizationandthusof informationloss.ù · µ ³û¹ 1 29 8 2 32 31 11 24 30 27ú · µ ³û¹ 1 31 8 24 5 29 2 11 3 10ù · µ Ù�¹ 1 29 2 8 32 11 15 18 26 31ú · µ Ù�¹ 17 16 12 23 9 22 7 21 20 14ù · µ æ�¹ 8 32 30 11 29 1 2 9 31 17ú · µ æ�¹ 5 2 32 30 27 10 19 17 6 25ù · µ í ¹ 29 24 2 31 15 32 12 11 14 10ú · µ í ¹ 32 2 31 27 4 6 12 24 8 29ù · µ ð�¹ 8 32 30 29 24 22 11 2 15 14ú · µ ð�¹ 24 1 27 18 5 13 2 28 20 8ù · µ ñ�¹ 2 24 31 22 29 30 8 9 18 10ú · µ ñ�¹ 11 27 15 10 5 4 32 6 29 25

Table6.2: Comparisonof anchorsof thefull datasetbetweenthemethodusedbyFreeman( ú · µ ö ¹ ) andthemethodpresentedin this book(

ù · µ ö ¹ )

92 CHAPTER6

129

82

32

1370.81+

370.81i237.94+

306.61i243.33+

225.41i252.20+

177.74i181.19+

186.72i29

306.61+237.94i

203.10+203.10i

199.54+142.99i

201.64+107.16i

150.33+120.33i

8225.41+

243.33i142.99+

199.54i148.35+

148.35i155.11+

118.43i110.01+

122.40i2

177.74+252.20i

107.16+201.64i

118.43+155.11i

128.36+128.36i

86.29+126.44i

32186.72+

181.19i120.33+

150.33i122.40+

110.01i126.44+

86.29i91.28+

91.28i31

114.67+187.96i

66.79+148.63i

77.01+

116.03i85.35+

97.45i55.48+

94.10i11

133.87+142.30i

85.12+116.89i

87.38+

87.38i91.90+

69.06i65.35+

71.60i24

97.75+

117.32i60.91+

95.20i64.62

+71.79i

68.45+58.20i

47.62+58.93i

3050.12+

81.04i29.30+

64.14i33.63

+50.01i

37.19+41.95i

24.26+40.58i

2744.02+

72.56i25.60+

57.35i29.57

+44.80i

32.80+37.64i

21.30+36.32i

Table6.3:Part1:B

ackrotatedw

eightedprojectorbasedcorrespondingtothe

firsteigenvalue31

1124

3027

1187.96+

114.67i142.30+

133.87i117.32+

97.75i81.04+

50.12i72.56

+44.02i

29148.63+

66.79i116.89+

85.12i95.20

+60.91i

64.14+29.30i

57.35+

25.60i8

116.03+77.01i

87.38+

87.38i71.79

+64.62i

50.01+33.63i

44.80+

29.57i2

97.45+85.35i

69.06+

91.90i58.20

+68.45i

41.95+37.19i

37.64+

32.80i32

94.10+55.48i

71.60+

65.35i58.93

+47.62i

40.58+24.26i

36.32+

21.30i31

65.37+65.37i

44.84+

68.69i38.21

+51.48i

28.29+28.29i

25.20+

25.20i11

68.69+44.84i

51.47+

51.47i42.58

+37.74i

29.61+19.59i

26.52+

17.22i24

51.48+38.21i

37.74+

42.58i31.45

+31.45i

22.18+16.67i

19.88+

14.68i30

28.29+28.29i

19.59+

29.61i16.67

+22.18i

12.24+12.24i

10.90+

10.90i27

25.20+25.20i

17.22+

26.52i14.68

+19.88i

10.90+10.90i

9.71+

9.71i

Table6.4:Part2:B

ackrotatedw

eightedprojectorbasedcorrespondingtothe

firsteigenvalue

CHAPTER6 93

129

82

3231

1124

10

388

+55

9i34

6+

239i

488

+36

4i18

5+

239i

212

+82

i17

7+

178i

128

+89

i29

559

+38

8i0

155

+14

6i13

2+

89i

126

+71

i91

+65

i69

+46

i21

8+

156i

823

9+

346i

146

+15

5i0

82+

20i

288

+26

8i88

+70

i16

4+

172i

55+

41i

236

4+

488i

89+

132i

20+

82i

039

+99

i16

3+

125i

22+

36i

17i

3223

9+

185i

71+

126i

268

+28

8i99

+39

i0

107

+67

i35

+34

i5

+10

i31

82+

212i

65+

91i

70+

88i

125

+16

3i67

+10

7i0

114

+11

9i53

+56

i11

178

+17

7i46

+69

i17

2+

164i

36+

22i

34+

35i

119

+11

4i0

29+

31i

2489

+12

8i15

6+

218i

41+

55i

1710

+5i

56+

53i

31+

29i

0

Tabl

e6.

5:S

ubm

atric

esba

sedo

nfir

stei

genv

ecto

r

94 CHAPTER6

129

28

32

1-364.31-

364.31i147.15+

267.28i144.20-

3.31i214.81+

161.81i-15.49

+75.78i

29267.28+

147.15i-127.76-

127.76i-81.48+

25.66i-148.86-

56.62i-3.68

-45.66i

2-3.31

+144.20i

25.66-81.48i

-28.56-28.56i

-8.78-

74.78i18.34-

11.52i8

161.81+214.81i

-56.62-148.86i

-74.78-8.78i

-99.26-99.26i

13.52-38.04i

3275.78-

15.49i-45.66-

3.68i-11.52+

18.34i-38.04+

13.52i-8.21

-8.21i

11-37.79+

12.25i23.52-

0.74i4.83

-10.02i

18.64-9.08i

4.66+

3.72i15

68.53+

31.26i-33.82-

29.08i-19.58

+7.83i

-37.70-11.17i

-1.75-

11.17i18

-26.13+2.65i

15.30+2.80i

4.52-

5.80i13.32-

3.27i2.49

+3.05i

2618.74-

10.80i-12.44+

3.05i-1.44

+5.88i

-8.90+

6.94i-2.90

-1.45i

31-0.76

+20.57i

3.82-

11.57i-4.07

-4.07i

-1.10-

10.69i2.64

-1.61i

Table6.6:Part1:B

ackrotatedw

eightedprojectorcorrespondingtothe

secondeigenvalue11

1518

2631

112.25

-37.79i

31.26+68.53i

2.65-

26.13i-10.80

+18.74i

20.57-

0.76i29

-0.74+

23.52i-29.08

-33.82i

2.80+

15.30i3.05

-12.44i

-11.57+3.82i

2-10.02+

4.83i7.83

-19.58i

-5.80+

4.52i5.88

-1.44i

-4.07-

4.07i8

-9.08+

18.64i-11.17

-37.70i

-3.27+

13.32i6.94

-8.90i

-10.69-1.10i

323.72

+4.66i

-11.17-1.75i

3.05+

2.49i-1.45

-2.90i

-1.61+

2.64i11

-2.17-

2.17i5.80

+0.25i

-1.70-

1.10i0.91

+1.40i

0.67-

1.44i15

0.25+

5.80i-7.79

-7.79i

0.97+

3.71i0.52

-3.12i

-2.78+

1.16i18

-1.10-

1.70i3.71

+0.97i

-0.95-

0.95i0.39

+1.03i

0.63-

0.84i26

1.40+

0.91i-3.12

+0.52i

1.03+

0.39i-0.64

-0.64i

-0.19+

0.84i31

-1.44+

0.67i1.16

-2.78i

-0.84+

0.63i0.84

-0.19i

-0.58-

0.58i

Table6.7:Part2:B

ackrotatedw

eightedprojectorcorrespondingtothe

secondeigenvalue

CHAPTER6 95

129

28

3211

1518

2631

10

388

+55

9i48

8+

364i

346

+23

9i18

5+

239i

177

+17

8i81

+12

0i73

+58

i89

+35

i21

2+

82i

2955

9+

388i

013

2+

89i

155

+14

6i12

6+

71i

69+

46i

80+

58i

4+

8i15

+10

i91

+65

i2

364

+48

8i89

+13

2i0

20+

82i

39+

99i

22+

36i

158

+8i

15+

5i16

3+

125i

823

9+

346i

146

+15

5i82

+20

i0

288

+26

8i16

4+

172i

027

+4i

988

+70

i32

239

+18

5i71

+12

6i99

+39

i26

8+

288i

035

+34

i0

00

107+

67i

1117

8+

177i

46+

69i

36+

22i

172

+16

4i34

+35

i0

024

+18

i11

119+

114i

1512

0+

81i

58+

80i

15i

00

00

00

32+

24i

1858

+73

i8

+4i

8+

8i4

+27

i0

18+

24i

00

048

+49

i26

35+

89i

10+

15i

5+

15i

9i0

11i

00

013

+8i

3182

+21

2i65

+91

i12

5+

163i

70+

88i

67+

107i

114

+11

9i24

+32

i49

+48

i8

+13

i0

Tabl

e6.

8:S

ubm

atric

esba

sedo

nse

cond

eige

nvec

tor

832

3011

291

29

3117

80

288

+26

8i21

6+

140i

164

+17

2i14

6+

155i

239

+34

6i82

+20

i12

+8i

88+

70i

53+

36i

3226

8+

288i

07

+8i

35+

34i

71+

126i

239

+18

5i99

+39

i10

1+

44i

107

+67

i7

+9i

3014

0+

216i

8+

7i0

018

+14

i39

+71

i21

+23

i0

20+

28i

9+

9i11

172

+16

4i34

+35

i0

046

+69

i17

8+

177i

36+

22i

39+

15i

119

+11

4i15

+9i

2915

5+

146i

126

+71

i14

+18

i69

+46

i0

559

+38

8i13

2+

89i

15+

10i

91+

65i

15+

10i

134

6+

239i

185

+23

9i71

+39

i17

7+

178i

388

+55

9i0

488

+36

4i82

+24

i21

2+

82i

77+

63i

220

+82

i39

+99

i23

+21

i22

+36

i89

+13

2i36

4+

488i

035

+25

i16

3+

125i

25+

18i

98

+12

i44

+10

1i0

15+

39i

10+

15i

24+

82i

25+

35i

030

+35

i5

3170

+88

i67

+10

7i28

+20

i11

4+

119i

65+

91i

82+

212i

125

+16

3i35

+30

i0

28+

15i

1736

+53

i9

+7i

9+

9i9

+15

i10

+15

i63

+77

i18

+25

i5i

15+

28i

0

Tabl

e6.

9:S

ubm

atric

esba

sedo

nth

irdei

genv

ecto

r

96 CHAPTER6

832

3011

29

8-5.12

-5.12i

8.84+0.51i

-27.39+31.84i

3.835.73i

20.55-22.69i

320.51

+8.84i

-7.68-7.68i

51.04-6.81i

1.17-8.37i-37.32+

4.01i30

31.84-

27.39-6.81+

51.04i-172.74-172.74i

-34.92+19.73i

122.65+129.07i

115.73

+3.83i

-8.37+

1.17i19.73-34.92i

-4.66-

4.66i-15.02+

25.08i29

-22.69+20.55i

4.01-37.32i129.07+

122.65i25.08

-15.02i

-91.76-91.76i

13.06

-4.30i0.70

+6.43i

-26.28-

15.88i-3.67+

3.46i18.85+

12.06i2

-11.05+13.62i

-0.99-

21.48i83.82+

58.16i12.89

-10.70i

-59.99-43.93i

9-2.74

-0.75i

3.12-1.55i-3.15+

16.21i2.42+

1.21i2.59

-11.75i

31-17.91+

10.15i8.12-23.90i

66.69+99.46i

18.66-6.17i

-46.74-73.71i17

1.31-

0.38i-0.89+

1.41i-2.76

-7.45i-1.30

+0.11i

1.87+5.48i

Table6.10:part1:B

ackrotatedw

eightedprojectorcorrespondingtothe

thirdeigenvalue

12

931

17

8-4.30

+3.06i

13.62-

11.05i-0.75

-2.74i

10.15-17.91i-0.38

+1.31i

326.43

+0.70i

-21.48-

0.99i-1.55

+3.12i

-23.90+8.12i

1.41-0.89i30

-15.88-26.28i58.16+

83.82i16.21-3.15i

99.46+66.69i

-7.45-

2.76i11

3.46-3.67i

-10.70+

12.89i1.21

+2.42i

-6.17+

18.66i0.11-1.30i

2912.06+

18.85i-43.93-

59.99i-11.75+

2.59i-73.71

-46.74i

5.48+1.87i

1-2.73

-2.73i9.62+

8.48i1.88-0.87i

14.14+5.14i

-0.99-

0.11i2

8.48+

9.62i-30.13-

30.13i-6.43

+2.50i

-45.83-20.02i3.27+

0.58i9

-0.87+

1.88i2.50-6.43i

-0.79-

0.79i-0.02

-8.09i0.13+

0.52i31

5.14+14.14i

-20.02-45.83i-8.09

-0.02i

-41.51-

41.51i3.32+

2.03i17

-0.11-

0.99i0.58+

3.27i0.52

+0.13i

2.03+3.32i

-0.18-

0.18i

Table6.11:part2:B

ackrotatedw

eightedprojectorcorrespondingtothe

thirdeigenvalue

CHAPTER6 97

1 5 10 15 20 25 30

1

5

10

15

20

25

30

524

0

Figure6.16:Densityplot of realpartof eigenprojectorcorrespondingto µ ³

1 5 10 15 20 25 30

1

5

10

15

20

25

30

67

-67

Figure6.17:Densityplot of imaginarypartof eigenprojectorcorrespondingto µ ³

98 CHAPTER6

1 5 10 15 20 25 30

1

5

10

15

20

25

30

293

-515

Figure6.18:Densityplot of realpartof eigenprojectorcorrespondingto µ Ù

1 5 10 15 20 25 30

1

5

10

15

20

25

30

104

-104

Figure6.19:Densityplot of imaginarypartof eigenprojectorcorrespondingto µ Ù

CHAPTER6 99

1 5 10 15 20 25 30

1

5

10

15

20

25

30

178

-244

Figure6.20:Densityplot of realpartof eigenprojectorcorrespondingto µ æ

1 5 10 15 20 25 30

1

5

10

15

20

25

30

42

-42

Figure6.21:Densityplot of imaginarypartof eigenprojectorcorrespondingto µ æ

100 CHAPTER6

1 5 10 15 20 25 30

1

5

10

15

20

25

30

69

-47

Figure6.22:Densityplot of realpartof eigenprojectorcorrespondingto µ ð

1 5 10 15 20 25 30

1

5

10

15

20

25

30

15

-15

Figure6.23:Densityplot of imaginarypartof eigenprojectorcorrespondingto µ ð

CHAPTER6 101

1 5 10 15 20 25 30

1

5

10

15

20

25

30

96

-137

Figure6.24:Densityplot of realpartof eigenprojectorcorrespondingto µ í

1 5 10 15 20 25 30

1

5

10

15

20

25

30

26

-26

Figure6.25:Densityplot of imaginarypartof eigenprojectorcorrespondingto µ í

102 CHAPTER6

1 5 10 15 20 25 30

1

5

10

15

20

25

30

34

-30

Figure6.26:Densityplot of realpartof eigenprojectorcorrespondingto µ ñ

1 5 10 15 20 25 30

1

5

10

15

20

25

30

13

-13

Figure6.27:Densityplot of imaginarypartof eigenprojectorcorrespondingto µ ñ

CHAPTER6 103

6.2 Subsetof EIES

The subsetof the EIES datasetTab.(5.3) consistsof seven prominentmembersof the original set. For the purposeof showing the advantagesof the enhancedmethodof eigensystemanalysis,the resultsobtainedby Freemanand reportedin [Fre97] will be used. In this paperFreemanwantedto show a hierarchyofmembersbasedon their eigenvectorcentrality. But to calculatethis eigenvectorcentralityhehadto dichotomizetheadjacency matrix. Thishedid,with theeffect,thatamatrixentrybecameü ö ´ Ñý¿ if thememberÐ wrotemoremailsto memberþthenviceversa,and Ë otherwise.Thenhehadto symmetrizedueto theproblemsdescribedin chapter2. Thisnow ledto theresult,thatWellmanwith ID »+Á becamethetop rankedmember.

As canbeseen,whenthemethodpresentedin thisbookis used,FreemanwithID ¿ himself is themostprominentmember. Freemanhimselfstatedin his paper,thatthedichotomizationledto theeffect,thatthemostgarrulousmemberbecamethemostprominent,which in hiscasewasWellmanwith ID »�Á . With themethodpresentedhere,nodichotomizationis necessaryandtherelevanceof themembersis solelybasedon their communicationbehavior, inboundaswell asoutbound.

Theoriginaladjacency matrixisgivenin Freeman[Fre97, p.11].Thecomplexmatrix ú composedasdescribedin chapter4 thenis givenin Eq.(6.1).As canbeseen,thegraphcorrespondingto this matrix is complete.Still therearepatternsvisible.

úØÑÿ���������Ë ¿^¿���� ¾+Ó Ý ¿vÎ��2¿�� Ý Á^Â�­¿v»^Î Ý �+Â��^Î Ý Â^Â�� »�Â Ý ¾+Ó��2¿^¿\¾ ݾ+Ó�2¿^¿�� Ý Ë Ó��­¿\Ë Ý ���r»^» Ý ��� Á Ý Ó Ý ¿���� »�Ó Ý¿���2¿vÎ Ý ¿\Ë��rÓ Ý Ë ¿���­¿vÎ Ý Â��Ó Ý Â��rÂ Ý Ó�� � Ý¿v»^Î��rÁ+Â Ý »^»���� Ý ¿vÎ��2¿�� Ý Ë �^Î���^Î Ý ¿v»��rÁ Ý Â+Ó�rÂ�� Ý�^Î�� �+Â Ý Á� � Ý Ó��rÂ Ý �+Î��^Î Ý Ë ¾��r¾ Ý ¿\Ë��2¿�� Ý»+Â�rÂ^Â Ý Ó Â�rÂ Ý Á�­¿v» Ý ¾� ¾ Ý Ë Â^Â�� Ó�� Ý¿^¿\¾��r¾�Ó Ý »�Ó��2¿�� Ý ��� Ó Ý Â��� Â+Ó Ý ¿����­¿\Ë Ý Ó����rÂ+Â Ý Ë

�����������(6.1)

6.2.1 Eigensystemof the EIES Subset

The eigensystemof the complex Hermitian matrix (ù Ñ ú Ç���È ø Ê ) is given in

Tabs.(6.17)and(6.18).In Fig.(6.28)all eigenvaluesaregivensortedby theiramount.Againfor better

visibility of symmetryFig.( 6.29) shows the positive and negative eigenvaluesrearranged.It is clearlyvisible from Fig.( 6.29)thata starlike structureemerges.

104 CHAPTER6

Fig.( 6.30) basedon Tab.( 6.12)shows, that while the largesteigenvaluecoversalreadymorethan �+ËqÌ of the variance,the two largesteigenvaluescover aboutÁ^ÂqÌ . Only thefirst two subspaceswill bediscussed.Thediversityof patternshasbeendiscussedin section 6.1.

1 2 3 4 5 6 7j

-1

-0.5

0

0.5

1

PSfragreplacements

µ��

Figure6.28:Eigenspectrumof theEIESdatasubsetsortedby amount

1 1.5 2 2.5 3 3.5 4

-1

-0.5

0

0.5

1

PSfragreplacements

µ��� ¼�µ ��

j,f

Figure6.29:Eigenspectrumof theEIESsubsetsortedby absolutevalue

The distribution of the eigenvector componentswithin the first and secondsubspaceis givenin Figs.(6.31)and( 6.33).Togetherwith thephasedistributionin Figs.(6.32and 6.34)thesamebehavior asin thecompletedatasetcanbeseen.Namely, that thereis a stronglyconnectedpatternvisible in the first subspace,whereasthesecondsubspaceis predominantlyastarlikepattern.

CHAPTER6 105Ð Coveredvarianceper µ ö Cumulativecoveredvariance��� "! Æ Ä Ü �$# "! Æ Ä Ü 1 0.64 0.642 0.29 0.933 0.05 0.984 0.02 1.05

ã Ë 1.06

ã Ë 1.07

ã Ë 1.0

Table6.12:Cumulativecoveredvarianceof eigenvaluesfrom Tab.( 6.18)

1 2 3 4 5 6 7 8k

0.2

0.4

0.6

0.8

1

PSfragreplacements

% Ù& ô ö

Figure6.30:Cumulativecoveredvariancegivenin Tab.( 6.12)

The orthogonalprojectorof the first subspaceis againdecomposedinto therealpart,givenin Fig.( 6.35)andimaginarypart,givenin Fig.( 6.36). As canbeseenin Fig.( 6.35)theauthorswith IDs  and � havea low patternsimilarity withtheanchor(authorwith ID ¿ ). On theotherhandit canbeseenin Fig.( 6.36)thattheauthorwith ID ¿ sharesa moreinboundorientedpatternwith theauthorwithID » andamoreoutboundrelationshipto authorswith IDs Ó and Î . In additiontheauthorwith ID » shows a high inboundorientedcommunicationwith theauthorswith IDs Ó and Î .

Theseresultsreflectthe rearrangedmatrix given in Tab.( 6.13). The secondprojectorgivenin Figs.(6.37)and( 6.38)showsagaintheauthorwith ID ¿ astheanchorof astarwith ahighpatternsimilarity with theauthorswith IDs »½¼�Ó and Î .This resultis supportedby therearrangedadjacency matrix in Tab.( 6.14)wherethe inboundcharacterof Freeman(the authorwith ID ¿ ) is visible andalsothestrongconnectionto White (ID » ), Bernard(ID Ó ) andWellman(ID Î ).

106 CHAPTER6

14

72

56

3F

reeman

10

93+

127i84

+118i

115+

84i53

+57i

33+

23i17

+16i

Bernard

4127

+93i

034

+35i

22+

5i57

+57i

12+

9i17

+15i

Wellm

an7

118+

84i35

+34i

024

+15i

15+

10i45

+33i

5+

4iW

hite2

84+

115i5

+22i

15+

24i0

5+

9i4i

4+

10iD

oreian5

57+

53i57

+57i

10+

15i9

+5i

08

+8i

4+

3iM

ullins6

23+

33i9

+12i

33+

45i4

8+

8i0

3+

3iA

lba3

16+

17i15

+17i

4+

5i10

+4i

3+

4i3

+3i

0

Table6.13:S

ubmatrixbasedon

firsteigenvector1

24

75

63

Freem

an1

0115

+84i

93+

127i84

+118i

53+

57i33

+23i

17+

16iW

hite2

84+

115i0

5+

22i15

+24i

5+

9i4i

4+

10iB

ernard4

127+

93i22

+5i

034

+35i

57+

57i12

+9i

17+

15iW

ellman

7118

+84i

24+

15i35

+34i

015

+10i

45+

33i5

+4i

Doreian

557

+53i

9+

5i57

+57i

10+

15i0

8+

8i4

+3i

Mullins

623

+33i

49

+12i

33+

45i8

+8i

03

+3i

Alba

316

+17i

10+

4i15

+17i

4+

5i3

+4i

3+

3i0

Table6.14:S

ubmatricesbasedon

secondeigenvector

CHAPTER6 107

If the partial sumof the first two subspacesù ³>ô Ù

is taken (Tab.( 6.15),mul-tiplied with theeigenvaluesandthenrotatedback,it is clearly visible that com-paredto theoriginalmatrixF in Eq.(6.1) this is already, aspredictedaveryclosematch.Theremainingpartialsum

ù æXô 'asgivenin Tab.( 6.16)is decidedlysmaller

in every entry, andthusservesasa correctionfor thefirst partialsum. Themaincorrectionis takingplacedueto thecommunicationof Bernard(ID Ó ).

108 CHAPTER6

1 2 3 4 5 6 7 8l

0.2

0.4

0.6

0.8

1

PSfragreplacements

± ²�³(´�±

Figure6.31:Distributionof theabsolutevalueof theeigenvectorcomponentsforÛ¸³

1 2 3 4 5 6 7 8l

-3.14

-1.57

0

1.57

3.14

PSfragreplacements

¶¸· ²�³(´ç¹

Figure6.32:Distributionof thephase¶Å· Û¸³(´ · µ ³û¹X¹

CHAPTER6 109

12

34

56

71

0.19

+0.

19i

113

+82

i17

+15

i10

0+

130i

51+

55i

33+

22i

80+

120i

282

+11

3i2.

9+

2.9i

4.8

+9.

3i7

+17

i12

+24

i9

+17

i7

+14

i3

15+

17i

9.3

+4.

8i2.

3+

2.3i

10+

11i

6.5

+6.

8i4.

4+

3.7i

8.6

+9.

5i4

130

+10

0i17

+7i

11+

10i

26+

26i

28+

26i

20+

16i

24+

22i

555

+51

i24

+12

i6.

8+

6.5i

26+

28i

19+

19i

13+

11i

23+

25i

622

+33

i17

+9i

3.7

+4.

4i16

+20

i11

+13

i7.

1+

7.1i

15+

18i

712

0+

80i

14+

7i9.

5+

8.6i

22+

24i

25+

23i

18+

15i

21+

21i

Tabl

e6.

15:P

artia

lsum

(*),+-1

23

45

67

1-0

.19

-0.

19i

2.5

+2.

4i0.

44+

0.85

i-3

.3-

3.4i

2.0

+2.

5i0.

13+

0.62

i1.

24-

0.68

i2

2.4

+2.

5i-2

.9-

2.9i

-0.8

0+

0.70

i-1

.8+

5.4i

-7.-

15.i

-9.-

13.i

7.7

+9.

5i3

0.85

+0.

44i

0.70

-0.

80i

-2.3

-2.

3i5.

4+

6.5i

-3.5

-2.

8i-1

.4-

0.6i

-4.6

-4.

5i4

-3.4

-3.

3i5.

4-

1.8i

6.5

+5.

4i-2

6.-

26.i

29.+

31.i

-7.8

-7.

3i10

.+13

.i5

2.5

+2.

0i-1

5.-

7.i

-2.8

-3.

5i31

.+29

.i-1

9.-

19.i

-5.0

-2.

6i-1

3.-

10.i

60.

62+

0.13

i-1

3.-

9.i

-0.6

-1.

4i-7

.3-

7.8i

-2.6

-5.

0i-7

.1-

7.1i

19.+

27.i

7-0

.68

+1.

24i

9.5

+7.

7i-4

.5-

4.6i

13.+

10.i

-10.

-13

.i27

.+19

.i-2

1.-

21.i

Tabl

e6.

16:P

artia

lsum

(*.,+/

110 CHAPTER6

011.00

-0.67-0.27

-0.160.07

0.04-0.01

21 30.62

0.760.056-0.005i

-0.073-0.021i0.027-0.039i

0.15-0.001i-0.059-0.026i

21 40.33+

0.06i-0.39-0.05i

-0.018-0.057i0.55+

0.07i0.12-0.16i

0.69-0.106-0.059i

2150.098-0.008i

-0.011-0.012i-0.121+

0.003i0.015+

0.004i-0.16+

0.03i0.118-0.037i

0.97

21 60.45-0.07i

-0.38+0.06i

0.65-0.20+

0.03i-0.35+

0.15i-0.18+

0.1i-0.001+

0.015i

2170.29-0.02i

-0.079-0.011i-0.55+

0.02i0.39-0.02i

-0.47+0.25i

-0.32+0.21i

-0.14+0.02i

2180.17+

0.01i-0.013-0.032i

0.22+0.001i

0.550.56

-0.51-0.11i0.15+

0.05i

219

0.40-0.06i-0.33+

0.07i-0.45+

0.03i-0.53+

0.13i0.42-0.11i

-0.16-0.04i-0.002+

0.014i

Table6.17:E

igensystemfor:

with; <=> ?@

01

1.00-0.67

-0.27-0.16

0.070.04

-0.01

A ;A BC ;D A ;A BC ;D A ;A BC ;D A ;A BC ;D A ;A BC ;D A ;A BC ;D A ;A BC ;D

21 3

0.620

0.760

0.056-0.09

0.076-2.86

0.047-0.96

0.15-0.02

0.064-2.73

21 4

0.340.18

0.39-3.00

0.06-1.88

0.440.15

0.20-0.92

0.690

0.12-2.63

215

0.098-0.09

0.016-2.32

0.1213.12

0.0160.29

0.162.93

0.12-0.30

0.970

21 6

0.45-0.15

0.383.00

0.650

0.212.99

0.382.75

0.202.64

0.0151.67

217

0.29-0.07

0.080-3.00

0.563.10

0.39-0.04

0.532.64

0.382.56

0.143.0

3

218

0.170.05

0.034-1.97

0.220.02

0.550

0.560

0.52-2.94

0.160.34

219

0.41-0.15

0.342.93

0.453.07

0.552.91

0.44-0.25

0.16-2.89

0.0141.7

3

Table6.18:E

igensystemof:w

ith; <CA ;A EBC ;DD

CHAPTER6 111

1 2 3 4 5 6 7 8l

0.2

0.4

0.6

0.8

1

PSfragreplacements

± ²ÚÙ>´>±

Figure6.33:Distributionof theabsolutevalueof theeigenvectorcomponentsforÛ�Ù

1 2 3 4 5 6 7 8l

-3.14

-1.57

0

1.57

3.14

PSfragreplacements

¶¸· ²ÚÙ>´Ï¹

Figure6.34:Distributionof phase¶¸· Û�Ù>´ · µ Ù�¹û¹

112 CHAPTER6

1 3 5 7

1

3

5

7

131

3

Figure6.35:Densityplot of realpartof eigenprojectorcorrespondingto µ ³

1 3 5 7

1

3

5

7

17

-17

Figure6.36:Densityplot of imaginarypartof eigenprojectorcorrespondingto µ ³

CHAPTER6 113

1 3 5 7

1

3

5

7

67

-131

Figure6.37:Densityplot of realpartof eigenprojectorcorrespondingto µ Ù

1 3 5 7

1

3

5

7

12

-12

Figure6.38:Densityplot of imaginarypartof eigenprojectorcorrespondingto µ Ù

114 CHAPTER6

6.2.2 Comparisonof subsetresultswith dichotomizedandsym-metrizied eigensystem

Tab.( 6.19) shows, that the anchorsdiffer dependingon the methodused. Therepresentationis againasin Tab.( 6.2). While themethodusedby Freemanstillshows thesameanchorin thefirst subspace,healreadyshowsWellman(ID Î ) astheanchorof thesecondsubspace.Freemanin his paperexplainsthatthis mightbedueto thedichotomization.Bernard(ID Ó ) whoseemsto bealessactivewriterbut avery importantmemberof thecommunityis clearlyvisiblewhenthemethodproposedin this bookis used.Not so,whenthemethodusedby Freemanis used.

H( µ ³ ) 1 4 7 2 5 6 3F(µ ³ ) 1 3 6 4 5 2 7

H( µ Ù ) 1 2 4 7 5 6 3F(µ Ù ) 7 2 4 5 3 6 1

H( µ æ ) 4 5 7 6 3 2 1F(µ æ ) 1 5 2 6 7 3 4

H( µ í ) 6 7 2 5 4 1 3F(µ í ) 6 5 7 2 4 3 1

H( µ ð ) 6 5 7 4 2 3 1F(µ ð ) 6 7 2 3 5 4 1

H( µ ñ ) 2 6 5 4 7 1 3F(µ ñ ) 4 3 5 6 2 7 1

H( µ ' ) 3 6 5 2 1 4 7F(µ ' ) 4 3 1 7 2 6 5

Table6.19:Comparisonof anchorsfor theEIESsubset

As couldbeshown, themethodis capableto presenta view on thecommuni-cationpatternsbasedon inboundandoutboundtraffic within a group.Comparedto the standardmethodsit revealsa clearerpicture. In the examplesusedsomeinsightscould be gainedinto the structureof the respective groups.For the fullEIESdatasetthefollowing resultscanbestated:F Sincetheentriesof theadjacency matrix alsoincludemessagessentto all

othermembers,thusdiluting thepersonto personcommunication,it canbestatedthat the first two subspacescontaina lot of datanoisedueto thesemessages“to no onein particular”.F Freeman(ID ¿ ) is theanchorof thefirst two subspaces.He is thecenterofaninboundorientedstarandalsoanimportantmemberof a largeconnected

CHAPTER6 115

subgroup.Thefact thatFreemanwasthe“administrator”of thecommuni-cationsystemmakesthatplausible.F Wellman(ID »+Á ) alsois a very importantmemberof thegroup.For onehecommunicatesvery extensively (outboundoriented)andalsoservesasananchorfor subspacethree.He alsoseemsto betheconnectionbetweenthegroupsaroundFreemanandBernard.F Whenthenext subspaces,especiallysubspaces and � areinspected,theyreveal that Bernard(ID ¾ ) is a very importantmemberof the community.He is also the centerof a star and an importantmemberof a connectedsubgroup.

For thesubsetthefollowing resultscanbestated:F Thefirst two subspacesalreadycovermorethan Á^ËqÌ of thevarianceof thedata.F Freeman(ID ¿ ) is againtheanchorof a inboundorientedstarandalsotheimportantmemberof thealmostcompletegraph.

116 CHAPTER6

Chapter 7

Conclusionand Futur eApplications

SNA providesa rich toolbox to analyzeandinterpretnetworks betweenhumanactors.The focusof this book is to presenta generalizedmethodfor oneof thetool setsavailablein SNA, namelytheeigenvectorcentrality. Thestandardmethodneedsproperdatapreparationbeforethe resultscan be interpretedin a usefulway. This is dueto thefactthatneitherimaginaryeigenvalueshave founda goodinterpretationin social sciencesnor have imaginaryeigenvectors. Both canbethe resultof an eigensystemcalculatedfor an asymmetricadjacency matrix. Tocircumvent theseproblemsdatahadto bemadesymmetric,which meantlossofinformation.

Theuseof theproposedgeneralizedapproachbasedon theeigensystemanal-ysis of complex Hermitianadjacency matricesobviously solvesthe problemofhaving to adjustasymmetricdatasoasto befit to beanalyzedwith standardrealvaluedeigensystemanalysis. The suggestedsolution is basedon the fact, thatcomplex numberscanstoremoreinformationthanreal numbersandthusno exanteassumptionsor filters have to beappliedto thedatathatmight distort there-sultsobtained.In Hilbert spacethenorm,andthusthedistances,is definedby theinnerproduct.Thereis no needto definesimilaritiesor any otherconstructusedin thestandardmethodswhendealingwith asymmetricdata. This is usedwhenrepresentinginboundandoutboundtraffic by therealpartandtheimaginarypartof thecomplex matrix entry. Theeigensystemsof suchmatricesallow for moredetailedinterpretationof theunderlyingcommunicationpatternsaswasupto nowpossible.Sinceall eigenvaluesarereal,anddependon the traffic volume,it caneasilybeassumed,that they canbe interpretedasdistinct “energy levels” withinthegroup.Sinceeigenvectorsof Hermitianmatricesdefineanorthogonalsystemit canfurtherbeassumed,thattheeigenvectorscanbeviewedasindependentwithrespectto their inherentcommunicationpatternstructure.Theeigenvectorcom-ponentsrepresentthemembersof thegroup. Thestatusor rankof eachmemberis givenby theabsolutevalueof theeigenvectorcomponentat eachenergy level.

117

118 CHAPTER7

The phasecontainsinformation aboutthe directionalsimilarity of the memberwith respectof thepatterngovernedby theanchor. Thuseventheanchorof thecompletegroupcanbe identified. He/Sheis representedasthe highestabsolutevalueeigenvectorcomponentbelongingto thehighestabsolutevalueeigenvalue.In addition the possibility to constructcompletesubspacesgives the chancetoanalyzein moredetailthesubstructuresof thegroup.

7.1 Futur eApplications

7.1.1 Mark ets

As a motivatingexamplefor further applicationstake a look at forecastingmar-kets, especiallyforecastingmarkets like Internetbasedpolitical stock markets.Eachtradecanbeviewedasa communicationbetweena traderanda share.Alltransactionsareloggedin theorderbookandarethusavailable.As caneasilybeseenany suchmarket canbe viewed asa bipartitegraph,andthusthespectrumwill bequalitatively liketheonegivenin Eq.(3.63).Theefficiency of marketsde-pendstronglyon theinformationflow within themarket. And this againdependson themarketstructure.

Preliminaryexperimentalresults,obtainedon a political stockmarket hostedat the Chair for InformationServicesandElectronicMarketsat the UniversitatKarlsruhe(TH) (seeTab.( 7.1 for resultsof the election),RMSE (Root Mean

SquareError)= G �IH ÜJ ), suggestthattheexpectedbehavior canbemadevisible.A stockmarket canbeseenasa bipartitegraphasin Fig.( 7.1). Thegraph K ÑLNM ¼PORQ>¼POTS*U consistsof vertices ORQ representingthe tradersin the market and O�Srepresentingtheshares.

In % Result Result Result OfficialPSM (normalized) Sudwest-Presse result

CDU 39,10 39,18 39,27 44,80SPD 33,00 33,07 31,68 33,30FDP 8,90 8,92 9,51 8,10B90/Grune 9,00 9,02 8,99 7,70REP 7,20 7,21 7,79 4,40Sonstige 2,60 2,61 2,76 1,70Summe 99,80 100,00 100,00 100,00

RMSE 4,356 4,280 4,947

Table7.1: Forecastedresultsvs. actualresultsof 2001diet electionsin Baden-Wurttemberg

CHAPTER7 119

TraderShareMarkets

Figure7.1: Stockmarket asbipartite-graph

If for examplethe market is dominatedby a monopolist,this could be rep-resentedas a star graphlike structure,which -as we have seen- can easily bedetectedwith the method. If on the otherhandthe market is an oligopoly, therepresentationasagraphwouldbeaconnectedif notevenacompletegraph.If itis now takeninto account,thattransactions,likecommunication,arerecordedasatwo-way flow, it is feasibleto usethemethodof Hermitianadjacency matricestoanalyzemarkettransactionsandthusgaininsightinto marketstructureandmarketefficiency.

It wasfound that if a shareis the anchorof a star, thenthe traderswith thenext highestabsoluteeigenvectorcomponentsarethosethat tradeheavily in theshare.On theotherhand,if a traderis theanchorof a star, thesharethathe/shetradesmostoften in, is identifiableby it´s absoluteeigenvectorcomponent.Theeigenvaluesagainshow thevolume,if oneusestheexchangedamountsof moneyasthe weightson the edges

M. Thusan edge Ç�VÑ Ë if a traderboughtor sold

shares.As our market offeredalsoa primary market, wheretraderscould buyportfolios, we alsohadthe opportunityto checkfor arbitragetraders.This waspossibleby checkingfor the eigenvectorwith the portfolio asanchoror for theindex of theportfolio in any eigenvector. Thehighestrankedtraderin thateigen-vectorcouldbe identifiedasanarbitragetrader. It is alsopossibleto show, howefficient information is distributedwithin this market. If the absolutevaluesofthe eigenvectorcomponentsshow an uniform distribution , then as wasshownin chapter 4 this pointstowardsa completegraph. This leadsto the interpreta-tion of equallydistributedinformationwithin thatmarket. Informationefficiencyis a prerequisitefor market efficiency. This shouldbe further investigated.Theeigenspectralanalysisover time couldalsoyield someinsight into stockmarket

120 CHAPTER7

behavior.A market,wherenetworking playsanimportantrole hasalwaysbeenthejob

market. As Arrow and Borzekowski point out in their recentdiscussionpaper( [AB04]) the network effect is morerelevantwithin a groupof similar situatedworkers.

7.1.2 Organization Analysis

Anotherapplicationareais thatof theanalysisof organizationalbehavior. Krack-hardtandHanson[KH93] show in theirarticlehow theknowledgeaboutthesocialnetworkswithin a company canhelpmanagementto useresourcesbetter. Ahujaet al. [AGC03] show, that the centralityof a groupmember, measuredin emailexchangeis a very goodpredictorof performanceof thesaidmember. They alsohadthe alreadymentionedproblemof asymmetricdata. If now the analysisofasymmetricdatacanbe achieved, and the statusof a membercalculatedbasedon asymmetriccommunicationbehavior, thenthe predictionaboutperformancecouldimprove.

7.1.3 Viral Mark eting

Viral Marketing,amodernword for “wordof mouth” whichwith theInternetbe-camea word-of-mouse(seeHelm [Hel00]), hasgainedwide interestin the pastyearsasa meansto usesocialnetworksto market new products.SubramaniandRajagopalan[SR03b] for examplesuggesta framework basedon socialnetworkconsiderationsto enhancetheoutcomeof viral marketingefforts. As of now viralmarketingcampaignsaremoreor lessanecdotalin planningandoutcome,sincethe underlyingmechanicsarenot yet well investigated.While the generalchar-acteristicsof epidemiologyarewell known, themechanicsthathelpto “spread”amarketing“virus” arestill underinvestigation.But if onecouldusethecommuni-cationwithin aknown scale-freenetwork andanalyzeit with theappropriatetools(e.g. theoneproposedin this work), onemight find a way to setup viral market-ing campaignsmoreeffectively. DomingosandRichardson[DR01] havealreadyintroduceda way to calculatethe network valueof customers.Network valueis in this paperdefinedas the expectedprofit from salesto other customers shemay influenceto buy, the customers thosemay influenceand so on recursively.This makesnetwork analysisapplicablein CustomerRelationshipManagement(CRM). But to find out who getsinfluencedto buy by whom is a very difficulttask, and Domingosand Richardsonmodel this network as a Markov randomfield asagoodfirst approximation.

CHAPTER7 121

7.2 Outlook

Someinterestingquestionsfor furtherinvestigationarefor exampletheevolutionof suchnetworks,andhow they canbeanalyzedor evenpredictedwith thehelpofthis method.Thepossibilityto applywell known methodsof perturbationtheoryfrom physicsto the Hermitianmatricesusedhere,might bring new insight intothedynamicsof grouppatternsandstructures.

Another interestingquestionwould be the comparisonwith a context basedanalysisof email traffic. It could well be, that the structurewe find throughthemereexistenceof communicationmight give a clueasto the topicsdiscussedinthat substructure.This is alreadybeinginvestigatedin the Internetlink researchcommunity.

As could be shown the proposedmethodnot only yields more insight intoasymmetricand thus more generalizedcommunicationstructures,but also hassomepotentialfor furtherapplicationsin severalfields. With theproblemof in-formationoverflow andtheresultingneedto structurethis informationit is safetopredictthatthesemethodswill gainevermorerelevance.

122 CHAPTER7

Appendix

123

124 CHAPTER7 WX1

23

45

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0.01

8-

0.03

6i0.

27-

0.02

i-0

.009

-0.

048i

-0.1

9+

0.07

i-0

.078

+0.

003i

0.03

2+0.

011i

h f kj 0.1

4+

0.03

i0.

26-

0.01

i0.

034

-0.

013i

-0.1

3-

0.14

i-0

.11

+0.

15i

-0.0

17-

0.07

1i-0

.001

13+

0.00

034i

0.11

2+0.

071i

Tabl

e7.

3:F

ullE

IES

data

sete

igen

syst

em:sut to

s )v

126 CHAPTER7 WX17

1819

2021

2223

24-0.016

0.014-0.013

-0.0110.010

0.0086-0.0071

0.0061

YX Z0.023-

0.031i-0.018-

0.013i-0.021

-0.009i-0.024+

0.022i-0.021

-0.012i

-0.031-0.021i

0.0050-0.0069i

-0.014+0.035i

YX [0.093-

0.024i0.0086+

0.0013i-0.023-

0.034i-0.122-0.027i

-0.066-

0.018i0.021

-0.050i

-0.067+0.019i

0.041-

0.020i

YX\-0.073+

0.001i0.21

+0.05i

-0.28-

0.11i0.34

-0.26+

0.04i0.052

+0.032i

-0.29+

0.08i-0.028

-0.091i

YX ]0.09

+0.22i

0.048-

0.059i-0.22

+0.08i

-0.047+0.026i

-0.042-

0.029i-0.16

+0.08i

-0.010-0.030i

-0.002+0.051i

YX^-0.088+

0.030i0.11

-0.19i

-0.011+0.028i

-0.15-

0.08i-0.16

+0.02i

0.050-

0.015i0.14

-0.31i

0.50YX_

0.43-0.133+

0.046i-0.06

-0.21i

-0.19-

0.13i-0.022

-0.072i

0.0135-0.0031i

-0.06+

0.16i-0.07

+0.18i

YX`-0.049+

0.055i0.0001+

0.032i-0.053

-0.010i

0.04+

0.29i-0.06

-0.20i

0.014-

0.068i0.04

+0.15i

0.35+

0.23i

YXa-0.018-

0.017i-0.0005-0.0035i0.008+

0.015i0.053

-0.013i0.0098+

0.0034i0.0047+0.0073i

0.018+0.005i

0.00006-0.0026i

YXb0.076

+0.026i

-0.13+

0.09i0.027

-0.035i

-0.003-0.111i

0.050-

0.017i0.068

+0.065i

-0.004-0.022i

-0.023+0.015i

YX Zc-0.21

+0.32i

-0.127+0.001i

-0.16-

0.02i0.010+

0.054i-0.036

-0.034i

0.15-

0.15i0.060

-0.023i

-0.066-

0.017i

YX ZZ

-0.16-

0.01i-0.09

+0.21i

0.085+0.062i

0.15-

0.10i0.037+

0.036i0.043

+0.064i

0.0072-0.0001i

-0.034-

0.017i

YX Z[

0.047+

0.067i0.079

+0.050i

-0.098-

0.004i-0.16

+0.04i

0.42-0.069+

0.116i-0.14

+0.13i

0.27-

0.18i

YX Z\

0.050-0.128i

0.39-0.010+

0.044i0.31

-0.05i

-0.18-

0.06i0.13

+0.08i

-0.31+

0.07i0.036+

0.061i

YX Z]

0.071+

0.116i0.26

+0.07i

0.27-

0.19i0.17

+0.20i

0.08+

0.34i-0.01

+0.25i

0.39-0.30

-0.03i

YX Z^

-0.24+

0.03i-0.004-

0.062i0.03

+0.18i

0.13-

0.08i0.28

-0.21i

-0.33+

0.05i-0.015-

0.059i0.082+

0.082i

YX Z_

-0.05-

0.22i-0.19

+0.21i

-0.32+

0.12i-0.037-

0.111i-0.13

+0.29i

0.11+

0.24i0.15

-0.13i

-0.0052+0.0128i

YX Z`

-0.15+

0.05i0.006-

0.17i0.39

-0.19+

0.21i-0.19

+0.15i

0.23+

0.05i-0.18

-0.17i

-0.06+

0.17i

YX Za

0.107-0.021i

0.27-

0.13i-0.12

-0.17i

-0.17+

0.16i-0.012+

0.018i-0.21

-0.06i

0.18-

0.15i-0.02

-0.17i

YX Zb

0.17-

0.33i0.19

-0.11i

0.30+

0.03i0.095

-0.048i

0.16+

0.05i-0.30

+0.12i

-0.16+

0.09i0.070

-0.008i

YX [c

0.16-

0.06i-0.039-

0.061i0.002+

0.089i0.15

+0.004i

-0.040-

0.090i0.36

-0.022+0.020i

-0.033-

0.102i

YX [Z

0.0001+0.048i

-0.10+

0.11i0.080

-0.056i

-0.029+0.064i-0.034+

0.112i0.15

+0.22i

-0.18+

0.19i0.21

+0.06i

YX [[

0.16-

0.14i-0.010-

0.027i0.19

+0.03i

-0.069-0.094i

-0.15-

0.10i0.007

-0.084i

-0.067-0.055i

0.017-

0.009i

YX [\

0.12-

0.26i-0.013-

0.058i0.17

+0.05i

-0.050+0.085i

0.005-

0.050i0.18

-0.28i

-0.028-0.075i

0.005-

0.047i

YX []

-0.035+0.034i

-0.009-0.018i

-0.008+0.020i

0.044-

0.012i-0.097

-0.043i

0.007-

0.020i-0.0031+

0.0128i-0.033

-0.012i

YX [^

-0.083+0.114i

0.12-

0.16i-0.17

+0.21i

-0.15+

0.22i-0.018+

0.050i-0.044+

0.056i-0.16

+0.29i

-0.16+

0.22i

YX [_

-0.041+0.017i

-0.139-0.016i

0.059+0.063i

0.067-

0.007i-0.04

-0.22i

-0.05-

0.21i0.040

-0.025i

-0.25-

0.03i

YX [`

0.014+

0.077i0.18

-0.25i

-0.054+0.029i

-0.05+

0.24i-0.14

+0.08i

0.049-

0.035i0.072

-0.076i

-0.032+0.058i

YX [a

-0.15-

0.04i0.006

+0.033i

0.063+0.092i

0.22+

0.13i-0.07

-0.15i

-0.011-0.123i

0.15+

0.03i0.026+

0.118i

YX [b

0.010+

0.018i-0.005+

0.018i-0.014

-0.009i

0.039+0.019i

0.054+0.009i

0.0003+0.0139i

0.024-

0.009i0.002

-0.027i

YX\c

0.027+

0.093i0.24

-0.18i

-0.038-

0.052i-0.048-0.073i

-0.039-

0.040i-0.021-

0.110i-0.001+

0.071i0.066

-0.003i

YX\ Z

-0.042-0.008i

0.063+

0.041i0.112+

0.003i-0.003-

0.097i0.094

-0.022i

-0.032+0.033i

-0.071+0.042i

0.017-

0.092i

YX\ [

0.011-0.024i

-0.073+0.075i

0.0001+0.017i0.027+

0.025i0.050

-0.024i

0.022+

0.022i0.022

-0.024i

0.000132-0.000028i

Table7.4:F

ullEIE

Sdataseteigensystem

:039to046

CHAPTER7 127

egf

2526

2728

2930

3132

-0.0

052

0.00

45-0

.003

50.

0025

0.00

17-0

.000

900.

0002

50.

0000

74

h f i -0.012

1+0.

0017

i0.

012+

0.02

0i0.

0021

+0.

0053

i0.

021

-0.

006i

-0.0

043+

0.00

64i-0

.018

+0.

005i

-0.0

16+

0.01

4i0.

011

+0.

014i

h f j -0.0

16+

0.04

8i0.

002+

0.03

4i-0

.048

+0.

045i

0.01

16-

0.00

32i

-0.0

10-

0.02

2i0.

001

+0.

015i

0.07

0-0.

036i

0.00

27-

0.00

93i

h f k 0.06

8-0.

070i

0.50

-0.0

74+

0.10

4i-0

.117

-0.

068i

0.12

+0.

19i

-0.1

8+

0.06

i-0

.17

-0.

15i

-0.3

2+

0.05

i

h f l -0.0

58-

0.00

5i-0

.13

+0.

13i

-0.0

13+

0.03

5i-0

.031

-0.

016i

-0.0

01-

0.04

7i-0

.110

+0.

014i

-0.0

41+

0.00

8i-0

.050

+0.

064i

h f m

0.48

-0.0

65-

0.08

9i0.

11-

0.27

i-0

.098

+0.

059i

-0.0

4+

0.15

i-0

.084

+0.

073i

-0.2

4+

0.05

i-0

.025

+0.

008i

h f n 0.1

4-

0.10

i-0

.02

+0.

19i

0.13

-0.

10i

-0.1

6+

0.28

i0.

23+

0.16

i0.

18-

0.12

i-0

.27

-0.

07i

-0.0

09-

0.06

9i

h f o -0.

19-

0.31

i-0

.019

-0.

019i

-0.0

4+

0.32

i-0

.081

-0.

101i

-0.2

0-

0.03

i0.

41-0

.20

+0.

04i

0.13

+0.

08i

h f p -0.00

07-

0.02

2i-0

.010

1+0.

0010

i-0.0

053+

0.00

67i0

.009

1-0.

0007

i-0

.009

+0.

015i

0.01

6+

0.00

8i-0

.002

5-0.

0051

i-0.

0048

-0.

0039

i

h f q 0.06

8+

0.04

4i0.

089

-0.

049i

0.07

1-

0.08

3i-0

.048

+0.

005i

-0.0

13+

0.02

9i-0

.027

-0.

051i

-0.0

72-

0.01

1i0.

050

+0.

025i

h f ir 0.0

08-

0.03

0i0.

042

-0.

064i

0.01

7-

0.00

2i0.

028+

0.01

5i-0

.010

+0.

031i

0.04

1-0.

057i

0.04

7+

0.00

3i0.

060

-0.

053i

h f ii 0.0

04-

0.06

1i-0

.039

-0.

071i

0.01

5-

0.08

7i-0

.018

-0.

025i

0.01

3+0.

027i

0.07

8-0.

067i

-0.0

27+

0.03

4i-0

.033

-0.

022i

h f ij 0.1

3+

0.18

i0.

13-

0.16

i0.

120

-0.

023i

-0.0

9-

0.17

i0.

06-

0.20

i0.

29-

0.17

i0.

19+

0.08

i-0

.34

-0.

07i

h f ik 0.0

71+

0.00

1i-0

.27

-0.

08i

-0.0

7-

0.13

i0.

05+

0.16

i-0

.17

-0.

09i

0.18

+0.

10i

0.06

+0.

13i

0.04

-0.

14i

h f il

0.15

-0.

20i

0.08

1-

0.04

3i0.

0036

+0.

0062

i0.

057+

0.08

3i-0

.022

+0.

068i

0.09

8-0.

041i

-0.0

82+

0.08

3i0.

062

-0.

113i

h f im -0.0

98-

0.05

1i0.

13+

0.07

i-0

.091

-0.

102i

-0.0

36+

0.10

8i0.

020+

0.05

8i-0

.29

+0.

06i

-0.1

15+

0.07

6i0.

34-

0.01

i

h f in 0.0

01+

0.14

1i-0

.24

+0.

01i

-0.2

4+

0.05

i-0

.036

+0.

071i

-0.1

8-

0.03

i0.

003

+0.

053i

0.15

-0.

08i

-0.0

65-

0.04

3i

h f io -0.

25+

0.16

i0.

012

-0.

049i

0.00

4-

0.10

5i-0

.24

-0.

01i

0.11

-0.

19i

-0.0

7-

0.19

i-0

.13

+0.

13i

-0.1

8+

0.11

i

h f ip 0.0

08+

0.05

5i0.

017

-0.

112i

0.06

+0.

21i

0.08

-0.

15i

0.09

7-

0.08

1i-0

.12

+0.

19i

0.14

-0.

04i

0.00

9+

0.02

9i

h f iq 0.0

20-

0.00

3i-0

.09

+0.

12i

0.01

3+0.

033i

0.00

4-

0.05

8i-0

.045

+0.

022i

-0.0

44+

0.03

1i-0

.083

+0.

028i

-0.0

67+

0.04

5i

h f jr 0.3

4+

0.18

i-0

.014

+0.

073i

-0.0

8+

0.23

i0.

18-

0.36

i0.

31-

0.17

i-0

.04

-0.

16i

-0.1

3+

0.14

i0.

40

h f ji -0.

08-

0.29

i-0

.23

-0.

12i

0.47

0.35

-0.

14i

0.04

6+0.

048i

-0.3

7+

0.03

i0.

08-

0.23

i-0

.009

+0.

018i

h f jj -0.

18+

0.10

i0.

006+

0.04

3i-0

.11

+0.

18i

-0.0

94-

0.02

2i0.

027

-0.

083i

-0.1

5+

0.05

i0.

045-

0.07

6i-0

.05

+0.

14i

h f jk -0.1

04+

0.04

0i0.

29-

0.36

i0.

001

-0.

19i

0.14

+0.

07i

-0.0

7+

0.30

i0.

19+

0.07

i0.

22-

0.25

i0.

18-

0.04

i

h f jl -0.0

08-

0.03

5i-0

.042

+0.

021i

-0.0

17+

0.01

5i-0

.034

-0.

004i

0.01

3-

0.00

7i0.

012-

0.01

0i-0

.015

-0.

015i

-0.0

52+

0.03

7i

h f jm 0.0

38+

0.11

0i0.

121+

0.05

3i0.

001

-0.

33i

0.06

3-

0.03

7i0.

0049

+0.

0018

i-0

.06

-0.

15i

0.11

7+

0.00

5i0.

15-

0.30

i

h f jn 0.0

8+

0.14

i0.

005+

0.04

7i0.

21-

0.06

i0.

48-0

.25

-0.

03i

0.05

7+

0.12

4i-0

.25

+0.

30i

-0.3

9+

0.05

i

h f jo -0.0

60+

0.00

5i-0

.067

+0.

066i

0.04

7+0.

036i

0.03

5+0.

007i

-0.0

031+

0.00

68i0

.008

+0.

013i

-0.0

005+

0.00

62i-0

.003

7+0.

0008

i

h f jp 0.0

94+

0.01

3i-0

.12

+0.

22i

0.20

-0.

01i

-0.0

7+

0.25

i0.

550.

13+

0.22

i0.

41-0

.13

+0.

09i

h f jq -0.0

10-

0.01

7i-0

.005

-0.

021i

-0.0

03+

0.01

9i0.

014

-0.

005i

-0.0

056-

0.00

23i0

.011

+0.

012i

0.01

11-

0.00

29i

-0.0

11-

0.01

1i

h f kr 0.0

21+

0.01

2i-0

.066

+0.

069i

-0.0

56+

0.03

5i0.

095+

0.03

1i-0

.069

+0.

021i

-0.0

54+

0.05

0i0.

100-

0.02

7i0.

084

-0.

030i

h f ki 0.010

2+0.

0065

i0.

019

-0.

022i

-0.0

35-

0.02

1i-0

.035

+0.

010i

0.01

6-

0.04

1i0.

005-

0.01

9i0.

039-

0.00

4i0.

017

-0.

011i

h f kj 0.008

2-0.

0054

i0.

076

-0.

044i

0.02

3+0.

006i

-0.0

17-

0.00

02i

0.02

6+0.

009i

-0.0

06+

0.03

4i-0

.015

-0.

0002

i0.

029

-0.

001i

Tabl

e7.

5:F

ullE

IES

data

sete

igen

syst

em:s -w to

s .-

128 CHAPTER7

WX

11

0.30.3

0.240.24

0.170.17

YX Z

0.450.31

-0.1-0.024

0.340.03

-0.0360.35

YX [

0.015-0.1

-0.2-0.2

-0.0110.015

-0.22-0.028

YX\

-0.14-0.39

0.033-0.14

0.39-0.16

0.0340.091

YX ]

-0.0190.36

-0.034-0.25

0.05-0.23

0.06-0.21

YX^

-0.330.18

-0.018-0.087

0.26-0.11

-0.086-0.15

YX_

0.020.12

-0.0960.25

-0.26-0.2

-0.0310.046

YX`

0.29-0.24

0.140.0061

0.26-0.48

-0.11-0.0021

YXa

-0.057-0.032

-0.099-0.0075

-0.140.14

0.0920.47

YXb

0.14-0.22

-0.12-0.34

-0.0480.043

-0.051-0.15

YX Zc

-0.0360.076

0.190.53

0.065-0.2

-0.00097-0.033

YX ZZ

-0.0061-0.1

0.0180.05

-0.36-0.28

-0.0920.28

YX Z[

0.150.19

0.19-0.2

-0.170.044

-0.00430.00085

YX Z\0.19

0.220.02

0.016-0.016

0.034-0.023

0.056

YX Z]-0.0064-0.072

0.16-0.3

-0.15-0.013

-0.0370.15

YX Z^0.17

0.0190.0078

-0.18-0.39

-0.3-0.075

-0.016

YX Z_0.23

-0.22-0.22

0.250.15

0.10.066

-0.049

YX Z`0.004

-0.059-0.2

0.11-0.03

0.0550.036

-0.29YX Za

0.2-0.081

-0.120.026

-0.160.43

0.064-0.1

YX Zb-0.13

-0.22-0.37

0.0370.026

-0.14-0.023

-0.028

YX [c0.031

0.058-0.25

-0.14-0.14

-0.0380.26

0.023

YX [Z0.0046

0.19-0.33

-0.0730.11

-0.27-0.019

0.27

YX [[-0.042

-0.12-0.03

0.29-0.14

0.057-0.12

-0.02

YX [\0.052

-0.120.1

-0.210.095

0.17-0.063

0.14

YX []0.16

-0.230.48

-0.043-0.024

0.0210.022

0.076

YX [^

-0.072-0.059

0.18-0.16

-0.05-0.022

-0.22-0.19

YX [_

-0.250.063

0.037-0.065

0.080.062

0.140.24

YX [`

0.0220.0073

-0.0970.067

0.095-0.06

0.049-0.16

YX [a

0.29-0.075

-0.067-0.08

-0.073-0.18

-0.048-0.28

YX [b

-0.31-0.16

-0.12-0.024

-0.2-0.036

-0.190.2

YX\c

0.0220.25

0.0810.048

-0.028-0.021-0.35

-0.092

YX\ Z

0.13-0.016

-0.120.084

0.140.24

-0.680.069

YX\ [

-0.0980.055

-0.0044-0.12

0.088-0.099

-0.22-0.14

Table7.6:F

ullEIE

Sdataseteigensystemaccordingto

Freem

an03

to0d

CHAPTER7 129

exf 0.15

0.15

0.08

80.

088

0.06

70.

067

0.04

30.

043

h f i -0.012

0.1

0.23

-0.0

76-0

.065

-0.0

180.

056

-0.0

12

h f j -0.25

-0.2

10.

074

-0.2

6-0

.18

-0.2

1-0

.054

-0.2

6

h f k -0.15

0.27

-0.1

50.

038

0.06

50.

055

-0.0

15-0

.024

h f l 0.18

-0.1

7-0

.13

-0.2

10.

078

-0.0

27-0

.052

0.01

1

h f m 0.048

-0.0

12-0

.14

0.07

80.

220.

092

-0.0

6-0

.095

h f n 0.027

0.27

-0.5

3-0

.006

9-0

.1-0

.12

-0.3

1-0

.064

h f o 0.054

-0.0

015

0.17

0.12

0.19

0.11

-0.0

65-0

.042

h f p 0.083

-0.0

071

0.14

-0.1

50.

0004

1-0

.047

-0.2

90.

013

h f q -0.18

-0.4

7-0

.16

0.01

-0.0

075

0.00

27-0

.069

0.16

h f ir -0.14

-0.5

20.

14-0

.006

30.

110.

13-0

.12

-0.1

2

h f ii -0.15

-0.0

470.

011

0.21

0.06

6-0

.11

0.27

0.12

h f ij 0.19

-0.0

120.

056

-0.2

20.

20.

160.

250.

24

h f ik -0.2

-0.1

1-0

.23

0.00

76-0

.065

0.14

0.16

-0.4

4

h f il -0.15

-0.0

087

-0.1

90.

044

-0.0

40.

410.

07-0

.17

h f im -0.1

0.14

0.19

-0.0

045

0.25

0.16

-0.2

3-0

.26

h f in 0.31

-0.2

1-0

.38

0.13

0.14

-0.1

60.

380.

089

h f io 0.22

0.16

0.31

-0.0

760.

028

-0.0

049

0.07

1-0

.5

h f ip -0.017

0.08

30.

025

0.34

0.00

660.

39-0

.095

-0.0

79

h f iq 0.22

-0.0

73-0

.018

-0.1

9-0

.17

0.31

-0.1

40.

0095

h f jr -0.041

-0.1

50.

210.

41-0

.036

-0.3

-0.1

40.

095

h f ji 0.048

-0.0

53-0

.11

0.06

-0.3

90.

10.

21-0

.06

h f jj -0.24

0.03

40.

026

-0.3

6-0

.053

0.11

0.17

0.15

h f jk 0.29

-0.1

-0.1

7-0

.017

0.24

-0.2

1-0

.04

-0.2

4

h f jl 0.45

-0.1

4-0

.10.

0007

7-0

.43

-0.1

2-0

.076

-0.1

7

h f jm -0.027

-0.0

130.

150.

081

-0.2

8-0

.15

0.06

7-0

.11

h f jn 0.15

-0.1

9-0

.013

0.01

60.

086

0.34

-0.1

90.

093

h f jo 0.089

-0.0

270.

090.

13-0

.19

0.34

0.2

0.07

7

h f jp 0.099

0.18

-0.0

031

-0.2

4-0

.083

0.12

-0.1

80.

3

h f jq 0.3

-0.0

270.

13-0

.17

0.11

0.03

0.3

-0.0

92

h f kr 0.11

0.01

80.

0057

0.42

-0.0

750.

015

-0.2

0.11

h f ki 0.015

-0.0

57-0

.044

-0.0

170.

037

0.03

4-0

.19

0.11

h f kj -0.12

0.13

0.07

20.

22-0

.046

-0.0

450.

210.

079

Tabl

e7.

7:F

ullE

IES

data

sete

igen

syst

emac

cord

ingt

oF

reem

an

sut tos )v

130 CHAPTER7

WX

0.030.03

0.0210.021

0.0120.012

0.0120.012

YX Z

0.0510.046

-0.066-0.049

-0.061-0.057

-0.0260.34

YX [

-0.28-0.16

-0.17-0.22

-0.22-0.24

-0.25-0.065

YX\

0.190.13

-0.24-0.039

0.017-0.074

-0.014-0.1

YX ]

-0.0690.068

-0.18-0.0065

0.023-0.016

-0.034-0.21

YX^

-0.0078-0.29

0.10.16

-0.0750.086

0.120.53

YX_

-0.053-0.068

-0.021-0.058

-0.11-0.092

-0.0940.13

YX`

-0.19-0.3

0.0210.041

0.150.036

0.015-0.15

YXa

-0.0790.2

0.14-0.13

-0.0160.022

-0.0230.35

YXb

0.310.12

-0.0160.14

-0.0850.14

0.120.31

YX Zc

-0.0830.23

-0.0660.0066

0.13-0.12

-0.0710.094

YX ZZ

0.24-0.32

-0.250.044

-0.035-0.046

-0.0680.13

YX Z[

0.0019-0.12

-0.093-0.16

0.0240.16

0.05-0.003

YX Z\

0.36-0.0044-0.026

0.0910.24

-0.16-0.015-0.078

YX Z]-0.55

0.047-0.18

0.0140.055

0.19-0.037

0.15

YX Z^0.22

0.20.32

-0.028-0.3

0.0650.077

-0.15

YX Z_-0.19

-0.0091-0.041-0.36

0.009-0.026

-0.0320.12

YX Z`0.15

-0.15-0.32

-0.017-0.045

0.28-0.11

0.18

YX Za-0.036

-0.064-0.049-0.0027

0.18-0.17

-0.0170.018

YX Zb0.083

0.130.046

-0.160.2

0.210.12

-0.035YX [c

-0.13-0.16

-0.0660.24

0.0690.045

-0.14-0.06

YX [Z0.047

-0.0230.17

0.0440.13

0.120.073

-0.16

YX [[0.0037

-0.210.26

0.110.13

0.34-0.33

0.023

YX [\0.02

-0.0420.36

0.30.2

0.0047-0.42

-0.064

YX []0.1

0.0220.004

0.21-0.13

0.15-0.16

0.039

YX [^0.013

-0.190.39

-0.390.27

-0.20.24

0.18

YX [_

0.31-0.25

-0.1-0.33

-0.17-0.11

-0.4-0.13

YX [`

-0.11-0.085

0.280.24

-0.52-0.26

-0.120.015

YX [a

0.0710.0074

-0.10.12

0.29-0.37

-0.250.21

YX [b

-0.0250.19

-0.0950.27

0.045-0.41

0.140.06

YX\c

-0.00710.042

-0.069-0.16

0.140.23

-0.15-0.043

YX\ Z

0.047-0.091

-0.0960.11

-0.20.099

0.12-0.19

YX\ [

0.0410.46

0.029-0.14

-0.0490.13

-0.480.13

Table7.8:F

ullEIE

Sdataseteigensystemaccordingto

Freem

an039

to046

CHAPTER7 131

exf 0.004

0.00

40.

0028

0.00

280.

0006

0.00

060.

0000

170.

0000

17

h f i 0.04

0.04

0.00

35-0

.07

0.33

-0.0

077

0.36

-0.0

17

h f j -0.13

-0.1

4-0

.17

-0.1

30.

19-0

.18

-0.1

9-0

.010

h f k 0.16

-0.0

78-0

.28

0.01

10.

140.

33-0

.04

-0.3

5

h f l 0.11

0.12

-0.2

70.

18-0

.21

-0.0

260.

31-0

.45

h f m -0.12

-0.0

53-0

.32

0.17

0.12

-0.1

3-0

.14

0.14

h f n 0.15

-0.0

630.

380.

022

0.19

0.00

540.

0045

-0.2

3

h f o 0.12

-0.1

60.

250.

048

-0.0

99-0

.36

0.00

27-0

.033

h f p 0.16

0.02

5-0

.21

0.02

2-0

.43

0.01

7-0

.29

-0.2

0

h f q 0.057

-0.2

50.

26-0

.15

-0.0

75-0

.031

0.11

-0.1

4

h f ir -0.15

0.05

80.

033

0.06

90.

170.

26-0

.1-0

.13

h f ii -0.23

0.4

-0.1

4-0

.086

-0.1

-0.0

630.

08-0

.14

h f ij 0.099

-0.0

70.

110.

0004

60.

340.

11-0

.53

-0.2

1

h f ik 0.44

0.06

8-0

.12

-0.0

14-0

.11

-0.1

6-0

.24

0.17

h f il 0.042

0.14

0.08

2-0

.005

5-0

.052

0.25

0.19

0.18

h f im -0.11

-0.0

71-0

.17

0.08

30.

160.

110.

079

0.10

h f in 0.09

-0.0

35-0

.14

0.06

3-0

.11

0.05

50.

022

0.11

h f io -0.015

-0.0

230.

2-0

.035

-0.2

0.23

-0.0

0094

-0.0

95

h f ip -0.25

-0.1

1-0

.12

0.23

0.09

7-0

.24

0.03

4-0

.39

h f iq 0.014

0.44

-0.0

26-0

.21

0.28

-0.2

7-0

.011

0.02

9

h f jr 0.35

-0.0

58-0

.059

0.29

0.21

0.25

-0.0

690.

11

h f ji -0.42

-0.2

80.

077

0.14

-0.1

30.

21-0

.17

-0.0

64

h f jj 0.22

-0.2

3-0

.23

0.1

0.06

4-0

.002

30.

25-0

.099

h f jk -0.19

0.2

0.08

3-0

.16

0.12

0.11

0.00

54-0

.13

h f jl -0.081

0.01

8-0

.18

0.14

0.15

-0.0

73-0

.006

9-0

.003

0

h f jm 0.14

0.15

0.06

90.

110.

021

0.23

0.12

-0.1

7

h f jn 0.021

-0.1

30.

150.

098

-0.0

40.

023

0.15

0.22

h f jo 0.22

0.16

0.00

86-0

.25

-0.0

960.

11-0

.1-0

.19

h f jp -0.14

0.01

2-0

.10.

037

-0.1

50.

21-0

.093

0.29

h f jq 0.14

-0.2

10.

140.

210.

13-0

.092

0.16

0.02

8

h f kr 0.052

-0.2

1-0

.26

-0.5

4-0

.05

0.07

2-0

.015

-0.0

14

h f ki 0.058

0.24

0.05

50.

34-0

.094

0.18

-0.0

790.

036

h f kj 0.019

0.15

0.18

0.34

-0.0

21-0

.24

-0.1

4-0

.017

Tabl

e7.

9:F

ullE

IES

data

sete

igen

syst

emac

cord

ingt

oF

reem

an

s -w to

s .-

132 CHAPTER7

12

34

56

78

16.5

+6.5i

470+

340i24.+

4.i68.+

55.i19.+

26.i64.+

66.i40.+

14.i390

+220i

2340

+470i

29.+29.i

5.7+

1.6i18.+

25.i10.1

+5.9i

10.+18.i

7.0-

1.8i40

+150i

34.+

24.i1.6

+5.7i

0.30+

0.30i0.3+

1.9i0.25

+0.65i0.08

+1.5i

0.47+

0.32i0.1

+9.4i

455.+

68.i25.+

18.i1.9

+0.3i

5.9+

5.9i2.4

+1.9i

4.4+

5.4i2.8

+0.2i

25.+31.i

526.+

19.i5.9

+10.1i0.65

+0.25i1.9

+2.4i

0.67+

0.67i1.2

+2.3i

0.81+

0.34i9.+

13.i6

66.+64.i

18.+10.i

1.5+

0.08i5.4

+4.4i

2.3+

1.2i3.9

+3.9i

2.1-

0.3i21.+

25.i7

14.+40.i

-1.8+

7.0i0.32+

0.47i0.2+

2.8i0.34

+0.81i

-0.3+

2.1i0.40

+0.40i

-2.+15.i

8220

+390i

150+

40i9.4

+0.1i

31.+25.i

13.+9.i

25.+21.i

15.-2.i

120+

120i9

22.+90.i

24.+24.i

1.9+

0.8i4.3

+7.7i

1.9+

2.8i3.2

+6.7i

3.0+

1.0i15.+

36.i10

39.+50.i

37.+36.i

2.6+

0.7i7.0

+7.9i

2.3+

2.9i5.6

+8.1i

3.9+

1.4i34.+

38.i11

200+

170i54.+

58.i5.1

+1.2i

16.+17.i

6.3+

4.7i11.+

16.i6.7

+1.0i

68.+95.i

12-7.+

24.i28.+

26.i1.6

+0.6i

3.6+

5.2i1.1

+2.3i

3.3+

5.4i2.7

+1.3i

20.+22.i

133.+

21.i3.+

15.i0.54

+0.61i0.5

+3.2i

0.14+

1.13i0.1

+3.0i

0.77+

0.98i2.+

16.i14

4.+40.i

17.+19.i

1.24+

0.59i2.6+

4.8i1.0

+1.9i

2.1+

4.5i2.0

+0.9i

12.+22.i

15100.+

70.i14.+

16.i1.7

+0.3i

6.3+

5.7i2.5

+1.2i

3.9+

5.0i2.0

-0.1i

25.+35.i

1649.+

67.i22.+

32.i2.0

+0.9i

5.1+

7.9i1.9

+2.6i

3.4+

7.4i2.9

+1.3i

22.+41.i

1765.+

74.i12.+

19.i1.5

+0.6i

4.3+

6.2i1.9

+1.7i

2.5+

5.3i1.9

+0.5i

15.+35.i

1861.+

75.i4.0

+8.1i

0.99+

0.38i3.1+

4.6i1.6

+1.1i

1.5+

3.4i1.15

-0.08i

8.+27.i

199.+

36.i3.4

+7.4i

0.48+

0.37i0.8+

2.7i0.49

+0.90i

0.4+

2.1i0.74

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20-1.+

27.i2.6

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0.35+

0.37i0.3+

2.2i0.23

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0.59+

0.45i-0.3+

10.4i21

5.+16.i

6.+15.i

0.60+

0.53i0.9+

3.0i0.24

+1.07i

0.5+

2.9i0.87

+0.89i

5.+15.i

226.+

41.i17.+

35.i1.5

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2.5+

7.2i0.7

+2.7i

1.7+

7.0i2.3

+2.0i

13.+34.i

233.+

27.i8.2

+9.6i

0.64+

0.34i1.3+

2.7i0.54

+1.01i

1.0+

2.4i1.04

+0.47i4.9

+12.3i

2472.+

120.i82.+55.i

5.1+

0.8i15.+

14.i5.4

+5.4i

12.+14.i

8.1+

1.4i69.+

66.i25

33.+38.i

-0.2+

4.6i0.44+

0.23i1.4+

2.3i0.73

+0.49i

0.5+

1.7i0.44

+0.01i

3.+14.i

2631.+

79.i2.1

+2.5i

0.66+

0.35i1.6+

3.8i1.2

+1.1i

0.7+

2.5i0.94

-0.17i

0.2+

20.i27

43.+93.i

32.+27.i

2.4+

0.7i6.3

+8.3i

2.6+

2.9i4.9

+7.5i

3.7+

0.8i26.+

41.i28

8.+22.i

1.3+

5.1i0.29

+0.26i0.5

+1.7i

0.28+

0.54i0.15+

1.40i0.41+

0.26i0.7

+9.0i

29570

+390i

53.+51.i

7.9+

0.4i32.+

25.i14.+

4.i19.+

21.i8.4

-3.0i

120+

170i30

69.+70.i

28.+49.i

2.7+

1.4i6.8

+10.6i

2.2+

3.5i4.4

+10.4i

3.7+

2.2i32.+

56.i31

80+

200i104.+

88.i7.0

+2.1i

18.+23.i

6.7+

8.7i15.+

22.i11.1

+3.2i

82.+106.i

32260

+170i88.+

100.i7.5

+1.8i

24.+23.i

8.2+

6.9i17.+

23.i9.9

+2.6i

110+

130i

Table7.10:Part1

ofpartialsumy3z 4

ofthefirsttw

ow

eightedprojectorsofthefull

EIE

Sdataset

CHAPTER7 133

910

1112

1314

1516

190

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24.i

36.+

37.i

58.+

54.i

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28.i

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3.i

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17.i

16.+

14.i

32.+

22.i

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1.9i

0.7

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6i1.

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7.7

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7.i

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5.2i

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8+

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1.7i

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6+

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9.3i

3125

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.+11

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5+

3.6i

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9.i

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28.i

23.+

22.i

3232

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.i60

.+64

.i24

.+9.

i12

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0.5i

20.+

8.i

18.+

27.i

30.+

18.i

Tabl

e7.

11:P

art2

ofpa

rtia

lsum

{ ),+- ofthe

first

two

wei

ghte

dpro

ject

orso

fthe

full

EIE

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tase

t

134 CHAPTER7

1718

1920

2122

2324

174.+

65.i75.+

61.i36.+

9.i27.-

1.i16.+

5.i41.+

6.i27.+

3.i120.+

72.i2

19.+12.i

8.1+

4.0i7.4

+3.4i

7.1+

2.6i15.+

6.i35.+

17.i9.6

+8.2i

55.+82.i

30.6

+1.5i0.38

+0.99i0.37

+0.48i0.37

+0.35i0.53

+0.60i1.2

+1.5i0.34

+0.64i

0.8+

5.1i4

6.2+

4.3i4.6

+3.1i

2.7+

0.8i2.2

+0.3i

3.0+

0.9i7.2

+2.5i

2.7+

1.3i14.+

15.i5

1.7+

1.9i1.1

+1.6i

0.90+

0.49i0.81+

0.23i1.07+

0.24i2.7+

0.7i1.01+

0.54i5.4

+5.4i

65.3

+2.5i

3.4+

1.5i2.1

+0.4i

1.8+

0.08i2.9

+0.5i

7.0+

1.7i2.4

+1.0i

14.+12.i

70.5

+1.9i-0.08

+1.15i0.37

+0.74i0.45

+0.59i0.89

+0.87i2.0

+2.3i0.47

+1.04i

1.4+

8.1i8

35.+15.i

27.+8.i

13.3+

1.2i10.4

-0.3i

15.+5.i

34.+13.i

12.3+

4.9i66.+

69.i9

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6.2i4.4

+4.2i

2.6+

1.5i2.2

+0.9i

2.6+

2.0i6.1

+5.1i

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2.0i9.+

19.i10

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7.3i6.4

+6.2i

3.6+

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+0.5i

2.9+

1.1i7.1

+2.7i

3.2+

1.4i15.+

16.i11

15.+12.i

10.1+

9.2i7.1

+2.6i

6.2+

1.1i8.8

+1.8i

22.+6.i

7.9+

3.7i44.+

41.i12

4.4+

5.5i4.5

+4.7i

2.3+

1.1i1.8

+0.4i

1.2+

1.1i2.9

+2.3i

1.7+

0.9i5.9

+9.3i

130.7

+3.1i

0.6+

2.6i0.70

+0.98i0.69

+0.62i0.61

+0.78i1.5

+1.9i0.62

+0.92i

1.5+

6.8i14

3.3+

4.4i3.0

+3.4i

1.7+

1.0i1.4

+0.5i

1.3+

1.2i3.1

+2.7i

1.4+

1.1i4.8

+10.5i

155.5

+3.2i

2.9+

2.3i2.4

+0.7i

2.2+

0.3i3.7

+0.4i

9.0+

1.5i3.0

+1.3i

18.+15.i

165.3

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3.9+

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0.9i2.9

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7.1+

3.6i2.8

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13.+18.i

174.2

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2.3+

3.0i2.0

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0.7i2.9

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7.0+

3.0i2.3

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12.+16.i

183.0

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0.55i2.6

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2.8i1.7

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191.2

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1.9+

7.3i20

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1.9i0.55+

1.27i0.44+

0.58i0.42+

0.42i0.50+

0.75i1.1+

1.9i0.33+

0.73i0.4

+5.7i

211.0

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0.9+

2.6i0.82

+0.86i0.75

+0.50i0.63

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0.58i0.73+

0.33i0.77+

0.72i1.8+

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0.70i2.4

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12.i14.+

10.i7.3

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5.7+

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2.4i30.+

30.i25

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0.62i0.56+

0.44i0.60+

0.33i1.30+

0.45i3.1+

1.4i0.84+

0.79i4.8

+7.2i

262.2

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+0.38i0.81

+0.62i0.81

+0.57i

1.9+

1.3i4.2

+3.6i

1.0+

1.4i4.5

+13.1i

277.3

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5.8+

4.7i3.3

+1.4i

2.7+

0.7i3.4

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8.0+

4.6i3.0

+2.0i

14.+21.i

280.6

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+0.86i0.36

+0.43i0.37

+0.30i0.57

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+1.3i0.38

+0.57i

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4.7i29

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11.9+

6.0i11.0

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0.8i48.+

6.i15.+

6.i98.+

76.i30

6.6+

9.3i4.9

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3.8+

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3.7+

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23.i31

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9.8+

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11.i8.3

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18.i15.+

16.i10.6

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9.1+

1.2i11.3

+1.6i

28.+5.i

11.3+

4.3i62.+

50.i

Table7.12:Part3

ofpartialsumy3z 4

ofthefirsttw

ow

eightedprojectorsofthefull

EIE

Sdataset

CHAPTER7 135

2526

2728

2930

3132

138

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53.i

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88.+

104.

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0.35

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9i1.

4+

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8+

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7+

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13:P

art4

ofpa

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first

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full

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t

136 CHAPTER7

12

34

56

78

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Table7.14:Part1

ofthepartialsumy5z 8

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CHAPTER7 137

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138 CHAPTER7

1718

1920

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+0.27i0.036+

0.006i-0.92+

0.20i0.40+

0.27i-2.1

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0.4i1.6

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0.80i-2.5

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17-0.15

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0.25+

1.20i-0.82

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0.18i-0.74

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1.20+

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0.61i-0.28

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2.i-9.8

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-57.-34.i

Table7.16:Part3

ofthepartialsumy5z 8

ofthefirsttw

ow

eightedprojectorsofthefull

EIE

Sdataset

CHAPTER7 139

2526

2728

2930

3132

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Tabl

e7.

17:P

art4

ofth

epa

rtia

lsum

{ .,+v ofthe

first

two

wei

ghte

dpro

ject

orso

fthe

full

EIE

Sda

tase

t

140 CHAPTER7

List of Figures

3.1 Two componentvectorin | Ù . . . . . . . . . . . . . . . . . . . . 243.2 Two componentvectorin theGaussplane . . . . . . . . . . . . . 243.3 Multiplication of two complex numbers} and~ . . . . . . . . . . 253.4 Nondirectionalstargraph . . . . . . . . . . . . . . . . . . . . . 373.5 Nondirectionalcompletegraph. . . . . . . . . . . . . . . . . . . 38

4.1 Communicationbehavior beforerotation . . . . . . . . . . . . . . 434.2 Communicationbehavior afterrotation. . . . . . . . . . . . . . . 454.3 Equivalentto non-directionalstargraphcorrespondingto Eq.(4.5) 484.4 Symmetricalperturbationof non-directionalstargraphrepresented

in Eq.(4.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.5 Asymmetricperturbationof anon-directionalstar . . . . . . . . . 514.6 Asymmetricstargraph . . . . . . . . . . . . . . . . . . . . . . . 544.7 Perturbedstargraphcorrespondingto Eq.(4.15) . . . . . . . . . . 564.8 Completesymmetricbidirectionalgraphcorrespondingto Eq.(4.26) 614.9 Completedirectionalcomplex graph . . . . . . . . . . . . . . . . 624.10 A network consistingof two subgroupsandsymmetricbilateral

communication . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.1 EIES:In-degreesortedby amount . . . . . . . . . . . . . . . . . 695.2 EIES:Inboundnormalizedstrengthsortedby amount . . . . . . . 695.3 EIES:Out-degreesortedby amount . . . . . . . . . . . . . . . . 705.4 EIES:Outboundnormalizedstrengthsortedby amount . . . . . . 705.5 EIESsubset:In-degree;k: authorID . . . . . . . . . . . . . . . . 745.6 EIES subset:Inboundnormalizedstrengthsortedby amount;k:

authorID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.7 EIESsubset:Out-degreesortedby amount. . . . . . . . . . . . . 765.8 EIESsubset:Outboundnormalizedstrengthsortedby amount . . 76

6.1 Eigenspectrumof thecompleteEIESdatasetsortedby value . . . 806.2 Eigenspectrumof the completeEIES datasetsortedby absolute

value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

141

142 CHAPTER7

6.3 Cumulativecoveredvarianceby eigenvaluesµ ö . . . . . . . . . . 816.4 Absolutevaluedistribution for theeigenvectorcorrespondingto µ ³ 826.5 Phasedistribution of the eigenvectorcomponentscorresponding

to µ ³ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.6 Distributionof

± Û�Ù>´>±. . . . . . . . . . . . . . . . . . . . . . . . . 84

6.7 Distributionof¶Å· Û�Ù>´Ï¹

. . . . . . . . . . . . . . . . . . . . . . . . 856.8 Eigenvectorcomponentdistribution for the3rd eigenvector . . . . 866.9 Eigenvectorcomponentdistribution for the3rd eigenvector . . . . 866.10 Eigenvectorcomponentdistribution for the5theigenvector . . . . 876.11 Phasedistribution for the5theigenvector . . . . . . . . . . . . . 886.12 Eigenvectorcomponentdistribution for the4theigenvector . . . . 896.13 Phasedistribution for the4theigenvector . . . . . . . . . . . . . 896.14 Eigenvectorcomponentdistribution for the6theigenvector . . . . 906.15 Phasedistribution for the6theigenvector . . . . . . . . . . . . . 906.16 Densityplot of realpartof eigenprojectorcorrespondingto µ ³ . . 976.17 Densityplot of imaginarypartof eigenprojectorcorrespondingtoµ ³ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.18 Densityplot of realpartof eigenprojectorcorrespondingto µ Ù . . 986.19 Densityplot of imaginarypartof eigenprojectorcorrespondingtoµ Ù . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.20 Densityplot of realpartof eigenprojectorcorrespondingto µ æ . . 996.21 Densityplot of imaginarypartof eigenprojectorcorrespondingtoµ æ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.22 Densityplot of realpartof eigenprojectorcorrespondingto µ ð . . 1006.23 Densityplot of imaginarypartof eigenprojectorcorrespondingtoµ ð . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.24 Densityplot of realpartof eigenprojectorcorrespondingto µ í . . 1016.25 Densityplot of imaginarypartof eigenprojectorcorrespondingtoµ í . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.26 Densityplot of realpartof eigenprojectorcorrespondingto µ ñ . . 1026.27 Densityplot of imaginarypartof eigenprojectorcorrespondingtoµ ñ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.28 Eigenspectrumof theEIESdatasubsetsortedby amount . . . . . 1046.29 Eigenspectrumof theEIESsubsetsortedby absolutevalue . . . . 1046.30 Cumulativecoveredvariancegivenin Tab.( 6.12) . . . . . . . . . 1056.31 Distribution of theabsolutevalueof theeigenvectorcomponents

forÛ¸³

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.32 Distributionof thephase

¶¸· Û¸³(´ · µ ³û¹X¹ . . . . . . . . . . . . . . . . 1086.33 Distribution of theabsolutevalueof theeigenvectorcomponents

forÛ�Ù

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.34 Distributionof phase

¶Å· Û�Ù>´ · µ Ù�¹X¹ . . . . . . . . . . . . . . . . . . 111

CHAPTER7 143

6.35 Densityplot of realpartof eigenprojectorcorrespondingto µ ³ . . 1126.36 Densityplot of imaginarypartof eigenprojectorcorrespondingtoµ ³ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.37 Densityplot of realpartof eigenprojectorcorrespondingto µ Ù . . 1136.38 Densityplot of imaginarypartof eigenprojectorcorrespondingtoµ Ù . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.1 Stockmarket asbipartite-graph . . . . . . . . . . . . . . . . . . 119

144 CHAPTER7

List of Tables

3.1 Definitionsof role,status,rank,prestige . . . . . . . . . . . . . . 173.2 OrderandSynonymity of role,status,rank,prestige. . . . . . . . 183.3 Eigensystemof astargraphin Fig.( 3.4) . . . . . . . . . . . . . . 373.4 Eigensystemof acompletegraphin Fig.( 3.5) . . . . . . . . . . . 38

4.1 Communicationbehavior representation. . . . . . . . . . . . . . 424.2 Communicationbehavior representationin thematrix

ù. . . . . 45

4.3 Eigensystemforù��

in Eq.(4.6) . . . . . . . . . . . . . . . . . . 494.4 Eigensystemfor the perturbedstar in Eq.( 4.7) after rotation in

Fig.( 4.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.5 Eigensystemfor theperturbedstarwith asymmetriccommunica-

tion givenin Fig.( 4.5) . . . . . . . . . . . . . . . . . . . . . . . 524.6 Eigensystemfor theperturbedstarasin Fig.( 4.5) in reverseddi-

rection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.7 Cumulativecoveredvarianceof eigenvaluesfrom Tab.( 4.6) . . . . 534.8 Eigensystemfor

ù�� ðin Eq.(4.14)with � ÑTü�� ÝIß . . . . . . . . 55

4.9 Eigensystemforù�� ð

in Eq.(4.14)with � Ñ ± � ± Ç È�������� . . . . . . 564.10 Eigensystemfor

ù�� ðg�in Eq.(4.16)with � Ñ6ü�� ÝIß . . . . . . . . 57

4.11 Eigensystemforù�� ðg�

in Eq.(4.16)with � Ñ ± � ± Ç�È�������� . . . . . . 574.12 Cumulativecoveredvarianceof eigenvaluesfrom Tab.( 4.6) . . . . 574.13 Eigensystemof

ù�� í in Eq.(4.27) . . . . . . . . . . . . . . . . . 624.14 Eigensystemof matrix

ù�� í,� in Eq.(4.29) . . . . . . . . . . . . . 634.15 Eigensystemfor

ù�� ðg�in Eq.(4.16)with � Ñ ± � ± Ç È�������� . . . . . . 63

4.16 Eigensystemof therotatedmatrix � in Eq.(4.30) . . . . . . . . . 65

5.1 Full EIESdataset . . . . . . . . . . . . . . . . . . . . . . . . . 685.2 Outbound,inboundstrength,centralityandprestigeof the EIES

completedataset . . . . . . . . . . . . . . . . . . . . . . . . . . 725.3 Numberof messagesexchangedbetweenmembersin the EIES

subset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.4 Outboundandinboundstrengthfor the EIES subsetnormalized

by thetotal numberof messages,aswell ascentralityandprestige 77

145

146 CHAPTER7

5.5 Eigensystemof the EIES subsetas calculatedwith the methodusedby Freeman . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.1 Cumulativecoveredvarianceof eigenvaluesfrom Tabs.(7.2)-(7.5) 826.2 Comparisonof anchorsof the full datasetbetweenthe method

usedby Freeman( ú · µ ö ¹ ) andthemethodpresentedin this book(ù · µ ö ¹ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.3 Part 1: Back rotatedweightedprojectorbasedcorrespondingtothefirst eigenvalue . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.4 Part 2: Back rotatedweightedprojectorbasedcorrespondingtothefirst eigenvalue . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.5 Submatricesbasedonfirst eigenvector . . . . . . . . . . . . . . . 936.6 Part1: Backrotatedweightedprojectorcorrespondingto thesec-

ondeigenvalue . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.7 Part2: Backrotatedweightedprojectorcorrespondingto thesec-

ondeigenvalue . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.8 Submatricesbasedonsecondeigenvector . . . . . . . . . . . . . 956.9 Submatricesbasedon third eigenvector . . . . . . . . . . . . . . 956.10 part1: Backrotatedweightedprojectorcorrespondingto thethird

eigenvalue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.11 part2: Backrotatedweightedprojectorcorrespondingto thethird

eigenvalue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.12 Cumulativecoveredvarianceof eigenvaluesfrom Tab.( 6.18) . . . 1056.13 Submatrixbasedon first eigenvector . . . . . . . . . . . . . . . . 1066.14 Submatricesbasedonsecondeigenvector . . . . . . . . . . . . . 1066.15 Partial sum

ù ³>ô Ù. . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.16 Partial sumù æXô '

. . . . . . . . . . . . . . . . . . . . . . . . . . 1096.17 Eigensystemfor

ùwith � ÑTü�� Ý<ß . . . . . . . . . . . . . . . . 110

6.18 Eigensystemofù

with � Ñ · ± � ± ¼ ¶¸· � ¹X¹ . . . . . . . . . . . . . . 1106.19 Comparisonof anchorsfor theEIESsubset . . . . . . . . . . . . 114

7.1 Forecastedresultsvs. actualresultsof 2001dietelectionsin Baden-Wurttemberg . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.2 Full EIESdataseteigensystem:µ ³ to µT� . . . . . . . . . . . . . 1247.3 Full EIESdataseteigensystem:µ�� to µ ³(ñ . . . . . . . . . . . . . 1257.4 Full EIESdataseteigensystem:µ ³�' to µ Ù í . . . . . . . . . . . . 1267.5 Full EIESdataseteigensystem:µ Ù>ð to µ æ>Ù . . . . . . . . . . . . 1277.6 Full EIESdataseteigensystemaccordingto Freemanµ ³ to µT� . . 1287.7 Full EIESdataseteigensystemaccordingto Freemanµ�� to µ ³(ñ . 1297.8 Full EIESdataseteigensystemaccordingto Freemanµ ³�' to µ Ù í . 1307.9 Full EIESdataseteigensystemaccordingto Freemanµ Ù>ð to µ æ>Ù . 131

CHAPTER7 147

7.10 Part 1 of partialsumó ³>ô Ù

of thefirst two weightedprojectorsofthefull EIESdataset . . . . . . . . . . . . . . . . . . . . . . . . 132

7.11 Part 2 of partialsumó ³>ô Ù

of thefirst two weightedprojectorsofthefull EIESdataset . . . . . . . . . . . . . . . . . . . . . . . . 133

7.12 Part 3 of partialsumó ³>ô Ù

of thefirst two weightedprojectorsofthefull EIESdataset . . . . . . . . . . . . . . . . . . . . . . . . 134

7.13 Part 4 of partialsumó ³>ô Ù

of thefirst two weightedprojectorsofthefull EIESdataset . . . . . . . . . . . . . . . . . . . . . . . . 135

7.14 Part1 of thepartialsumó æXô ñ

of thefirst two weightedprojectorsof thefull EIESdataset . . . . . . . . . . . . . . . . . . . . . . 136

7.15 Part2 of thepartialsumó æXô ñ

of thefirst two weightedprojectorsof thefull EIESdataset . . . . . . . . . . . . . . . . . . . . . . 137

7.16 Part3 of thepartialsumó æXô ñ

of thefirst two weightedprojectorsof thefull EIESdataset . . . . . . . . . . . . . . . . . . . . . . 138

7.17 Part4 of thepartialsumó æXô ñ

of thefirst two weightedprojectorsof thefull EIESdataset . . . . . . . . . . . . . . . . . . . . . . 139

148 CHAPTER7

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