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Modeling and Optimization Approaches for Ensuring Robustness in Networked and Financial Systems

Permanent Link: http://ufdc.ufl.edu/UFE0042699/00001

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Title: Modeling and Optimization Approaches for Ensuring Robustness in Networked and Financial Systems
Physical Description: 1 online resource (145 p.)
Language: english
Creator: VEREMYEV,ALEXANDER F
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: ATTACK -- CDO -- CLIQUE -- CLUSTER -- DESIGN -- ENTROPY -- GRAPH -- NETWORK -- OPTIMIZATION -- RANDOM -- ROBUST
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
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theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Our work develops theoretical principles and optimization models for ensuring robustness in networked and financial systems. Robustness is an important characteristic of complex systems arising in diverse fields. In many cases, robustness can be interpreted as the ability of the system to preserve integrity when it faces natural or man-made disruptions. In addition, from mathematical modeling prospective, robustness can be viewed as the ability of a model to remain valid under varying assumptions, parameters and initial conditions. In the first part of this work we consider robustness issues in complex networked systems, which are crucial in a variety of applications. In many situations, one of the key robustness requirements is the connectivity between each pair of nodes through a path that is short enough, which makes a network cluster more robust with respect to potential network component disruptions. Several new compact linear 0-1 programming models for identifying such network clusters (referred to as k-clubs) are developed in this work. Moreover, we introduce a new related concept referred to as R-robust k-club, which naturally arises from the aforementioned formulations and extends the standard definition of a k-club by explicitly requiring that there must be at least R distinct paths of length at most $k$ between all pairs of nodes. As a natural extension of the work mentioned above, we develop optimal network design and enhancement strategies that can provably provide certain robustness and attack tolerance characteristics. In the definitions of attack tolerance used in this study, it is generally required that a network has a guaranteed ability to maintain not only the overall connectivity, but also specific (more restrictive) connectivity properties (in particular, small diameter) after multiple failures of network components (nodes and/or edges), regardless of whether these failures are random or targeted. Attack tolerance properties of known "robust" network configurations are also investigated in this study. In certain special cases, the aforementioned concept of R-robust k-club has been proved to be optimal in terms of both attack tolerance and construction costs. We also consider another type of robust network clusters that is based on relaxing the requirement on edge density (i.e., the ratio of the number of edges in the cluster to the maximum possible number of edges). These clusters are referred to as quasi-cliques. We obtain several important results in this context, including the most compact known linear mixed-integer formulation for the maximum quasi-clique problem. Moreover, we analyze the asymptotic behavior of these clusters in uniform random graphs and derive explicit analytical formulas that characterize the size of the maximum quasi-cliques in a graph depending on its parameters. We also prove that there exists an abrupt jump (first-order phase transition) of the order of magnitude of the maximum quasi-clique, which to our knowledge is the first mathematically rigorous proof of such behavior. From another perspective, we also consider the problems of optimal choice of "robust" clusters in financial systems. Particularly, we analyze the problem of finding optimal Collateralized Debt Obligations (CDO) from a pool of different assets. In this context, the optimal choice of these clusters (CDOs) is considered to be robust if it does not significantly change under different model assumptions. We consider a synthetic CDO with the goal to build a maximally profitable CDO. In addition to "standard" CDOs, so called "step-up" CDOs are also investigated. The performed case study is based on the time to default scenarios for obligors (instruments) generated by Standard & Poor?s CDO Evaluator. It shows that step-up CDOs can save about 25%-35% of tranche spread payments (i.e., profitability of CDOs can be boosted about 25%-35%). Several optimization models are developed from the bank originator prospective. Moreover, we introduce the "entropy" approach for pricing CDOs. Numerical experiments show that the model is robust to the choice of different model parameters.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by ALEXANDER F VEREMYEV.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Uryasev, Stanislav.
Local: Co-adviser: Boginski, Vladimir L.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-10-31

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042699:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042699/00001

Material Information

Title: Modeling and Optimization Approaches for Ensuring Robustness in Networked and Financial Systems
Physical Description: 1 online resource (145 p.)
Language: english
Creator: VEREMYEV,ALEXANDER F
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: ATTACK -- CDO -- CLIQUE -- CLUSTER -- DESIGN -- ENTROPY -- GRAPH -- NETWORK -- OPTIMIZATION -- RANDOM -- ROBUST
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Our work develops theoretical principles and optimization models for ensuring robustness in networked and financial systems. Robustness is an important characteristic of complex systems arising in diverse fields. In many cases, robustness can be interpreted as the ability of the system to preserve integrity when it faces natural or man-made disruptions. In addition, from mathematical modeling prospective, robustness can be viewed as the ability of a model to remain valid under varying assumptions, parameters and initial conditions. In the first part of this work we consider robustness issues in complex networked systems, which are crucial in a variety of applications. In many situations, one of the key robustness requirements is the connectivity between each pair of nodes through a path that is short enough, which makes a network cluster more robust with respect to potential network component disruptions. Several new compact linear 0-1 programming models for identifying such network clusters (referred to as k-clubs) are developed in this work. Moreover, we introduce a new related concept referred to as R-robust k-club, which naturally arises from the aforementioned formulations and extends the standard definition of a k-club by explicitly requiring that there must be at least R distinct paths of length at most $k$ between all pairs of nodes. As a natural extension of the work mentioned above, we develop optimal network design and enhancement strategies that can provably provide certain robustness and attack tolerance characteristics. In the definitions of attack tolerance used in this study, it is generally required that a network has a guaranteed ability to maintain not only the overall connectivity, but also specific (more restrictive) connectivity properties (in particular, small diameter) after multiple failures of network components (nodes and/or edges), regardless of whether these failures are random or targeted. Attack tolerance properties of known "robust" network configurations are also investigated in this study. In certain special cases, the aforementioned concept of R-robust k-club has been proved to be optimal in terms of both attack tolerance and construction costs. We also consider another type of robust network clusters that is based on relaxing the requirement on edge density (i.e., the ratio of the number of edges in the cluster to the maximum possible number of edges). These clusters are referred to as quasi-cliques. We obtain several important results in this context, including the most compact known linear mixed-integer formulation for the maximum quasi-clique problem. Moreover, we analyze the asymptotic behavior of these clusters in uniform random graphs and derive explicit analytical formulas that characterize the size of the maximum quasi-cliques in a graph depending on its parameters. We also prove that there exists an abrupt jump (first-order phase transition) of the order of magnitude of the maximum quasi-clique, which to our knowledge is the first mathematically rigorous proof of such behavior. From another perspective, we also consider the problems of optimal choice of "robust" clusters in financial systems. Particularly, we analyze the problem of finding optimal Collateralized Debt Obligations (CDO) from a pool of different assets. In this context, the optimal choice of these clusters (CDOs) is considered to be robust if it does not significantly change under different model assumptions. We consider a synthetic CDO with the goal to build a maximally profitable CDO. In addition to "standard" CDOs, so called "step-up" CDOs are also investigated. The performed case study is based on the time to default scenarios for obligors (instruments) generated by Standard & Poor?s CDO Evaluator. It shows that step-up CDOs can save about 25%-35% of tranche spread payments (i.e., profitability of CDOs can be boosted about 25%-35%). Several optimization models are developed from the bank originator prospective. Moreover, we introduce the "entropy" approach for pricing CDOs. Numerical experiments show that the model is robust to the choice of different model parameters.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by ALEXANDER F VEREMYEV.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Uryasev, Stanislav.
Local: Co-adviser: Boginski, Vladimir L.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-10-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042699:00001


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MODELINGANDOPTIMIZATIONAPPROACHESFORENSURINGROBUSTNESSINNETWORKEDANDFINANCIALSYSTEMSByALEXANDERF.VEREMYEVADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011AlexanderF.Veremyev 2

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IdedicatethisthesistomyparentsRaisaandFedor,andmydearAnna. 3

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ACKNOWLEDGMENTS IamverythankfultomyadvisorsProf.StanUryasevandProf.VladimirBoginskifortheirsupportduringmydoctoratestudyingattheUniversityofFlorida.WiththeirresearchguidanceandinvaluablehelpIwasabletogrowonbothprofessionalandpersonallevels.Iwouldliketoexpressmygratitudetoothermembersofmydoctoratecommittee,Prof.PanosPardalosandProf.LiqingYanfortheircontributiontomyresearch.IwouldalsoliketoexpressmygreatestappreciationtomycolleaguesfromtheRiskManagementandFinancialEngineeringlabandCenterforAppliedOptimization.Intensivediscussions,exchangeofideasandjointresearchwithmyfellowgraduatestudentsfromthesetwolabshelpedmesignicantlytoachievemygoals.Also,Iwouldliketothankmyfamilyandfriends,whosupportedandencouragedmeinallofmybeginnings. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 LISTOFSYMBOLS .................................... 10 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 14 2LARGEROBUSTNETWORKCLUSTERSIDENTIFICATION .......... 20 2.1PreliminaryFormulationsofMaximum2,3,k-clubProblems ........ 21 2.1.1Maximum2-clubProblemFormulation ................ 21 2.1.2Maximum3-clubProblemFormulation ................ 22 2.1.3Maximumk-clubProblemFormulation ................ 23 2.2CompactLinearIntegerProgrammingFormulationoftheMaximumk-clubProblem .................................... 24 2.3CompactLinear0-1FormulationoftheMaximumk-clubProblem ..... 26 2.4R-robustk-clubs ................................ 29 2.4.1MaximumR-robustk-clubProblem .................. 30 2.4.2ImportantSpecialCase:R-robust2-clubs .............. 32 2.4.3ErrorandAttackTolerancePropertiesofR-robust2-clubs ..... 32 2.5ComputationalExperiments .......................... 34 3OPTIMALATTACKTOLERANTNETWORKDESIGNSTRATEGIES ...... 39 3.1Notations,Denitions,andPreviousWork .................. 43 3.2AttackToleranceCharacteristicsofCliquesandCliqueRelaxations .... 46 3.2.1AttackToleranceofDensity-BasedCliqueRelaxations(-cliques) 47 3.2.2AttackToleranceofDegree-BasedCliqueRelaxations(k-plexes) 48 3.3NecessaryandSufcientConditionsforAttackToleranceofDiameter-BasedCliqueRelaxations(k-clubs) ......................... 49 3.4Linear0-1FormulationsofDesignandEnhancementProblemsfor2-clubswithGuaranteedAttackToleranceProperties ................ 54 3.4.1ComputationalComplexityoftheConsideredProblems ....... 56 3.4.2OptimizationProblemFormulations .................. 56 3.4.3IllustrativeExamples .......................... 59 3.5ExactAnalyticalSolutionsforOptimal2-club/2-coreandR-robust2-clubNetworkDesignProblems ........................... 62 5

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4DENSITY-BASEDCLIQUERELAXATIONSANDTHEIRASYMPTOTICBEHAVIORINRANDOMGRAPHS ............................... 68 4.1DerivationofAsymptoticBehaviorProperties ................ 68 4.2LinearMixed-IntegerFormulationsoftheMaximum-cliqueProblem ... 79 4.3ComputationalExperiments .......................... 82 5IMPLIEDCOPULACDOPRICINGMODEL:ENTROPYAPPROACH ...... 86 5.1ConventionalCopulaandtheImpliedCopula ................ 88 5.2ImpliedCopula:EntropyApproach ...................... 92 5.3CaseStudy ................................... 102 6OPTIMALSTRUCTURINGOFCDOCONTRACTS ................ 113 6.1Step-upCDOBackground ........................... 114 6.2OptimizationModels .............................. 116 6.2.1OptimizationofAttachement/DetachmentPoints(withFixedPoolofAssets) ................................ 116 6.2.2SimultaneousOptimizationofCDOPoolandCreditTranching ... 119 6.2.3Simplication1:ProblemDecompositionforLargeSizeProblems 121 6.2.4Simplication2:MinimizationofLowerandUpperBounds ..... 123 6.3CaseStudy ................................... 125 6.3.1Case1a ................................. 126 6.3.2Case1b ................................. 126 6.3.3Case2 .................................. 127 6.3.4Case3 .................................. 128 7CONCLUSION .................................... 132 7.1DissertationContribution ........................... 132 7.2FutureWork ................................... 134 APPENDIX AMINIMALNUMBEROFEDGESINATTACKTOLERANT2-CLUBS ....... 135 BEQUIVALENTREPRESENTATIONOFOBJECTIVEINTHEOPTIMALCDOCONTRACTSTRUCTURINGMODEL ....................... 140 REFERENCES ....................................... 142 BIOGRAPHICALSKETCH ................................ 145 6

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LISTOFTABLES Table page 4-1Maximum-cliquesizesintheuniformrandomgraphsforn=100,and=90% 84 4-2Maximum-cliquesizesintheuniformrandomgraphsforn=100,and=85% 85 5-1Marketquotesfor5,7,10-yeariTraxxonDecember20,2006 .......... 105 6-1Attachmentpoints(indecimals)ofthe5-periodCDOcontractobtainedfromoptimizationProblem1a,Problem1bandProblem2foreverym. ........ 131 6-2Attachmentpoints(indecimals)ofthe5-periodCDOcontractobtainedfromoptimizationProblem3foreverym(u=2.5%,l=0.93and0.97). ....... 131 7

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LISTOFFIGURES Figure page 2-1Generalviewoftheprotein-proteininteractionnetwork(S.Cerevisiae) ..... 36 2-2Maximumk-clubsintheS.Cerevisiaenetwork .................. 37 2-3Maximum2-robustk-clubsinpower-lawgraph(n=100) ............. 38 3-1Example:2-connectedgraphand2-robust2-club ................. 65 3-2Example:Astronglyedgeattacktolerant4-clubwhichisnota2-robust4-club 65 3-3Optimaldesignof2-club/2-coreforn=10,11nodes ............... 66 3-4Optimaldesignof2-club,2-robust2-club,3-robust2-club,and4-robust2-clubforn=10nodes ................................... 66 3-5Optimalenhancementofpower-lawnetworkinstancewith=0.5,n=20 .. 67 4-1Relativesizeofthemaximum-cliquesintheuniformrandomgraphsforxed=70%anddifferentn,p .............................. 84 5-1ExampleofaCCCdistribution,wl=30andwr=50 ............... 105 5-2Distributionsofthecollateralhazardrate,asimpliedin5-yeariTraxxtranchespreads,obtainedfromtheHullandWhitemodelforI=100,300 ........ 106 5-3Distributionsofthecollateralhazardrate,asimpliedin5-yeariTraxxtranchespreads,obtainedfromtheHullandWhitemodelforI=500,1000 ....... 107 5-4Distributionsofthecollateralhazardrate,asimpliedin5-yeariTraxxtranchespreads,obtainedfromourmodelforI=100,...,1000 .............. 108 5-5Distributionsofthecollateralhazardrate,asimpliedin5-yeariTraxxtranchespreads,obtainedfromourmodelinCCCclassforI=100,...,1000 ...... 109 5-6Distributionsofthecollateralhazardrate,asimpliedin5,7and10-yeariTraxxtranchespreads,obtainedfromourmodelforI=100 .............. 110 5-7Distributionsofthecollateralhazardrate,asimpliedin5-yeariTraxxtranchespreadsindifferentdates,obtainedfromourmodelforI=100 ......... 111 5-8Distributionsofthecollateralhazardrate,asimpliedin5,7and10-yeariTraxxtranchespreadsintwodifferentdates,obtainedfromourmodelforI=100 .. 112 6-1SyntheticCDOspreadsowstructure ....................... 130 6-2CDOattachmentanddetachmentpointsstructure ................ 130 8

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A-1Illustrationofthenon-overlappingsetsofedgesandnodesusedintheproofofTheorem A.1 .................................... 139 A-2Illustrationofthenon-overlappingsetsofedgesandnodesusedintheproofofTheorem A.2 .................................... 139 9

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LISTOFSYMBOLS,NOMENCLATURE,ORABBREVIATIONS AoRDAPSG AmericanoptimalDecisionPortfolioSafeguardCDO CollateralizedDebtObligationsCDS CreditDefaultSwapE SetofarcsE[] ExpectedvalueG NetworkgraphLP Linearprogrammingmax() Maximummin() Minimumo() LittleonotationO() BigOnotationPfg Probabilitys.t. SubjecttoV SetofverticesZ+ Setofpositiveintegernumbersde Roundupbc Rounddown8 Forall;forany;foreach9 Thereexists;thereis;thereareP Denotesthesummationofaseriesofterms)]TJ /F6 7.97 Tf 5.61 -4.38 Td[(nk DenotesthebinomialcoefcientCkndenedasCkn=n! (n)]TJ /F6 7.97 Tf 6.59 0 Td[(k)!k!foranypositiveintegerskn 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyMODELINGANDOPTIMIZATIONAPPROACHESFORENSURINGROBUSTNESSINNETWORKEDANDFINANCIALSYSTEMSByAlexanderF.VeremyevMay2011Chair:StanUryasevCochair:VladimirBoginskiMajor:IndustrialandSystemsEngineering Ourworkdevelopstheoreticalprinciplesandoptimizationmodelsforensuringrobustnessinnetworkedandnancialsystems.Robustnessisanimportantcharacteristicofcomplexsystemsarisingindiverseelds.Inmanycases,robustnesscanbeinterpretedastheabilityofthesystemtopreserveintegritywhenitfacesnaturalorman-madedisruptions.Inaddition,frommathematicalmodelingprospective,robustnesscanbeviewedastheabilityofamodeltoremainvalidundervaryingassumptions,parametersandinitialconditions. Intherstpartofthisworkweconsiderrobustnessissuesincomplexnetworkedsystems,whicharecrucialinavarietyofapplications.Inmanysituations,oneofthekeyrobustnessrequirementsistheconnectivitybetweeneachpairofnodesthroughapaththatisshortenough,whichmakesanetworkclustermorerobustwithrespecttopotentialnetworkcomponentdisruptions.Severalnewcompactlinear0-1programmingmodelsforidentifyingsuchnetworkclusters(referredtoask-clubs)aredevelopedinthiswork.Moreover,weintroduceanewrelatedconceptreferredtoasR-robustk-club,whichnaturallyarisesfromtheaforementionedformulationsandextendsthestandarddenitionofak-clubbyexplicitlyrequiringthattheremustbeatleastRdistinctpathsoflengthatmostkbetweenallpairsofnodes. 11

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Asanaturalextensionoftheworkmentionedabove,wedevelopoptimalnetworkdesignandenhancementstrategiesthatcanprovablyprovidecertainrobustnessandattacktolerancecharacteristics.Inthedenitionsofattacktoleranceusedinthisstudy,itisgenerallyrequiredthatanetworkhasaguaranteedabilitytomaintainnotonlytheoverallconnectivity,butalsospecic(morerestrictive)connectivityproperties(inparticular,smalldiameter)aftermultiplefailuresofnetworkcomponents(nodesand/oredges),regardlessofwhetherthesefailuresarerandomortargeted.Attacktolerancepropertiesofknownrobustnetworkcongurationsarealsoinvestigatedinthisstudy.Incertainspecialcases,theaforementionedconceptofR-robustk-clubhasbeenprovedtobeoptimalintermsofbothattacktoleranceandconstructioncosts. Wealsoconsideranothertypeofrobustnetworkclustersthatisbasedonrelaxingtherequirementonedgedensity(i.e.,theratioofthenumberofedgesintheclustertothemaximumpossiblenumberofedges).Theseclustersarereferredtoasquasi-cliques.Weobtainseveralimportantresultsinthiscontext,includingthemostcompactknownlinearmixed-integerformulationforthemaximumquasi-cliqueproblem.Moreover,weanalyzetheasymptoticbehavioroftheseclustersinuniformrandomgraphsandderiveexplicitanalyticalformulasthatcharacterizethesizeofthemaximumquasi-cliquesinagraphdependingonitsparameters.Wealsoprovethatthereexistsanabruptjump(rst-orderphasetransition)oftheorderofmagnitudeofthemaximumquasi-clique,whichtoourknowledgeistherstmathematicallyrigorousproofofsuchbehavior. Fromanotherperspective,wealsoconsidertheproblemsofoptimalchoiceofrobustclustersinnancialsystems.Particularly,weanalyzetheproblemofndingoptimalCollateralizedDebtObligations(CDO)fromapoolofdifferentassets.Inthiscontext,theoptimalchoiceoftheseclusters(CDOs)isconsideredtoberobustifitdoesnotsignicantlychangeunderdifferentmodelassumptions.WeconsiderasyntheticCDOwiththegoaltobuildamaximallyprotableCDO.InadditiontostandardCDOs, 12

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socalledstep-upCDOsarealsoinvestigated.Theperformedcasestudyisbasedonthetimetodefaultscenariosforobligors(instruments)generatedbyStandard&PoorsCDOEvaluatorR.Itshowsthatstep-upCDOscansaveabout25%-35%oftranchespreadpayments(i.e.,protabilityofCDOscanbeboostedabout25%-35%).Severaloptimizationmodelsaredevelopedfromthebankoriginatorprospective.Moreover,weintroducetheentropyapproachforpricingCDOs.Numericalexperimentsshowthatthemodelisrobusttothechoiceofdifferentmodelparameters. 13

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CHAPTER1INTRODUCTION Themainobjectiveofourstudyistodevelopmodelsforensuringrobustnessinnetworkedandnancialsystems.Large-scalecomplexnetworksplayacrucialroleinagreatvarietyofareasnowadays.Althoughasignicantamountofworkhasbeendoneonstudyingstructuralpropertiesofnetworksintermsoftheirconnectivity,theresearchonvariouscharacteristicsofcomplexnetworksisfarfromcomplete.Inadditiontothelarge-scaleandcomplexnatureofnetworks,oneoftenneedstodealwithuncertainpotentialdisruptionsthatcaninterferewiththeoperationofanetworkedsystem.Theseissuescanbecausedbyavarietyoffactors,includingman-madeandnaturaldisruptions,whichmayresultinfailuresofcomponents(nodesand/oredges)inthenetwork. Anaturalapproachtotakeintoaccountpotentialmultiplenetworkcomponentfailuresistoconsiderrobustnetworkclustersthatensureasufcientdegreeofrobustconnectivitybetweennodes.Theconventionaldenitionofconnectivity(e.g.,theexistenceofapathbetweeneverytwonodes)maynotprovidetherequiredrobustnesscharacteristics,sincealongpathbetweenapairofnodescanmaketheconnectionvulnerable,especiallyifeverynodeand/oredgeinthepathcanpotentiallyfail.Inthiscontext,theshorterthepathbetweeneverypairofnodesis,themorerobustthecorrespondingnetworkstructurebecomes(althoughspecialcasesofvulnerablenetworkswithshortconnectivitypathscanstillbeconstructed).However,therobustnesscharacteristicscanbesubstantiallyimprovediftherearemultipledistinctpathsbetweeneverypairofnodes.Thiswouldensurethatanetworkclusterstaysconnectedevenifsomenodesand/oredgesaredeletedfromthenetwork. Clearly,acliqueisaveryrobustnetworkstructureintheaforementionedcontext,sinceeverytwonodesinacliquearedirectlyconnectedbyanedge.Itiseasytoseethatthedeletionofanynumberofnodesfromacliquewouldnotviolatetheclique 14

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structureoftheremainingnodes.Moreover,acliquewithnnodesisguaranteedtoremainaconnectednetworkifatmostn)]TJ /F5 11.955 Tf 11.96 0 Td[(2edgesarerandomlydeleted. Therehasbeenalotofworkrelatedtovariousaspectsofndinglargecliquesinnetworks(foranextensivereviewofthemaximumcliqueproblemsee[ 6 ]).However,inmostpracticalsituations,cliquesareoverlyrestrictivestructures,sinceitischallengingtoconstructanetworkwithallthepossibleconnectionsinthepresenceofobstaclesandotherlimitationsinreal-lifesituations.Therefore,severalconceptsreferredtoascliquerelaxationshavebeenintroduced.Themainideabehindtheseconceptsistorelaxcertainpropertiesofacliquewhilestillmaintainingsufcientconnectivityandrobustnesscharacteristicsoftheobtainednetworkstructures.Notethatinmanycasesthemaximumsizeofthesecliquerelaxationsissubstantiallylargerthanthemaximumsizeofcliquesinthesamenetwork,whichprovidesasignicantadvantageinsituationswhenlargerobustclustersneedtobeidentied. Theideasforthesecliquerelaxationsoriginallycomefromthestudyofsocialnetworks;however,thesedenitionscanbeutilizedinavarietyofotherapplicationareas,suchascommunication/informationexchangenetworks,energynetworks,etc.Therearethreemaindirectionsforpossiblerelaxationsofthecliquedenition: 1. Density-basedrelaxationsrelaxingtherequirementfortheedgedensityofacliquetobe1:quasi-cliques(-densesubgraphs,-cliques)[ 1 ]; 2. Degree-basedrelaxationsrelaxingtherequirementforthedegreeofeachnodeinacliqueofsizentoben)]TJ /F5 11.955 Tf 11.95 0 Td[(1:k-plexes[ 11 ]; 3. Path(diameter)-basedrelaxationsrelaxingtherequirementforthelengthofthepathbetweenanytwonodesinacliquetobe1:k-cliques[ 9 ]andk-clubs[ 10 ]. Aquasi-clique(alsoreferredtoasa-densesubgraph,or-clique)isasubgraphthathastheedgedensityofatleast,where2(0,1].Clearly,aquasi-cliquebecomesacliqueif=1.Ak-plexisasubgraphinwhichthedegreeofeachnodeisatleastn)]TJ /F3 11.955 Tf 13.29 0 Td[(k(assumingthatnisthenumberofnodesinthissubgraph).Ak-cliqueisasubgraphwherethelengthofthepathbetweenanytwonodesisatmostk(notethat 15

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inthisdenitionothernodesinthispatharenotrequiredtobelongtothek-clique),whereasak-clubisasubgraphthathasadiameterofatmostk(inthisdenitionforanypairofnodesink-clubthereexistsatleastonepathoflengthatmostksuchthatallthenodesinthispathbelongtothisk-club).Obviously,fork=1,ak-plex,ak-cliqueorak-clubwouldalsobeaclique. Althoughthesedenitionsareratherstraightforwardandintuitivelyclear,mathematicallyrigorousstudiesonrelatedoptimizationaspects(e.g.,mathematicalprogrammingformulationsforndingthelargestcliquerelaxationsingraphs)havestartedtoappearonlywithinthepastfewyears.Thecompactlinearmixed-integerformulationforthemaximumquasi-cliqueproblemwithO(n)variablesandconstraintshasbeendevelopedinthiswork.Themaximumk-plexproblemhasbeenaddressedby[ 4 ],wheretheyprovidethemostcompactformulationwithnvariablesandconstraints. Asubstantialpartofourworkconcentratesonthethirdtypeofcliquerelaxationsmentionedaboveanddevelopsnewcompactformulationsforthemaximumk-clubproblem.Duetotheaboveconsiderations,ak-clubcanbeviewedasatighterstructurethanak-clique;therefore,itismoreapplicableforconnectivityandclusteringproblemsonnetworkswhererobustnessissuesplayanimportantrole.Forinstance,inthecontextofreal-lifesparsenetworks(e.g.,communicationnetworks),identifyingthemaximumk-clubwouldmeanidentifyingthelargestpossibleclusterinanetworkthatcanserveasasystemofcommunicationhubsthatareconnectedandhaveshorttransmissionpathsbetweeneachother.Inmanyotherapplications(e.g.,network-baseddatamining),maximalk-clubswoulddenotelargetightlyconnectedclusters.Wedescribethenewcompactformulationsofthemaximumk-clubproblem.Asanalternativetotheapproachofdirectlyformulatingthelinear0-1problemwithO(nk)entitiesusedby[ 17 ]and[ 3 ],ourapproachisbasedonformulatingapreliminarynon-linear0-1problemforndingthemaximumk-club,andthenutilizingthestructureoftheproblemtoformulateitasalinear0-1problemwithO(kn2)entities.Wealsodene 16

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thenewconceptofanR-robustk-club,whichnaturallyarisesfromthedevelopedk-clubformulationandprovidesanadditionaldegreeofrobustnessfortheconsiderednetworkclusters.Inaddition,weconsideranimportantspecialcaseofanR-robust2-clubanddemonstrateitsattractivepropertiesoferrorandattacktolerance. Wealsodevelopandanalyzeoptimalattack-tolerantnetworkdesignandenhancementstrategiesthatcanprovablyprovidecertainrobustnesscharacteristics.Inthedenitionsofattacktoleranceusedinthiswork,wegenerallyrequirethatanetworkhasaguaranteedabilitytomaintainnotonlytheoverallconnectivity,butalsospecic(morerestrictive)connectivityproperties(inparticular,smalldiameter:d(G)k)aftermultiplefailuresofnetworkcomponents(nodesand/oredges),regardlessofwhetherthesefailuresarerandomortargeted.Thispropertyisreferredtoasstrongattacktolerance,whereasthepropertyofanetworktomaintainjusttheregularconnectivityafternode/edgefailures(withnoexplicitrestrictiononthediameter)isreferredtoasweakattacktolerance.Althoughoptimaldesignofanetworkwithasmalldiameter(inparticular,diameter2)isawell-studiedproblem,theobtainednetworkcongurationsoftendonothaveanyguaranteedattacktolerancecharacteristics.Weinvestigateattacktolerancepropertiesofknownrobustnetworkcongurations(e.g.,cliquerelaxations),aswellasprovenecessaryandsufcientconditionsforguaranteedweakandstrongattacktolerancepropertiesfornetworksofdiameter2.WedemonstratethattherecentlyintroducedconceptofanR-robust2-clubistheonlynetworkcongurationthatisguaranteedtohaveastrongattacktoleranceproperty(i.e.,maintainbothconnectivityanddiameter2)afteranyR)]TJ /F5 11.955 Tf 12.33 0 Td[(1edgesaredeleted.Furthermore,wedemonstratethatifalledgeshavethesameconstructioncost,theproblemofoptimalR-robust2-clubnetworkdesignhasanexactanalyticalsolutionthatrequiresO(Rn)constructededges,whichmakesthiscongurationasymptoticallyascost-efcientasaregularsparseconnectednetwork.Wealsopresentlinear0-1formulationsforrelatednetworkdesignandenhancementproblemswithdifferentedgeconstructioncosts,whichareNP-hard 17

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inthegeneralcase.Illustrativeexamplesareprovidedtodemonstratetheintroducedconceptsandresults. Wealsoinvestigatetheasymptoticbehaviorofdensity-basedcliquerelaxations(quasi-cliques)inuniformrandomgraphsG(n,p)(i.e,everyedgeinthegraphwithnverticesexistswithprobabilityp).Wederiveexplicitanalyticalformulasthatcharacterizethesizeofthemaximumquasi-cliqueinagraphdependingonitsparameters.Theasymptoticbehaviorofthemaximumcliquesizeintheuniformrandomgraphhasbeenextensivelystudiedin[ 43 ],[ 42 ].Weshowthatas(quasi-cliqueparameter)approaches1,asquasi-cliquebecomesclique,theanalyticalformulaforthemaximum-cliquesizereducestotheresultsobtainedin[ 43 ],[ 42 ].Therefore,ourndingsmightbeviewedasageneralizationofthepreviouslyobtainedresults.Wealsoprovethatthereexistsanabruptjump(rst-orderphasetransition)oftheorderofmagnitudeofthemaximumquasi-clique,whichtoourknowledgeistherstmathematicallyrigorousproofofsuchbehavior.Although,thederivedformulasarevalidfortheasymptoticbehaviorofthemaximum-cliquesizeintheuniformrandomgraphs,meaningthatnshouldbeverylarge,thecomputationalexperimentsshowthatforcertainrangesofpandtheseformulasareratherpreciseevenforn=100. Fromanotherperspective,wealsoconsiderrobustmodelingissuesinnancialsystems.Particularly,wefocusonCDOpricingandstructuring(clustering)models.Aso-calledimpliedcopulaCDOpricingmodelisconsideredforcalibratingobligorhazardrates.Tondtheprobabilitydistributionofthehazardrates,HullandWhitein[ 27 ]suggestedminimizingthesumofdeviationsfromno-arbitrageequationsandasmoothingterm.Weproposeanalternativeentropyapproachtotheimpliedcopulamodel.Thedistributionisfoundbymaximizingentropywithno-arbitrageconstraintsbasedonbidandaskpricesofCDOtranches.Toreducethenoise,anewclassofdistributionsisintroduced,so-calledCCCdistributions.Acasestudyshowsthatthe 18

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entropyapproachhasarobustperformance,whiletheHullandWhitemodelissensitivetothesmoothingcoefcientandthenumberofhazardratesonthegrid. TheobjectiveofCDOstructuringmodelistohelpabankoriginatortobuildamaximallyprotableCDO.WeconsideranoptimizationframeworkofstructuringCDOs.InadditiontostandardCDOswestudysocalledstep-upCDOs.InastandardCDOcontracttheattachment/detachmentspointsareconstantoverthelifeofCDO.Inastep-upCDOtheattachment/detachmentpointsmaychangeovertime.Weshowthatstep-upCDOscansaveabout25%-35%oftranchespreadpayments(i.e.,protabilityofCDOscanbeboostedabout25%-35%).Severaloptimizationmodelsaredevelopedfromthebankoriginatorprospective.WeconsiderasyntheticCDOwherethegoalistominimizepaymentsforthecreditriskprotection(premiumleg)whilemaintainingaspeciccreditrating(assuringthecreditspread)ofeachtrancheandmaintainingthetotalincomingCDSspreadpayments.Thecasestudyisbasedonthetimetodefaultscenariosforobligors(instruments)generatedbyStandard&PoorsCDOEvaluatorR[ 30 ]. 19

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CHAPTER2LARGEROBUSTNETWORKCLUSTERSIDENTIFICATION Amongpossiblegeneralizations(or,relaxations)ofaclique,ak-clubisadiameter-basedrelaxation,specically,ak-clubisasubsetofverticesSVsuchthatthediameterofinducedsubgraphG(S)isatmostk.Themaximumk-clubproblemiscomputationallychallenging,andthedecisionversionofthisproblemhasbeenshowntobeNP-complete[ 3 ]. Theonlypreviouslyknownmathematicalprogrammingformulationforthegeneralcase(k>2)ofthemaximumk-clubproblemhasbeendevelopedby[ 17 ]and[ 3 ].Fork>2,thislinear0-1formulationhassignicantdrawbacks:itisratherextensiveandrequiresO(nk)entities;moreover,formulatingtheconstraintsrequiresexplicitenumerationofallpossiblepathsoflengthatmostkbetweenallpairsofnodes.Thismakessolvingthemaximumk-clubproblemusingthisformulationratherchallengingandinmanycasescomputationallyintractableevenforsmallvaluesofk(e.g.,k=4,5,...).Clearly,askincreases,thenumberofentitiesinthisformulationwouldbecomeverylarge,asitbecomesO(nn)ifk=O(n). Here,wepresentanew,substantiallymorecompact,linear0-1formulationforthemaximumk-clubproblem,whichrequiresO(kn2)entities.ThisvalueapproachesO(n3)ifk=O(n);therefore,thepresentedformulationismuchmoreefcientinthegeneralcase.Asignicantadvantageofthisformulationisthefactthatthevalueoftheparameterkdoesnotsubstantiallychangethesizeandthestructureoftheproblemformulation.Approachesoflinearizingnon-linear0-1problemscanpotentiallybeefcientforcertainclassesofproblems(e.g.,linearizationtechniquesforquadratic/polynomial0-1[ 7 16 20 ]andfractional0-1[ 24 25 ]problemshavebeenaddressedintheliterature). Wealsoextendthisformulationtorequiretheexistenceofseveraldistinctpathsbetweenallpairsofnodes,whataddsextrarobustnessandattacktoleranceproperties 20

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totheconsideredk-clubstructures.ThisconsiderationmotivatesustointroducethenewdenitionofanR-robustk-clubandtodevelopacompactlinearintegerformulationforthemaximumR-robustk-clubproblem. 2.1PreliminaryFormulationsofMaximum2,3,k-clubProblems 2.1.1Maximum2-clubProblemFormulation DenotebyG=(V,E)asimpleundirectedgraphwiththesetofnvertices(nodes)V=f1,...,ngandthesetofedgesE.LetA=faijgni,j=1betheadjacencymatrixofG.Thisisannn0-1matrix,whereanelementaij=1ifthereisanedge(undirectedarc)betweennodesiandj,andaij=0otherwise.Now,consideraproblemofndingamaximum2-clubinthisgraph.SupposewepicksomesubgraphGs,andwewanttocheckwhetherthissubgraphisa2-club.Forthesepurposeswedenex=(x1,...,xn)tobea0-1vectorwithxi=1ifnodeibelongstoGs,andxi=0otherwise.ThesubgraphGsisa2-clubifitsdiameterisatmost2.Inotherwords,everypairofnodes(i,j)isconnecteddirectly,orthroughsomeothernodek.Suchaconnectionofnodes(i,j)canbeeasilyformulatedintermsoftheconstraint aij+nXk=1aikakjxkxixj, whichisequivalenttoaij+nXk=1aikakjxkxi+xj)]TJ /F5 11.955 Tf 11.95 0 Td[(1. Thisisalinearconstraintwith0-1variables. Thus,theproblemofndingthemaximum2-clubinthegraphGcanbeformulatedas maxnXi=1xi subjecttoaij+nXk=1aikakjxkxi+xj)]TJ /F5 11.955 Tf 11.96 0 Td[(1, 21

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xi2f0,1g, wherei=1,...,n;j=i+1,...,n. Therearen(n)]TJ /F5 11.955 Tf 12.6 0 Td[(1)=2constraintsinthisformulation.Sinceaijis0or1,therearesomeunnecessaryconstraints,andwecanproceedfurthertosimplifyandreducethesetofconstraints.Let(i)=fj:aij=1gbeaneighborhoodofnodei.Usingthisdenition,theproblemformulationcanberewrittenasmaxnXi=1xi subjecttoXk2(i)\(j)xkxi+xj)]TJ /F5 11.955 Tf 11.95 0 Td[(1,xi2f0,1g, wherei=1,...,n;j=i+1,...,n;j=2(i). NotethatthenumberofconstraintsinthisformulationdependsontheedgedensityofthegraphG(V,E). 2.1.2Maximum3-clubProblemFormulation Usingthesamelogicasintheprevioussubsection,wecanformulatethemaximum3-clubproblemasfollows:maxnXi=1xi subjecttoaij+nXk=1aikakjxk+nXk=1nXm=1aikakmamjxkxmxi+xj)]TJ /F5 11.955 Tf 11.95 0 Td[(1,xi2f0,1g, wherei=1,...,n;j=i+1,...,n.Thisformulationcanbealsosimpliedusingthenotationofneighborhoodofnodei((i))aspresentedintheprevioussubsection.Thismightbeusefulforcomputationalpurposes,butforsimplicityofunderstandingwekeeptheformulationsintheformatabove. 22

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Thisistheproblemwithalinearobjectiveandquadraticconstraints.Inastandardandstraightforwardlinearizationapproach(amoreefcientalternativetothisapproachisproposedlater),onecanintroducenewvariableswijtolinearizethisproblemasfollows.Denewij=xixj.Notethatn(n)]TJ /F5 11.955 Tf 12.88 0 Td[(1)=2)]TJ /F3 11.955 Tf 12.88 0 Td[(nvariableswijareneededsincewij=wji,andwiiisneedless,but,forsimplicitywefurthersay,thatweneedO(n2)variablessinceweareinterestedintheorderofnumberofvariables.Theconstraintwij=xixjisequivalenttowijxi,wijxj,wijxi+xj)]TJ /F5 11.955 Tf 11.95 0 Td[(1. Now,themaximum3-clubproblemcanberepresentedasthefollowinglinear0-1problem:maxnXi=1xi subjecttoaij+nXk=1aikakjxk+nXk=1nXm=1aikakmamjwkmxi+xj)]TJ /F5 11.955 Tf 11.96 0 Td[(1,wijxi,wijxj,wijxi+xj)]TJ /F5 11.955 Tf 11.95 0 Td[(1,xi,wij2f0,1g, wherei=1,...,n;j=i+1,...,n.ThisformulationislinearandcontainsO(n2)0-1variablesandO(n2)constraints. 2.1.3Maximumk-clubProblemFormulation Usingthepreliminaryformulationsfortheabovespecialcases,weformulatethegeneralcaseofthemaximumk-clubproblem.Usingthesimilarlogicandnotationsasabove,themaximumk-clubproblemcanberepresentedas 23

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maxnXi=1xi subjecttoaij+nXk=1aikakjxk+nXk=1nXm=1aikakmamjxkxm+nXk=1nXm=1nXt=1aikakmamtatjxkxmxt+...+nXi1=1nXi2=1nXik)]TJ /F14 5.978 Tf 5.76 0 Td[(2=1nXik)]TJ /F14 5.978 Tf 5.76 0 Td[(1=1aii1ai1i2...aik)]TJ /F14 5.978 Tf 5.75 0 Td[(2ik)]TJ /F14 5.978 Tf 5.75 0 Td[(1aik)]TJ /F14 5.978 Tf 5.76 0 Td[(1jxi1...xik)]TJ /F14 5.978 Tf 5.76 0 Td[(1xi+xj)]TJ /F5 11.955 Tf 11.96 0 Td[(1,xi2f0,1g, wherei=1,...,n;j=i+1,...,n. Thisformulationcanbealsolinearizedandsimpliedusingthesamestandardapproachesasabove,andtheresultinglinearformulationwillhaveO(nk)]TJ /F7 7.97 Tf 6.58 0 Td[(1)variablesandconstraints.However,inthenextsection,weshowthatthesizeofthisformulationcanbesubstantiallyreducedbyapplyingamoreefcientlinearizationtechniquethatemploysthespecialstructureofk-clubs. 2.2CompactLinearIntegerProgrammingFormulationoftheMaximumk-clubProblem Inthissection,wedescribehowtolinearizethemaximumk-clubprobleminordertoobtainamorecompactformulation.Theideaofthislinearizationistodenenewvariablesw(l)ij(i,j=1,...,n;l=2,...,k),whichrepresentthenumberofdistinctpathsofdistancelfromnodeitojinthesubgraphGs(x)denedbyvectorx=(x1,...,xn).IfnodeiorjdoesnotbelongtoGs(x),thenw(l)ij=0.Notethatw(l)ij=w(l)ji.Thus,thereareonlyn+kn(n)]TJ /F5 11.955 Tf 12.27 0 Td[(1)=2variablesintheproblemformulation.WeshowthattherearealsoO((k)]TJ /F5 11.955 Tf 11.96 0 Td[(1)n2)constraints. Forl=2wecanwritew(2)ij=xixjnXk=1aikakjxk. 24

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Theaboveconstraintincorporatestheadditionalvariablesw(2)ij.Notingthatw(2)ijn,wecanlinearizeitasfollows:w(2)ijnXk=1aikakjxk+n(2)]TJ /F3 11.955 Tf 11.95 0 Td[(xi)]TJ /F3 11.955 Tf 11.95 0 Td[(xj),w(2)ijnXk=1aikakjxk)]TJ /F3 11.955 Tf 11.95 0 Td[(n(2)]TJ /F3 11.955 Tf 11.95 0 Td[(xi)]TJ /F3 11.955 Tf 11.95 0 Td[(xj),w(2)ijnxi,w(2)ij)]TJ /F3 11.955 Tf 21.92 0 Td[(nxi,w(2)ijnxj,w(2)ij)]TJ /F3 11.955 Tf 21.92 0 Td[(nxj. Forthislinearizationweneed3n(n)]TJ /F5 11.955 Tf 12.88 0 Td[(1)additionalconstraints.Notethatotheradditionalvariablesw(l)ijcanbefoundrecursively,sincew(l)ij=xinXk=1w(l)]TJ /F7 7.97 Tf 6.58 0 Td[(1)kjaik. Similarly,usingthefactthatw(l)ijnl)]TJ /F7 7.97 Tf 6.58 0 Td[(1wecanlinearizeitasw(l)ijnXk=1w(l)]TJ /F7 7.97 Tf 6.59 0 Td[(1)kjaik+nl)]TJ /F7 7.97 Tf 6.59 0 Td[(1(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xi),w(l)ijnXk=1w(l)]TJ /F7 7.97 Tf 6.59 0 Td[(1)kjaik)]TJ /F3 11.955 Tf 11.95 0 Td[(nl)]TJ /F7 7.97 Tf 6.59 0 Td[(1(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xi),w(l)ijnl)]TJ /F7 7.97 Tf 6.59 0 Td[(1xi,w(l)ij)]TJ /F3 11.955 Tf 21.92 0 Td[(nl)]TJ /F7 7.97 Tf 6.59 0 Td[(1xi. Puttingalltheseconstraintstogether,themaximumk-clubproblemcanbeformulatedasmaxnXi=1xi 25

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subjecttokXl=2w(l)ijxi+xj)]TJ /F5 11.955 Tf 11.95 0 Td[(1,i=1,...,n,j=2(i),w(2)ijnXk=1aikakjxk+n(2)]TJ /F3 11.955 Tf 11.95 0 Td[(xi)]TJ /F3 11.955 Tf 11.95 0 Td[(xj),w(2)ijnXk=1aikakjxk)]TJ /F3 11.955 Tf 11.95 0 Td[(n(2)]TJ /F3 11.955 Tf 11.95 0 Td[(xi)]TJ /F3 11.955 Tf 11.95 0 Td[(xj),w(2)ijnxi,w(2)ij)]TJ /F3 11.955 Tf 21.92 0 Td[(nxi,w(2)ijnxj,w(2)ij)]TJ /F3 11.955 Tf 21.92 0 Td[(nxj, andforl=3,...,kw(l)ijnXk=1w(l)]TJ /F7 7.97 Tf 6.59 0 Td[(1)kjaik+nl)]TJ /F7 7.97 Tf 6.59 0 Td[(1(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xi),w(l)ijnXk=1w(l)]TJ /F7 7.97 Tf 6.59 0 Td[(1)kjaik)]TJ /F3 11.955 Tf 11.95 0 Td[(nl)]TJ /F7 7.97 Tf 6.59 0 Td[(1(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xi),w(l)ijnl)]TJ /F7 7.97 Tf 6.59 0 Td[(1xi,w(l)ij)]TJ /F3 11.955 Tf 21.92 0 Td[(nl)]TJ /F7 7.97 Tf 6.59 0 Td[(1xi,xi2f0,1g,wij2Z+. wherei,j=1,...,n. 2.3CompactLinear0-1FormulationoftheMaximumk-clubProblem Inthissubsectionwefurtherrenethecompactlinearintegerprogrammingformulationdescribedaboveandtransformthatformulationtoanequivalentlinear0-1programmingproblem.Recallthatw(l)ij,(i,j=1,...,n;l=2,...,k)representthenumberofdistinctpathsoflengthlfromnodeitojinthesubgraphGs(x)denedbythe0-1vector(x1,...,xn).Itmeansthatw(l)ijisanon-negativeintegervariablewiththeupper 26

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boundnl)]TJ /F7 7.97 Tf 6.58 0 Td[(1.However,inthecontextofthestandardmaximumk-clubproblem,wedonotneedtoknowthenumberofdistinctpathsofdistancelfromnodeitojinthesubgraphGs(x).Inthissense,thesevariablescontainalotofunnecessaryinformation.Sinceweonlyneedtocheckifthereisatleastonepathoflengthlbetweennodesiandj,theonlyinformationweneedtoknowaboutthevariablew(l)ijiswhetherithasazerooranonzerovalue. Toaddressthisconsideration,wedene0-1variablesv(l)ij,(i,j=1,...,n;l=2,...,k)asfollows:v(l)ij=1ifthereexistsatleastonepathoflengthlfromnodeitojinthesubgraphGs(x)denedbyvector(x1,...,xn),andv(l)ij=0otherwise.Notethatv(l)ij=v(l)ji.Thus,thereareonlyn+kn(n)]TJ /F5 11.955 Tf 11.95 0 Td[(1)=2variablesinthenewformulation. Forl=2,wecanwritev(2)ij=minfxixjnXk=1aikakjxk,1g. Thisequalitycanbelinearizedasfollows:v(2)ijxi,v(2)ijxj,v(2)ijnXk=1aikakjxk,v(2)ij1 n nXk=1aikakjxk!+(xi+xj)]TJ /F5 11.955 Tf 11.96 0 Td[(2), wherev(2)ijisa0-1variableforany1i
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Similarly,wecanlinearizeitasv(l)ijxi,v(l)ijnXk=1aikv(l)]TJ /F7 7.97 Tf 6.58 0 Td[(1)kj,v(l)ij1 n nXk=1aikv(l)]TJ /F7 7.97 Tf 6.59 0 Td[(1)kj!+(xi)]TJ /F5 11.955 Tf 11.95 0 Td[(1), wherev(l)ijisa0-1variableforany2lkand1ii=1,...,n,v(2)ijxi,v(2)ijxj,v(2)ijnXk=1aikakjxk,v(2)ij1 n nXk=1aikakjxk!+(xi+xj)]TJ /F5 11.955 Tf 11.96 0 Td[(2), andforl=3,...,k;j>i=1,...,n,v(l)ijxi,v(l)ijnXk=1aikv(l)]TJ /F7 7.97 Tf 6.58 0 Td[(1)kj,v(l)ij1 n nXk=1aikv(l)]TJ /F7 7.97 Tf 6.59 0 Td[(1)kj!+(xi)]TJ /F5 11.955 Tf 11.95 0 Td[(1),xi,v(l)ij2f0,1g,i,j=1,...,n;l=2,...,k. 28

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Itshouldbenotedthatforatighterformulationforcomputationalpurposestheaboveconstraintsv(2)ij1 n nXk=1aikakjxk!+(xi+xj)]TJ /F5 11.955 Tf 11.96 0 Td[(2), andv(l)ij1 n nXk=1aikv(l)]TJ /F7 7.97 Tf 6.59 0 Td[(1)kj!+(xi)]TJ /F5 11.955 Tf 11.95 0 Td[(1), canbemadetighterusingthefollowingtransformationoftheright-handside:v(2)ij1 nPk=1aikakj nXk=1aikakjxk!+(xi+xj)]TJ /F5 11.955 Tf 11.95 0 Td[(2), andv(l)ij1 nPk=1aik nXk=1aikv(l)]TJ /F7 7.97 Tf 6.58 0 Td[(1)kj!+(xi)]TJ /F5 11.955 Tf 11.96 0 Td[(1), 2.4R-robustk-clubs Duetothenetworkrobustnessconsiderationsdiscussedintheintroductorysections,wementionedthattheexistenceofashortpathbetweenanytwonodesinak-club(forrelativelysmallvaluesofk)isausefulpropertyintermsofrobustnesscharacteristics;however,themaindrawbackofthestandarddenitionofak-clubisthattheconsideredpathsarenotrequiredtobedistinct,whichmeansthatk-clubscanstillbevulnerabletotargetedattacksthatdestroyappropriatenetworkcomponents.Toaddressthisdrawback,weproposetodeneanothertypeofnetworkstructures,whichhaveapropertythatmultipleshortpathsexistbetweenanypairofnodes.Moreformally,wedeneasubgraphGstobeanR-robustk-club(or,a(k,R)-club)ifthereareatleastRinternallynode-disjointpathsoflengthatmostkbetweeneverypairofnodesinthesubgraphGs.Itshouldbenotedthatalthoughnetworkclustersthathavethispropertyhavegoodattacktolerancecharacteristics,developingmathematicalprogrammingtechniquesforndingtheexactsolutionofthemaximum(k,R)-clubproblemisnotaneasytask.Torelaxthisdenition,onecanintroducealternativerequirementsfordisjoint 29

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paths,suchas:1)internallyedge-disjointpathsthatmaysharecommonnodes;or2)pathsthathaveadifferenceinatleastoneedge. Hereandfurtherinthischapter,weconsiderR-robustk-clubsinthecontextofthelatter(relaxed)denition,thatistwopathsareconsidereddistinctiftheyhaveadifferenceinatleastoneedge.Asitisshownbelow,theaforementionedformulationforthemaximumk-clubproblemcanbedirectlygeneralizedtothisdenitionofanR-robustk-club;however,itcannotbeextendedtoR-robustk-clubswithinternallynode-disjointpaths.Despitethedifcultiesindealingwiththegeneralcaseoftheproblem,itisimportanttonotethatforthespecialcaseofanR-robust2-club,alloftheabovedenitionsofdisjointpathsareequivalent,whichmeansthatinthiscaseonecanjustassumethattwopathsaredistinctiftheyhaveadifferenceinatleastoneedge,andthiswouldautomaticallyimplythatthesepathsarenode-disjointandedge-disjoint. 2.4.1MaximumR-robustk-clubProblem Inthissection,weintroduceacompactformulationforthemaximumR-robustk-clubproblemintherelaxedform.Weshowthatitcanbederivedfromthemaximumk-clubproblemformulationpresentedabove.Furthermore,weformallydiscusscertainrobustnesspropertiesoftheaforementionedspecialcaseofR-robust2-clubs,whichaddressissuesoferror/attacktolerance(i.e.,theresiliencytopotentialmultiplefailuresofnodesand/oredgesinanetwork). HereweprovideacompactlinearintegerformulationfortheproblemofndingamaximumR-robustk-clubinthecontextoftherelaxeddenitionmentionedabove.Recallthatwhenweformulatethemaximumk-clubproblemweusevariablesw(l)ij(i,j=1,...,n;l=2,...,k),whichrepresentthenumberofdistinctpathsofdistancelfromnodeitojinthesubgraphGs(x)denedbyvector(x1,...,xn).Thus,theproblemformulationofndingthemaximumR-robustk-clubisverysimilartotheproblemof 30

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ndingthemaximumk-clubwiththeonlydifferencethatwerequireaij+kXl=2w(l)ijR(xi+xj)]TJ /F5 11.955 Tf 11.96 0 Td[(1),i
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wij2Z+. wherei,j=1,...,n. Notethattherecursivemethodofcalculatingvariablesdoesnotallowustoextendthisformulationandrequireallpathstobenode-disjoint.Clearly,theexistenceofcertainnumberofnode-disjointpathsismoredesirableinpractice,sinceitguaranteesthatthisclusterhasthecertainlevelofattacktolerance.Inthenextsectionweconsideraspecialcasewithk=2wherethenodedisjointrequirementissatised. 2.4.2ImportantSpecialCase:R-robust2-clubs Asitwasnotedabove,inthecaseofR-robust2-clubs,anytwonodeshaveatleastRcompletelydistinctpathsconnectingthem;thatis,thesepathsdonothaveanyedges/nodesincommon.Asitisshowninthenextsubsection,R-robust2-clubshaveveryattractiveerrorandattacktoleranceproperties.Beforeweproceedwiththeseconsiderations,wepresenttheformulationofthemaximumR-robust2-clubproblem,whichinthisspecialcasehasonly0-1variables,ratherthanintegervariablesforthegeneralcaseofthemaximumR-robustk-clubproblem(intherelaxedform)consideredabove. Theformulationofthisproblemisrathercompactandcanbewrittenasfollows maxnXi=1xi subjecttoaij+nXk=1aikakjxkR(xi+xj)]TJ /F5 11.955 Tf 11.96 0 Td[(1),xi2f0,1g, wherei=1,...,n;j=i+1,...,n. 2.4.3ErrorandAttackTolerancePropertiesofR-robust2-clubs Inthissubsection,weconsiderindetailthepropertiesofanimportantspecialcaseofR-robustk-clubsanR-robust2-club.Asithasbeenmentionedbefore,themain 32

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attractivefeatureofR-robust2-clubsisthefactthatallRpathsbetweenanytwonodesarecompletelydistinct,thatis,theydonothaveanyedges/nodesincommon. Anillustrativeexampleofa2-robust2-clubisgiveninFigure 2-2A .Ifonecomparesthestructureofthis2-robust2-clubtothestructureoftheregular2-clubinthesamenetwork(Figure 2-2 ),itcanbeeasilyseenthatthedeletionofthecentralnodefromthe2-clubcompletelydestroystheconnectivityofthiscluster;however,thedeletionofanyonenodeoredgefromthe2-robust2-clubnotonlypreservestheconnectivityofthiscluster,butitalsodoesnotviolateits2-clubstructure(i.e.,alltheremainingnodesarestillconnectedthroughapathofatmosttwoedges). ThisobservationleadsustosomeinterestingconsiderationsregardingtheerrorandattacktoleranceofR-robust2-clubs.Attacksonnetworkscanbedenedastargeteddisruptionsthatattempttodestroycertaincomponentsofthenetwork(nodesoredges)inordertointerferewithnetworkconnectivity.Arelatednotionoferrors,whichessentiallyrepresentrandom(nottargeted)disruptionsofnetworkcomponents,canalsobeconsidered.Theabilityofanetworktomaintaincertainconnectivitycharacteristicsinthepresenceoferrorsand/orattacksisreferredtoastheerrorand/orattacktoleranceofanetwork.Awell-knownexperimentalstudyoferrorandattacktoleranceofpower-lawanduniformrandomnetworkswithrespecttonodefailuresispresentedin[ 2 ]. Clearly,theissueoferrorandattacktoleranceofanetworkisimportantinavarietyofapplications;moreover,theseissuesneedtobegeneralizedandconsideredwithrespecttobothnodefailuresandedgefailures.Inthiscontext,thedenitionofanR-robust2-clubisattractive,sinceitexplicitlyaddressestheerrorandattacktolerancepropertiesofthesenetworkclusters.Specically,thefollowingfactscanbeeasilyestablished. Proposition2.1. ThedeletionofanyonenodefromanR-robust2-clubguaranteesthattheremainingnodesandedgesformatleastan(R)]TJ /F5 11.955 Tf 11.96 0 Td[(1)-robust2-club. 33

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Proposition2.2. ThedeletionofanyoneedgefromanR-robust2-clubguaranteesthattheremainingnodesandedgesformatleastan(R)]TJ /F5 11.955 Tf 11.96 0 Td[(1)-robust2-club. Fromtheseobservations,amoregeneralstatementcharacterizingerrorandattackpropertiesofR-robust2-clubsimmediatelyfollows. Proposition2.3. Thedeletionofany(R)]TJ /F5 11.955 Tf 11.31 0 Td[(1)networkcomponents(nodesand/oredges)fromanR-robust2-clubguaranteesthattheremainingnodesandedgesforma2-club. Theserobustnesscharacteristicsareattractiveduetothefollowingconsiderations: 1. ErrorandattacktolerancepropertiesofR-robust2-clubsaresimilartothoseofcliques(thedeletionofmultiplenetworkcomponentsdoesnotviolatetheconnectivityofacluster);however,thesizeofR-robust2-clubsisusuallylargerthanthesizeofcliquesinreal-worldnetworks); 2. Theconnectivitypatternthatispreservedafterthedeletionof(R)]TJ /F5 11.955 Tf 12.65 0 Td[(1)networkcomponentsisa2-club(ratherthanjustaregularconnectedcomponent),whichensuresthatallnodesareconnectedbyashortpathevenaftermultiplenetworkcomponentfailures; 3. Thenumberofnetworkcomponentsthatcanbedeletedwithoutviolatingthe2-clubstructureoftheconsideredR-robust2-clubisdeterminedsolelybytheuser-denedparameterRanddoesnotdependonanyotherparameters,suchasthesizeofthisR-robust2-cluborthesizeoftheoriginalnetwork. 2.5ComputationalExperiments Inthissectionwepresentsomecomputationalexperimentsfortheoptimizationmodelsdescribedabove.Moreexperimentscanbefoundin[ 14 ].First,weconsiderareal-worldnetworkinstancethatrepresentsprotein-proteininteractionsoftheyeastS.Cerevisiae(Figure 2-1 ).Thisisasparsepower-lawnetworkwithapproximately2,000nodesandedges.Notethatthisnetworkhasbeenconsideredin[ 3 ],wherethemaximum2-cluband3-club(3-clique)areidentied.Weconductcomputationalexperimentsforthisnetworkandfoundthemaximum2,3,and4-clubs,aswellasthemaximum2-robust2-clubusingthedevelopedformulations.WeuseILOGCPLEX[ 21 ]softwaretorunoptimizationproblems,andPajek[ 23 ]todrawnetworks.For 34

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computationalpurposes,weusethefollowingpreprocessingproceduretodecreasethesizeoftheconsideredoptimizationproblemsfortheS.Cerevisiaenetwork. 1. Identifyconnectedcomponents.Sinceanyk-clubisaconnectedcluster,thenanytwonodesinthegraphwhicharedisconnectedcannotbelongtoanyk-club.Thus,anyk-clubisasubsetofaconnectedcluster.Theconnectedclusterscanbeidentiedinapolynomialtime. 2. Ignoreanynodethatcannotbelongtoalargek-club.Thisnetworkisatypicalexampleofasparsepower-lawgraph.Inthesegraphs,therearefewnodeswithlargedegreesandmanynodeswithverylowdegrees.Onecanndsomelargek-club(assumethatitssizeisNk)whichhasanodewiththelargestdegreeinthisnetworkanduseNkasalowerboundofthemaximumk-club.Therefore,anynodewhichhaslessthanNkk-distantneighbors(i.e.,nodesthatareconnectedwiththeconsiderednodethroughapathofatmostkedges)cannotbelongtothemaximumk-club.Wecanignorethesenodesandreducethesizeoftheoptimizationproblem. 3. IgnoreanynodewithadegreelessthanRinthemaximumR-robust2-clubproblem.SinceanynodeinR-robust2-clubshouldhaveadegreeofatleastR,thenwecanignoreanynodewhichhasadegreelessthanRandalsoreducethesizeoftheoptimizationproblem. TheresultsofthesecomputationsarepresentedinFigure 2-2 .Itcanbeeasilyobservedfromthisgurethatthemaximum2,3,and4-clubsareveryvulnerabletoattacksonnodes(e.g.,thedeletionofonlyonenodewouldviolatetheclusterconnectivity);however,themaximum2-robust2-clubwouldremaina2-clubifanyonenodeoredgeisdeleted. Wealsogeneratedoneinstanceofpower-law1networkwithn=100,=1andtheproposedlinearmodelsthere.Figure 2-3 showsthemaximumcliqueandthemaximum2-robustand3-robust2-clubsinthesamepower-lawgraph,anditcanbeseenthat2-robustand3-robust2-clubsaresubstantiallylargerthanthemaximumclique. 1Apowerlaw(scale-free)networkwithaparameterisanetworkwherethenumberofnodeswithadegreekisproportionaltok)]TJ /F13 7.97 Tf 6.58 0 Td[(. 35

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Figure2-1. Generalviewoftheprotein-proteininteractionnetworkoftheyeastS.Cerevisiae 36

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AMaximum2-club BMaximum(2,2)-club CMaximum3-club DMaximum4-club Figure2-2. Graphicalrepresentationofmaximum2,3,and4-clubs(A,B,andC),aswellasD)themaximum2-robust2-club,intheS.Cerevisiaenetwork.TheguresareobtainedbyusingPajeknetworkdrawingsoftware[ 23 ]. 37

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AMaximumclique BMaximum(2,2)-club CMaximum(2,3)club Figure2-3. GraphicalrepresentationofA)maximumclique(6nodes),B)2-robust2-club(19nodes),andC)3-robust2-club(15nodes)inthesamepower-lawnetworkwith100nodesand=1.2.Themaximum2-robustand3-robust2-clubsaresubstantiallylargerthanthemaximumclique,whiletheystillhavegoodrobustnesscharacteristics.TheguresareobtainedbyusingPajeknetworkdrawingsoftware[ 23 ]. 38

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CHAPTER3OPTIMALATTACKTOLERANTNETWORKDESIGNSTRATEGIES Thetaskofensuringefcientandreliableoperationofcriticalnetworkinfrastructure(e.g.,communication,supply,transportation,andothertypesofnetworks)playsanextremelyimportantroleinthecontextofpublicwelfare.Connectivityisanessentialcharacteristicofanyoperationalnetwork;however,theconventionaldenitionofconnectivity(i.e.,theexistenceofapathbetweeneverytwonodes)maynotprovidetherequiredrobustnesscharacteristics,sincelongpathsbetweennodesmaymakenetworksrathervulnerableandexpensivetooperate,especiallyifeverynodeand/oredgeinthepathcanpotentiallyexperienceatemporaryorpermanentfailure.Thefailuresofnetworkcomponentscanbecausedbynaturalorman-madedisruptions(e.g.,naturaldisasters,terroristattacks,etc.)Ingeneral,thefailurescanbeeitherrandom(nospecicnetworkcomponentsaretargeted;thesefailuresareoftenreferredtoaserrors)ortargeted(certaincriticalnetworkcomponents,e.g.high-degreenodesorhigh-loadedges,aretargeted;thesefailuresarereferredtoasattacks).Inthiscontext,theimportantissuesoferrorandattacktolerance(i.e.,theabilityofanetworktoremainoperationaleveninthepresenceofrandomortargeteddisruptions)needtobeefcientlyaddressedinthedesignandenhancementofmodernnetworkinfrastructures. Itshouldbenotedthatmanyreal-worldnetworksfollowcertainpatternsintheirdegreedistributions,andtheycanoftenbemodeledaspower-laworuniformrandomgraphs.Inparticular,manypubliclyavailabledatasetsonreal-worldnetworkinfrastructureconnectivitypatterns(e.g.,telecommunications/internet,airtransportation,etc.)suggestthatthepower-lawmodelisapplicabletocharacterizingthesenetworks.Someempiricalstudiesoferrorandattacktoleranceofpower-lawanduniformrandomgraphshavebeendoneinthepast(e.g.,[ 2 ]);however,comprehensiveexplicittheoreticaljusticationsandprovablyoptimalstrategiesforrobustnetworkdesignand 39

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enhancement(thatwouldsimultaneouslyguaranteeerror/attacktolerance,lowdiameter,andcostefciency)areyettobedeveloped. Ingeneral,inalotofpracticalapplicationsoneneedstoconsiderbothnodefailuresandedgefailures.Inmanysituations,edgefailuresaretheonesthatreceivemoreattention.Inparticular,thewell-knownN)]TJ /F5 11.955 Tf 11.97 0 Td[(1robustnesscriterionstatingthatanetworkshouldremainoperationalafterafailureofanyoneedgeisusedincertainapplicationareas,suchastheanalysisofpowergrid.Fromamathematicalperspective,edgefailuresdonotreducethesize(numberofnodes)oftheresidualnetwork;therefore,ifalarge-scalenetworkisrobustwithrespecttomultiple(sayuptoR)edgefailures,itguaranteesthatnoneofthenodesinthisnetworkwouldbecomeisolated.ThiscanbereferredtoastheN)]TJ /F3 11.955 Tf 12.55 0 Td[(Rrobustnesscriterion,or,intheterminologyusedlaterinthischapter,aweakedgeattacktolerancepropertyoflevelR.Intheanalysisbelow,weaddressbothedgeattacktoleranceandnodeattacktolerancecharacteristicsofdifferenttypesofnetworks.Inourdenitionsofattacktolerance,werequirethatanetworkisguaranteedtoatleaststayconnectedafteranattackonanyoneormultipleedges/nodes.Furthermore,weimposemorerestrictiverequirementsonnetworkconnectivitypatternsthatarereferredtoasstrongattacktolerance. Inthischapter,weaddresstheaforementionedissuesfromarigorousmodelingandoptimizationperspective.Inparticular,ourgoalistodevelopoptimalstrategiesfornetworkdesignandenhancementthatexplicitlytakeintoaccountcertainrobustness/attacktolerancecriteria(thatisformallydenedbelow),aswellasthetotalcostofconstructinganewnetworkorenhancing(upgrading)theexistingnetwork. Inabasicnetworkdesignproblem,nnodesneedtobeoptimallyconnectedbyasetofarcs/edgessothatthetotalarc/edgeconstructioncostisminimized.Althoughtheconstructioncostofeachpossibleedgecanbedifferentanddependonmanypracticalfactors,thetotalcostprimarilydependsonthenumberofconstructededges.Clearly,toensuretheoverallconnectivityoftheconstructednetwork,oneneedstobuildatleast 40

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n)]TJ /F5 11.955 Tf 12.72 0 Td[(1edges(hereandfurtherweassumeasimpleundirectedgraph,i.e.,ifmultipleedgesconnectthesamepairofnodes,thentheyarerepresentedasasingleedge,andalltheedgesareundirected,butdirectedarcscanalsobeincorporatedintoouranalysis).Twoextremecasesofpossibleconnectednetworkcongurationswithnnodesandn)]TJ /F5 11.955 Tf 12.45 0 Td[(1edgesareachain(allnodesconnectedconsecutively)andastar(alsoreferredtoasahub-and-spokecongurationwithonecentralhubnode),andanyotherconnectedcongurationwithn)]TJ /F5 11.955 Tf 12.19 0 Td[(1edgeswouldgenerallybeaspanningtreeintheconsiderednetwork.Whileneitherofthesecongurationsisattack-tolerant,thehub-and-spokeconguration(anditsmodications)isoftenpreferredinpracticeandusedinavarietyofapplications,sinceithasalowdiameter,i.e.,thelengthofthepath(numberofedges)betweenanytwonodesinthisnetworkisatmost2.Lowdiameter(e.g.,asmallnumberofintermediarynodesandedgesbetweenanypairofnodes)isimportantfordifferenttypesofnetworks;therefore,itisoftenconsideredasoneofthekeyrequirements.Anetworkclusterthatbydenitionisexplicitlyguaranteedtohaveadiameterofatmostkisreferredtoasak-club.Clearly,anyhub-and-spokestructurewithnnodesandn)]TJ /F5 11.955 Tf 11.98 0 Td[(1edgesisa2-club;therefore,a2-clubcanbenaturallyconsideredasacost-efcientconnectednetworkstructurethatalsosatisesthelow-diameterrequirement.However,asitcanbeeasilyseen,a2-clubcanbeveryvulnerabletonodeoredgefailures;hence,itusuallyhaspoorattacktoleranceproperties(thisisespeciallytrueforthehub-and-spokeconguration).Since2-clubsstillhaveattractivepropertiesofbothcostefciencyandlowdiameter,itisreasonabletoattempttodevelopanetworkcongurationthatpreservestheseadvantagesofa2-club,butatthesametimeisguaranteedtoremainconnected(oreventoremaina2-club)ifoneormultiplenodesoredgesfailduetoattacksorerrors.Itturnsoutthatsuchefcientnetworkcongurationsexist;moreover,theycanbeproventobeoptimalintermsofnecessaryandsufcientconditionsforsatisfyingcertainrobustnessandattacktolerancecharacteristics. 41

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Particularly,weconsidertwotypesofattacktoleranceproperties,whicharereferredtoasweakandstrongattacktolerance.Wesaythatanetwork(k-club)hasaweakattacktolerancepropertyifitremainsconnectedafterthedeletionofanyonenodeoredge,andanetwork(k-club)hasastrongattacktolerantpropertyifitremainsak-clubafterthedeletionofanyonenodeoredge.Wealsodeneweak/strongattacktoleranceoflevelRiftheaforementionedrequirementsholdafterthedeletionofanyRnodes/edges.Althoughthestrongattacktolerancepropertymaybemoreusefulinpractice,constructingthistypeofanetworkmaybemoreexpensive,sinceitmayrequiremoreedges.Aweakattacktolerancepropertymaybesufcientifoneexpectsonlytemporarynode/edgefailures,anditisenoughtojustpreservethebasicconnectivityofanetwork.Weprovethatanynetworkcongurationthatsimultaneouslysatisesconstructioncostefciency,lowdiameter(specically,diamG2),andedgeattacktolerancerequirementshastobeoneofthestructuresthatwedescribebelow(2-club/2-coreforweakattacktolerance,andR-robust2-clubforstrongattacktolerance). Althoughthesetheoreticalresultsgiveusbetterunderstandingoftherequirednetworkstructureforaweakorstrongattacktolerantproperty,theydonotprovidetheexactsolutionsforthecorrespondingoptimalnetworkdesignproblems.Forexample,ifanyonewantstooptimallydesigna2-club,whichremainsa2-clubafterthedeletionofanyR)]TJ /F5 11.955 Tf 12.42 0 Td[(1edges(thatis,levelR)]TJ /F5 11.955 Tf 12.42 0 Td[(1strongattacktolerance),theobtainedtheoreticalresultsindicatethatithastobeanR-robust2-club.Butthetheorydoesnotyetprovideanyinformationabouttheminimumnumberofedgesrequiredtodesignthistypeofnetwork.Thus,thenextstepistondthemostcost-efcientR-robust2-clubintermsoftotalconstructioncost.ApreviousstudyshowsthatthisproblemisgenerallyNP-hard. However,itturnsoutthatifanyonewantstodesignanoptimalR-robust2-club,andtheconstructioncostforeveryedgeisthesame,thenthereexistsanexactanalyticalsolutionoftheunderlyingoptimizationproblem.InSection 3.5 weprovideaproof 42

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ofthisfactandpresentoneoftheexactsolutionswhichisveryintuitiveandeasytoimplement.WeshowthatthisnetworkcongurationhasonlyO(Rn)edges.Thefactthatthisnetworkcongurationhasalowdiameter,astrongattacktolerancepropertyoflevelR)]TJ /F5 11.955 Tf 12.69 0 Td[(1(whereRisauser-denedparameter),andonlyalinearonnnumberofedges(O(Rn)=O(n)foranyxedR,withR<
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structure,thatis,theremainingnetworkstillhasdiameter2.Thisprovidesasignicantadvantageintheapplicationswheretheexistenceofshortandreliablepathsbetweenanytwonodesisofcriticalimportance(i.e.,inmilitarycommunicationnetworks).ThisalsoclearlydistinguishesthedenitionofR-robustk-clubs(andR-robust2-clubs)fromthewell-knowndenitionofaK-connectedgraph,whichalsopreservestheconnectivityafteranyK)]TJ /F5 11.955 Tf 12.3 0 Td[(1networkcomponentsaredestroyed,butitdoesnotimposeanyexplicitrestrictionsonthediameteroftheoriginalandtheremainingnetwork.Inthecaseoflarge-scalenetworks,R-robustk-clubnetworkdesignswouldprovidesubstantialadvantagescomparedtoK-connectednetworkdesigns,primarilyduetothefactthatR-robustk-clubsexplicitlyhelpavoidexcessivelylongpathsbetweennetworknodes.Figures 3-1A 3-1B giveasimpleillustrativeexampleofthedifferencebetweentheaforementioneddenitions. Severalpreviousstudiesattemptedtoaddresstheissueofconstructinganetworkwithsmalldiameterandtheexistenceofdisjointpathsbetweennetworknodes.In[ 12 ]and[ 13 ]theauthorsconsidertheproblemsofconstructingaK-connectedgraphwithminimumnumberofedgesandquasiminimaldiameter.However,inthedenitionofquasiminimaldiameter,thediameteroftheconstructedgraphdependsonthenumberofnodesn,whereasinthecaseofanR-robustk-club,thediameterkdoesnotdependonanyotherparameters.Moreover,theconstructedK-connectedgraphwithquasiminimaldiameterdoesnotnecessarilypreservethesamediameterafterthedeletionofuptoK)]TJ /F5 11.955 Tf 12.95 0 Td[(1nodes/edges(althoughitdoespreservethebasicconnectivity).Inarecentstudy[ 5 ],theauthorsconsideredtheproblemofndingaminimumcostsubgraphofGsuchthatfortwospeciednodessandtthereareatleastRedge-disjointpathsoflengthatmostkandobtainedtheresultsforcertainspecialcases.Adrawbackofthatproblemsetupisthatthedisjointpathsarefoundonlybetweenthegiventwonodessandt,whereasintheproblemsconsideredinthis 44

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chapter,weimposetherequirementthatanytwonodesareconnectedbymultipledisjointpathsoflengthatmostk. Intheterminologyusedinthisdissertation,werefertotheattacktolerancecharacteristicssimilartothoseofK-connectedgraphsastheweakattacktoleranceproperty,sinceitonlypreservesthebasicconnectivity,butdoesnotnecessarilypreservethelowdiameter,whereasthecasewhenagraphnotonlystaysconnected,butalsopreservesalowdiameter,isreferredtoasthestrongattacktoleranceproperty.Thesepropertiesareformallydenedbelow. Denition1. AconnectedgraphG(V,E)hasaweakedge(node)attacktolerancepropertyifitstaysconnectedafterthedeletionofanyoneedge(node). Denition2. AconnectedgraphG(V,E)withdiameterkhasastrongedge(node)attacktolerancepropertyifitstaysconnectedandmaintainsdiameterkafterthedeletionofanyoneedge(node). Asanaturalextensionofthesedenitions,onecanalsodenetheweak/strongattacktolerancepropertyoflevelRasfollows: Denition3. AconnectedgraphG(V,E)withdiameterkhasaweak/strongedge(node)attacktolerancepropertyoflevelRifitstaysconnected(andinthecaseofstrongattacktolerancemaintainsdiameterk)afterthedeletionofanyRedges(nodes). Obviously,inthisterminology,anyK-connectedgraphhasaweaknode/edgeattacktolerancepropertyoflevelK)]TJ /F5 11.955 Tf 12.59 0 Td[(1.However,aK-connectedgraphdoesnotgenerallyhaveanyguaranteedstrongattacktoleranceproperties.Investigatingstrongattacktolerancepropertiesofvarioustypesofnetworkcongurationsisclearlyasubjectthatneedstobeanalyzedfromarigorousperspective.Forinstance,accordingtotheaboveproposition,anyR-robust2-clubhasastrongnode/edgeattacktolerancepropertyoflevelR)]TJ /F5 11.955 Tf 12.61 0 Td[(1.AlessobviousfactthatisprovenlaterinthischapteristhatanR-robust2-clubistheonlypossiblenetworkcongurationofdiameter2thathasastrongedgeattacktolerancepropertyoflevelR)]TJ /F5 11.955 Tf 11.95 0 Td[(1. 45

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Furtherinthischapter,weproveseveralinterestingtheoreticalresultsonweakandstrongattacktolerancepropertiesoftheaforementionedcliquerelaxationsandspecicallyconcentrateonR-robust2-clubs,sincetheypossessauniquecombinationofattacktoleranceandlowdiameter.Inaddition,weanalyticallyshowthatconstructing2-club/2-cores,orR-robust2-clubsisprovablycost-efcientcomparedtoothercliquerelaxations,whichmakesthesecongurationsattractivefrommultipleperspectives. 3.2AttackToleranceCharacteristicsofCliquesandCliqueRelaxations Beforeweproceedwithfurtherdiscussionaboutcliquerelaxations,weshouldnotethatoutofallnetworkdesigncongurationsrepresentedbysimpleundirectedgraphs,onlyaclique(i.e.,acompletelyinterconnectednetworkwithnnodesandn(n)]TJ /F5 11.955 Tf 12.63 0 Td[(1)=2edges)isguaranteedtobeattack-tolerantregardlessofanyexternalparametersotherthann,sinceitwillremainacliqueifanynumberofnodes(upton)]TJ /F5 11.955 Tf 12.29 0 Td[(1)isdeleted,anditwillalsostayconnectedandhavediameterofatmost2ifanyn)]TJ /F5 11.955 Tf 12.4 0 Td[(2edgesfail.Notethatintermsoftheabovedenitionsacliquecanalsobeviewedasan(n)]TJ /F5 11.955 Tf 12.41 0 Td[(1)-robust2-club.Thus,ithasastrongattacktolerancepropertyoflevel(n)]TJ /F5 11.955 Tf 12.04 0 Td[(2)(intermsofnodeandedgeattacks).However,asithasbeenpointedoutabove,cliquesareoftentoorestrictiveand/orexpensivetoconstruct. Onthecontrary,abasicnetworkconnectivityrequirementisnotrestrictiveenoughtoguaranteeanyattacktoleranceproperties.Inattempttoprovideatradeoffbetweentherobustnessofcliquesandthecostefciencyofsparselyconnectednetworks,theaforementionedconceptsofcliquerelaxations(-cliques,k-plexes,k-clubs)havebeenintroduced.Inthissection,weexaminethesecliquerelaxationsintermsoftheirattacktolerancecharacteristics.Ourgoalistondoutspecicvaluesoftheparametersofthecliquerelaxationsthatguaranteeaspeciclevelofattacktolerance.Furtherinthissection,weonlyconsidertheweakattacktolerancethatensuresthattheremainingstructureisconnectedafterthedeletionofseveralnodes/edges. 46

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Ingeneral,itisnotclearifanyoftheconsiderednetworkcongurationscanbeguaranteedtosimultaneouslybeattack-tolerant,satisfythesmalldiameterrequirement,andhavealowconstructioncost.However,someresultsabouttheirweakattacktoleranceisprovenbelow. 3.2.1AttackToleranceofDensity-BasedCliqueRelaxations(-cliques) Thepropositionbelowshowsthatingeneral-cliques(<1)arenotattacktolerant.Specically,foranypossibleweshowthatthereexistssucha-cliquesothataremovalofonlyoneedgeornodedestroysitsconnectivity.Inadditiontothedensityrequirement,wealsorequirea-cliquetobeaconnectedgraph(asin[ 1 ]). Proposition3.1(-cliqueattacktolerance). 8<19a-cliqueG(V,E)suchthat 9e2E:G(V,Ene)isdisconnected. 9u2V:G(Vnu,E)isdisconnected. Proof. Theproofisbasedontheconstructinganexampleofaconnectedgraph(foranypossible<1)thatisa-clique,butlosesconnectivityaftertheremovalofonespecicedgeornode.Letusxsome(arbitrarilychosen)valueof(-cliqueparameter).NowconsideracliqueG(V,E)withjVj2=(1)]TJ /F4 11.955 Tf 12.57 0 Td[(),forsimplicity,weusen=jVjhenceforth.Thechoiceofthesizenisjustiedlater.Pickaparticularnodevandremoveanyn)]TJ /F5 11.955 Tf 12.4 0 Td[(2outofn)]TJ /F5 11.955 Tf 12.41 0 Td[(1edgesgoingfromthisnodeanddenotetheremainingstructureasG(V,E).Notethatthedegreeofnodevis1.Thus,ifweremovetheonlyremainingedgeoriginatingfromv(saythisedgeise=(v,u))thenthegraphG(V,Ene)isdisconnected.Also,ifweremoveanodeu,whichistheonlyonenodeconnecteddirectlytov,thenG(Vnu,E)isalsodisconnected. Tocompletetheproof,wealsoneedtoshowthatG(V,E)isindeeda-clique(thatis,ithasatleastn(n)]TJ /F7 7.97 Tf 6.59 0 Td[(1) 2edges).SincejEj=n(n)]TJ /F5 11.955 Tf 12.1 0 Td[(1)=2)]TJ /F5 11.955 Tf 12.1 0 Td[((n)]TJ /F5 11.955 Tf 12.1 0 Td[(2)andn2=(1)]TJ /F4 11.955 Tf 12.1 0 Td[(),or1)]TJ /F5 11.955 Tf 11.95 0 Td[(2=n,thenjEj=n(n)]TJ /F5 11.955 Tf 11.96 0 Td[(1) 2)]TJ /F5 11.955 Tf 11.95 0 Td[((n)]TJ /F5 11.955 Tf 11.96 0 Td[(2)=1)]TJ /F5 11.955 Tf 13.25 8.08 Td[(2(n)]TJ /F5 11.955 Tf 11.95 0 Td[(2) n(n)]TJ /F5 11.955 Tf 11.95 0 Td[(1)n(n)]TJ /F5 11.955 Tf 11.96 0 Td[(1) 2 47

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1)]TJ /F5 11.955 Tf 13.25 8.09 Td[(2(n)]TJ /F5 11.955 Tf 11.95 0 Td[(1) n(n)]TJ /F5 11.955 Tf 11.95 0 Td[(1)n(n)]TJ /F5 11.955 Tf 11.96 0 Td[(1) 2=1)]TJ /F5 11.955 Tf 13.25 8.09 Td[(2 nn(n)]TJ /F5 11.955 Tf 11.96 0 Td[(1) 2n(n)]TJ /F5 11.955 Tf 11.96 0 Td[(1) 2 Thus,G(V,E)isnotaweakly-clique,whichendstheproofofthisproposition. 3.2.2AttackToleranceofDegree-BasedCliqueRelaxations(k-plexes) Thenextpropositionshowsthatforcertainvaluesoftheparameterkofthek-plex(kn 29k-plexG(V,E)suchthatG(V,E)isdisconnected. Proof. First,notethatifweremoveanyedgefromk-plex,thentheremaininggraphisa(k+1)-plex,andifweremoveanyonenodefromak-plex,thentheremaininggraphisstillak-plex,butcontainsonlyn)]TJ /F5 11.955 Tf 11.65 0 Td[(1nodes.Formally,ifagraphG(V,E)isak-plex,then 8v2V,G(Vnv,E)isak-plex 8e2E,G(V,Ene)isa(k+1)-plex Thus,toproveourpropositionitisenoughtoshowthatanyk-plexwithk=n 2isconnected,andfork=n 2+1thereexistk-plexthatisdisconnected. Therststatementcanbeprovedbycontradiction.Supposethatthereexistsak-plexG(V,E)withk=n 2,whichisdisconnected.Thus,wecandivideitintotwosubgraphsG(A,Ea)andG(B,Eb),suchthatV=A[B,E=Ea[Eb,and8a2A,b2B,(a,b)=2E.Inotherwords,subgraphsG(A,Ea)andG(B,Eb)areisolated. 48

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SincejAj+jBj=jVj=n,thenoneofthesubgraphsshouldhavenomorethann 2numberofnodes;hence,themaximumdegreeofanyofitsnodecannotbegreaterthanorequalton 2.But,bydenitionofk-plex,anyitsnodehasadegreeofatleastn)]TJ /F3 11.955 Tf 11.95 0 Td[(k=n)]TJ /F8 11.955 Tf 11.95 9.69 Td[(n 2=n 2.Hence,allk-plexeswithk=n 2areconnected. Now,weshowthatthereexistsak-plexwithk=n 2+1thatisdisconnected.Considertwocompletegraphs(cliques)G(A,Ea)andG(B,Eb),wherejAj=n 2,andjBj=n 2.Then,obviouslyagraphG(V,E),whereV=A[B,E=Ea[Ebisadisconnectedk-plexwithk=n 2+1. Finally,itshouldbenotedthatintermsofcostefciency,itcanbeeasilyseenthatconstructinganoptimalquasi-cliqueforanyxed,orak-plexforanyxedkwouldingeneralrequireO(n2)edges,whichmakesthesecongurationsasymptoticallyasexpensiveasaclique.However,thisisnotthecaseforthediameter-basedcliquerelaxations(k-clubs),sinceevenfork=2ak-clubonnnodescanbeconstructedusingonlyn)]TJ /F5 11.955 Tf 12.71 0 Td[(1(O(n))edges.Inthenextsection,weinvestigatetheissueofguaranteedattacktoleranceofk-clubs.Clearly,ifonecanconstructanetworkthatrequiresonlyO(n)edges,haveasmalldiameter(ideally,diameter2),andhaveaguaranteedstrongattacktoleranceproperty,thiswouldbebenecialfrommultipleperspectives.Althoughtheserequirementsmayseemtoorestrictive,itturnsoutthatthesetypesofnetworkscanbedesigned.Thefollowingtwosectionsaddressmultipleaspectsoftheseproblems. 3.3NecessaryandSufcientConditionsforAttackToleranceofDiameter-BasedCliqueRelaxations(k-clubs) Inthissection,weanalyzeattacktolerancepropertiesofthethirdtypeofaforementionedcliquerelaxations:k-clubs.Clearly,k-clubs(inparticular,2-clubs)dohaveattractiveproperties,suchassmalldiameterandconstructioncostefciency,whichmakesthesecongurationsthesubjectofspecialinterestinthischapter.Intermsoftherequired 49

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numberofedges,designinganetworkwithasmalldiameter(i.e.,diameter2)wouldrequireO(n)insteadofO(n2)edges(asinthecaseofcliques,quasi-cliques,andk-plexes).However,thesestructuresgenerallylackattacktolerancepropertieswithrespecttonodeandedgeattacks(consider,forinstance,ahub-and-spokeconguration,whichisanoptimal2-clubintermsoftheminimalnumberofconstructededges). Despitethelackofattacktolerancepropertiesinthegeneralcase,weshowthatincertainspecialcases,weakandstrongattacktolerancepropertiesofk-clubscanbetheoreticallyguaranteed.Weformulatesufcientconditionsforak-clubtobetoleranttonodeandedgeattacks;moreover,wewillshowthatinthecaseofa2-club,necessaryandsufcientconditionsforguaranteededgeattacktolerancecanbeproven.Specically,wewillshowthattheaforementionedrecentlyintroducedconceptofanR-robustk-clubcanindeedprovideatleastthesufcientconditionsforstrongattacktolerancewithrespecttomultiplenode/edgefailures.Moreover,weprovethatinthecaseof2-clubs,anR-robust2-clubisanecessaryandsufcientcongurationthatisguaranteedtobestronglyattacktolerant(i.e.,maintaintheconnectivityanddiameter2)afteranyR)]TJ /F5 11.955 Tf 12 0 Td[(1edgesaredestroyed.Wealsoformulaterelatednecessaryandsufcientconditionsforweakattacktoleranceof2-clubs,whichcanbeutilizedwhenonlytheconnectivityoftheresidualnetworkafteranattackisthepropertyofinterest. Itturnsoutthattheconditionsderivedbelowareeasilyinterpretableandcanbeviewedasequivalentrepresentationsofattacktolerantk-clubs.Inthecaseof2-clubs,theyalsocanbeeasilyincorporatedintotheoptimizationproblemsfordesigningaminimum-costweaklyorstronglyattacktolerant2-club.Particularly,weshowthattorequirea2-clubtohaveaweakedgeattacktoleranceproperty,itisequivalenttorequirea2-clubtoalsobea2-core(i.e.,thedegreeofeachnodemustbeatleast2).Fora2-clubtohaveastrongedgeattacktoleranceproperty,itisequivalenttorequirea2-clubtobea2-robust2-club.Further,weformulateandproverelatedrequirementsforweak 50

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andstrongnodeattacktoleranceof2-clubs,aswellastherequirementsfork-clubsfork>2. Proposition3.3(Weakattacktolerancerequirementfor2-clubs). LetG(V,E)bea2-club,then8e2E,G(V,Ene)isconnectedifandonlyifG(V,E)isalsoa2-core. Proof. First,weprovethenecessarycondition,i.eshowthatif8e2E,G(V,Ene)isconnected,thenG(V,E)isa2-core.SupposethatG(V,E)isnota2-core.Bydenition,everynodein2-corehasadegreeofatleast2.Then,thereshouldexistatleastonenode(sayi)withadegree1.Letsayitisconnectedtosomeothernodej.Ifweremoveanedge(i,j)fromthegraphG(V,E),thenitwillhaveanisolatednodei,soG(V,Ene)isnotconnected.Thiscontradictionendstheproofofthenecessarycondition. Second,thesufcientconditionisifG(V,E)isa2-core,then8e2E,G(V,Ene)isconnected.Assumethatedge(i,j)isdeleted.ToprovethatG(V,E)isconnectedweneedtoshowthatthereexistsapathbetweenanypairofnodes(p,q):p,q2V. Considerthreedifferentcases: Twopairofnodes(p,q)and(i,j)havenocommonnodes,i.ep,q6=i,p,q6=j.SinceG(V,E)isa2-club,thenthereexistsapathoflengthatmost2betweennodespandq.Obviously,thispathcannotincludetheedge(i,j);therefore,nodespandqremainconnected. Twopairofnodes(p,q)and(i,j)haveonlyonecommonnode,letsayp=i,q6=j.SinceG(V,E)isa2-core,thanthedegreeofnodep=iisatleast2,sothereexistsatleastonenodes6=j,sothat(p,s)2E.Twopairofnodes(s,q)and(i,j)havenocommonnodes.Bytheproofofthepreviouscase,nodessandqareconnected.Thus,nodespandqareconnected.Note,thatthereexistsapathoflengthatmost3betweenpandq. Twopairofnodes(p,q)and(i,j)havetwocommonnodes,i.ep=i,q=j.SinceG(V,E)isa2-core,thenthedegreeofnodep=iisatleast2,sothereexistsatleastonenodes6=j,sothat(p,s)2E.Twopairofnodes(s,q)and(i,j)haveonlyonecommonnode,sobytheproofofthepreviouscase,nodessandqareconnectedwithapathoflengthatmost3.Then,nodespandqareconnectedwithapathoflengthatmost4. 51

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Thesecasesconsiderallpossiblepairsofnodespandq;thus,thetheoremisproved. ItshouldbenotedthatduringtheproofofthetheoremwealsoshowedthatifG(V,E)isa2-cluband2-core,thenafterdeletionofanyedge,itnotonlyremainsconnectedbutalsobecomesa4-club,inotherwords,theresidualnetworkmaintainsarelativelysmalldiameter,althoughitincreasescomparedtotheoriginalnetwork. Now,assumethattheexistenceofashortpathbetweenanytwonodesisacrucialpropertyforanetworkfunctioninginanadverseenvironment.Particularly,wewantanetworktomaintainconnectivityand2-clubstructureincaseofanyedgefailure.Thisattacktolerantpropertymaybecriticalincommunicationnetworkswheretheexistenceofshortpathsbetweenallnodeswithfewintermediariesisanimportantproperty.Thepropositionbelowprovidesnecessaryandsufcientconditionsthatthisnetworkshouldsatisfyinordertohavethisdesiredproperty.Also,inadditiontoedgeattacktolerancetheproposedcongurationisguaranteedtobenodeattacktolerant. However,thepropositionbelowcannotbedirectlygeneralizedtothecasewithR-robustk-clubsfork>2.Specically,R-robustk-clubs(k>2)onlyprovidesufcientconditionsforstrongattacktolerance,whichisdemonstratedlaterinthissection. Proposition3.4(Strongattacktolerancerequirementfor2-clubs). LetG(V,E)bea2-club,then 8e2E,G(V,Ene)isa2-clubifandonlyifG(V,E)isa2-robust2-club; 8v2V,G(Vnv,E)isa2-clubifG(V,E)isa2-robust2-club. Proof. TheproofthatifG(V,E)isa2-robust2-club,then8e2E,G(V,Ene)isa2-club,or8v2V,G(Vnv,E)isa2-clubfollowsimmediatelybydenition,sinceitrequirestheexistenceofatleast2disjointpathsoflengthatmost2betweenanypairofnodes.Obviously,thesetwopathscannothaveacommonedge,ornode.Thus,afterdeletionof 52

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anyedgeornodefroma2-robust2-clubtherewillbeatleastonepathoflengthatmost2betweenanypairofnodes,i.e.theremainingstructureisstilla2-club. Theotherstatementthatif8e2E,G(V,Ene)isa2-club,thenG(V,E)isa2-robust2-clubweshowbycontradiction.SupposethatG(V,E)isnota2-robust2-club;thus,thereexistsatleastonepairofnodeswhichisconnectedwithonlyonepathoflengthatmost2(theremayexistotherpathsbutwithgreaterlength).Thus,ifwedeleteanyedgefromthispath,thentheremainingstructurelosesa2-clubproperty.Thiscontradictionendstheproofoftheproposition. Thereadermightnoticethatwedonotclaimthatif8v2V,G(Vnv,E)isa2-club,thenG(V,E)isa2-robust2-club.Actually,itisnottrueingeneral.Figure 3-1A showsa2-clubwhichsatisesthisproperty,butisnota2-robust2-club,whichwasmentionedabove. Thepropositioncanbeeasilyextendedtothemoregeneralcasewhereanetworkisrequiredtomaintainconnectivityanddiameter2afteranyRedgesfailures.ThecorollarybelowshowsthatR-robust2-clubsnotonlysatisfythisrequirement,butalsotheyareoptimalinthiscase.Moreover,R-robust2-clubsprovidemixed(edgeand/ornode)attacktoleranceproperties. Corollary1. LetG(V,E)bea2-club,then8ER=fe1,...,eRgE,G(V,EnER)isa2-clubifandonlyifG(V,E)isan(R+1)-robust2-club. Corollary2. LetG(V,E)bea2-club,then8El=fe1,...,elgEand8Vm=fv1,...,vmgVsuchthatl+m=R,G(VnVm,EnEl)isa2-clubifG(V,E)isan(R+1)-robust2-club. TheproofsofthesecorollariesfollowimmediatelyextendingtheproofofProposition 3.4 AlthoughR-robustk-clubsprovidestrongattacktoleranceproperties,theyarenotoptimalinthesensethattheymaystillbequiterestrictive.ThecorollarybelowstatesthatR-robustk-clubs(k>2)provideonlyasufcientconditionforstrongedgeattack 53

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attacktoleranceoflevelR)]TJ /F5 11.955 Tf 12.81 0 Td[(1.Figure 3-2 showsanexampleofa4-clubthathasastrongedgeattacktoleranceproperty,butisnota2-robust4-club(therearenointernallynodedisjointpathsbetweenanypairofnodes). Corollary3. LetG(V,E)beak-club,then:8El=fe1,...,elgEand8Vm=fv1,...,vmgVsothatl+m=R,G(VnVm,EnEl)isak-clubifG(V,E)isan(R+1)-robustk-club. TheproofofthatstatementfollowsimmediatelyfromthedenitionofanR-robustk-club. Althoughstrongattacktolerancepropertiesmightseemquiterestrictive,inthenextsectionweshowthatonecanconstructa2-club/2-coreusingonly3n)]TJ /F7 7.97 Tf 6.59 0 Td[(3 2edges,andfortheconstructionofanR-robust2-clubonlyRn)]TJ /F5 11.955 Tf 12.47 0 Td[((R(R+1)=2)edgesareenough.Wealsoshowthatbothofthesenumbersaretheexactlowerboundsonthenumberofedgestodesignthesenetworkcongurations.NotethatbothofthesenumbersaretheorderofO(n)(O(Rn))ratherthanO(n2)forthecliquesandothercliquerelaxationsmentionedabove.Moreover,theseresultsalsoshowthatanR-robustk-clubcanbeconstructedusingatmostO(Rn)edges,whichmakesthesecongurationsnotasexpensiveasonemayassume.Theseissuesareaddressedinmoredetailinnextsections. 3.4Linear0-1FormulationsofDesignandEnhancementProblemsfor2-clubswithGuaranteedAttackToleranceProperties Intheprevioussectionsweanalyzetheoreticalaspectsofattacktolerancepropertiesfordifferenttypesofnetworks,withaspecialemphasisondiameter-basedcliquerelaxations(k-clubs).AlthoughnetworkswithaspeciedupperboundkonthediametercanbeconstructedusingO(n)edges(i.e.,thehub-and-spokecongurationfork=2),asitwasmentionedabove,noattacktolerancepropertiescanbeguaranteedforthesenetworksinageneralcase.Sincewehaveshownthatspecicnetworkcongurationscanguaranteeweakandstrongattacktoleranceproperties,anatural 54

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questionthatariseshereishowtodesignthesenetworksataminimumcost.Clearly,ifallthepossibleedgeshavethesameconstructioncost,thenthetotalcostisessentiallyequivalenttothetotalnumberofconstructededges.Ifeachedge(i,j)hasadifferentconstructioncostcij,thenthetotalcostcanobviouslybeexpressedasP(i,j)cijxij,wherexijarebinaryvariablesdenotingwhetheredge(i,j)isconstructed.Theconstraintsoftheseoptimizationproblemsexpresstheconditionsforthedesignednetworktohaveweakorstrongattacktoleranceproperties. Itturnsoutthatfork=2theoptimizationproblemsofdesigninganoptimal2-club/2-coreandR-robust2-clubcanbeeasilyformulated,althoughtheobtainedlinear0-1formulationsrequireO(n3)entities,whichmakestheseproblemscomputationallychallenging.Itshouldbenotedthatfork>2thesituationisevenworse(bothcomputationallyandtheoretically),sincethetaskofformulatingtheproblemofoptimalR-robustk-clubnetworkdesign/enhancementisachallengebyitself.Intheremainderofthechapter,weconcentrateonthecaseofk=2duetothefollowingreasons:1)networkswithsuchasmalldiameterareattractiveinmanypracticalapplications(i.e.,inmilitarycommunicationnetworks,whereasmallnumberofintermediariesbetweenallthenodesisanimportantproperty);2)despitethefactthattheconsideredoptimizationproblemsarehardtosolveinthegeneralcase,itispossibletoderiveexactanalyticaloptimalsolutionsincertainspecialcasesfork=2. Laterinthissection,wepresentthelinear0)]TJ /F5 11.955 Tf 13.63 0 Td[(1problemformulationsfor2-club/2-coreandR-robust2-clubnetworkdesignproblems,aswellasaddresstheircomputationalcomplexity.Notsurprisingly,theseproblemsaregenerallyNP-hard,whichismentionedinthenextsubsection.However,laterinthischapterweshowthatboththeoptimal2-club/2-corenetworkdesignandtheoptimalR-robust2-clubnetworkdesignproblemshaveexactanalyticalsolutionswhenallcij'sarethesame,whichmakestheseproblemseasytosolveinthisspecialcase.Moreover,weshowthattheseanalytical 55

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optimalsolutionsarecost-efcient,whichalsomakestheminterestingfromapracticalperspective. 3.4.1ComputationalComplexityoftheConsideredProblems Therecentpaper[ 8 ]showedthatgivenagraphG(V,E)theproblemofndingasupersetofedgesE0EsuchthatthegraphG0=(V,E0)hasdiameternogreaterthan2(or2-club)isNP-hard.IntheproblemsdescribedabovewerequirethenewgraphG0=(V,E0)notonlytohavediameternogreaterthan2,butalsotosatisfyotherrequirements,i.e.,theminimumdegreeofeverynodeforthe2-core,andonthenumberofpathsoflengthatmost2betweenanypairofnodes.TheNP-hardnessoftheconsideredproblemsfollowsfromthefactthattheseproblemsincludetheoneconsideredin[ 8 ]asaspecialcase,althoughwedonotprovidethedetailedNP-hardnessproofsineachspeciccase,sincethemainfocusofthisworkisonanalyticalsolutionsoftheconsideredproblemsratherthanonadetailedcomplexityanalysis. Next,weconsidertwotypesofproblems,whichareessentiallythesamefromtheformulationandcomputationalcomplexityperspective: ForagivensetofnodesVandanemptysetofedges,optimallydesignanetworkG(V,E)withdiameteratmost2andspeciedweak/strongattacktoleranceproperties; ForagivenexistingnetworkG(V,E),optimallyenhancethisnetwork(i.e.,addasupersetofedgesE0E)suchthattheresultingnetworkhasadiameterofatmost2andspeciedweak/strongattacktoleranceproperties. 3.4.2OptimizationProblemFormulations SupposethatwehaveanexistingnetworkrepresentedbyasimpleundirectedgraphG(V,E)(jVj=n)witha0-1adjacencymatrixA(A=(aij)ni,j=1).Then,ourgoalistoenhancethisnetworkwithsomeadditionaledges,sothatthenewnetworkwillbe2-club/2-core,orR-robust2-clubdependingonthedesiredtypeofattacktolerance.Assumealsothatanadditionofnonexistingedge(i,j)betweennodesi 56

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andjisassociatedwithaxedcostcij.Theobjectiveistominimizetotalcostofsuchanetworkenhancement.NotethatifmatrixAisanullmatrix,thenwehaveaproblemofoptimalnetworkdesign.Letxij,i,j=1,...,nbeabinaryvariablerepresentingthedecisionifedge(i,j)isconstructed.AmatrixX(X=(xij)ni,j=1)ofvaluesofthedenedvariablesrepresentsanadjacencymatrixofthedesirednetwork.Inorderfortheconstructednetworktobe2-club/2-core,twosetsofconstraintsneedtobesatised.Therstoneisthateverypairofnodesiandjhastobeconnecteddirectly,orthroughanintermediarynode.Itcanbewrittenasxij+nXk=1xikxkj1 foreverypair(i,j),thus,wehaven(n)]TJ /F5 11.955 Tf 12.09 0 Td[(1)=2constraintsintherstset.Thesecondoneinsuresthateverynodeihasatleastdegree2.ItcanbewrittenasnXi=1xij2 foreverynodej,thus,wehavenconstraintsinthesecondset.Theoptimizationproblemformulationisasfollows. ProblemA(optimal2-club/2-corenetworkdesign/enhancement)minxij,i6=j=1,...,nnXi=1nXj=i+1cijxij subjectto xij+nXk=1xikxkj1,nXi=1xij2,xij=xji,xijaij, 57

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xij2f0,1g, fori6=j=1,...,n. Next,weconsidertheproblemformulationforoptimaldesign/enhancementofR-robust2-club.Bydenition,anR-robust2-clubrequirestheexistenceofatleastRdifferentpathsoflengthatmost2betweenanypairofnodes(i,j).Thisrequirementcanbewrittenasxij+nXk=1xikxkjR foreverypairofnodes(i,j).Theoptimizationproblemcanbeeasilyformulatedasfollows: ProblemB(optimalR-robust2-clubnetworkdesign/enhancement)minxij,i6=j=1,...,nnXi=1nXj=i+1cijxij subjectto xij+nXk6=i,j;k=1xikxkjR,xij=xji,xijaij,xij2f0,1g, fori6=j=1,...,n. Notethattheseproblemformulationsareabitredundant,butforsimplicity,wekeepthemthisway. TheR-robust2-clubandtheregular2-clubconstraintsinbothproblemsarequadratic.Thesimplestlinearizationmethodleadsustothefollowinglinearizedproblem ProblemB(linearized)minwijk,xijnXi=1nXj=i+1cijxij 58

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subjectto xij+nXk6=i,j;k=1wikjR,wikjxik,wikjxkj,wikjxik+xkj)]TJ /F5 11.955 Tf 11.96 0 Td[(1,xij=xji,xijaij,wikj,xij2f0,1g, fork6=i6=j=1,...,n. ThelinearizationofProblemAisthesame. Itshouldbenotedthatthesimplelinearizationtechniqueonthediameterconstraintsproposedhereworkswellfortheconsideredproblemswithk=2;however,itmaybenecessarytousemoreadvancedlinearizationtechniquesforrelatedproblemsinthegeneralcasewithk>2.Forinstance,anefcientlinearizationtechniqueformulti-quadratic0-1problemshasbeenproposedin[ 7 ].Acompactlinearizationprocedurefortherelatedmaximumk-clubproblem(k>2),whichndsthemaximumk-clubinagivennetwork,hasbeenproposedin[ 14 ]. 3.4.3IllustrativeExamples Asmentionedabove,theconsideredoptimizationproblemsareNP-hard,anditturnsoutthattheyarecomputationallychallengingevenformoderatesizenetworks.Sincethefocusofthisworkisontheoreticalfoundationsforlow-diameterattack-tolerantnetworks,ratherthanoncomputationalalgorithms,wehaveconductedillustrativecomputationalexperimentsforrelativelysmallvaluesofntodemonstratethesolutionstructuresoftheconsiderednetworkdesignandenhancementproblems.Forillustrative 59

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purposes(thatisclariedandrevisitedfromananalyticalperspectiveinthenextsection)weassumethateachedgehasthesameconstructioncost. Inourrstexampleweassumethattherequirednetworkneedstobeconstructedfromscratch,inotherwords,theinitialgraphG(V,E)hasanemptysetofedges.WeusedILOGCPLEX[ 21 ]softwaretorunoptimizationproblems,andPajek[ 23 ]todrawnetworks.Atthebeginning,weidentiedanoptimal2-club/2-coreinagraphwithanevenandoddnumberofnodesn(n=10,11)usingthelinearizedformulationofProblemAproposedabove.WehavealsofoundoptimalR-robust2-clubsusingtheformulationofProblemBforn=10andR=1,2,3,4. Figures 3-3A and 3-3B showthesolutionsforoptimal2-club/2-coredesignforn=10,andn=11.Thesesolutionshaveaclearpatternwhichroughlycanbedescribedasfollows:onenode(hub)connectstoanyothernodesdirectly,andothernodesformpairs.Theoptimal2-club/2-corestructurecanbeviewedasanextensionofanoptimal2-club(Figure 3-4A ).Clearly,intheoptimal2-clubstructurethereisonenodeconnectingtoanyothernodedirectly(hub-and-spokeconguration).Thus,toupgradeanoptimal2-clubtoanoptimal2-club/2-coreoneneedstoconnectothernon-hubnodestoeachother,formingpairs.Suchanupgraderequiresonlyn)]TJ /F7 7.97 Tf 6.58 0 Td[(1 2additionaledgesandensuresthatthisnetworkhasaweakattacktoleranceproperty.Ingeneral,thispatternrequiresn)]TJ /F5 11.955 Tf 11.26 0 Td[(1+n)]TJ /F7 7.97 Tf 6.59 0 Td[(1 2edgestodesignanoptimal2-club/2-corewithnnodes.Althoughthisconclusioncomesempiricallyfromthesolutionsofoptimizationproblems,andthereisnoguaranteethatthesamepatternholdsforlargern,weformallyproveinthenextsectionthatnotonlythesenetworkstructuresareoptimalforlargern,butalsothatsuchanupgradeprocedurestillworks.Thus,anoptimal2-club/2-corehasasimplepattern,whichalsocanbeviewedasanupgradednetworkfromanoptimal2-club,i.e.,oneneedstoaddaspecicnumberofextraedgestoensurethattheupgradednetworkisoptimalamongallthepossible2-club/2-cores.Notethatanoptimal2-club/2-corehasonlyaweakedgeattacktoleranceproperty,whatmeansthatitjust 60

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staysconnectedifanyoneedgeisdestroyed.Theobservedpatternlacksanodeattacktoleranceproperty,i.e.,thenetworkwillbedisconnectedifthehubnodeisdestroyed. Figure 3-4 showsthesolutionsforoptimalR-robust2-clubdesignforR=1,2,3,4,andn=10.Thesesolutionsalsohaveeasilyinterpretablepatterns.IntheoptimalR-robust2-clubthereareRnodes(hubs)directlyconnectedtoanyothernodes,soRnodesinthisnetworkhaveadegreen)]TJ /F5 11.955 Tf 12.76 0 Td[(1andn)]TJ /F3 11.955 Tf 12.75 0 Td[(RothernodeshaveadegreeR.Notethatanupgradeprocedurecanalsobeclearlyidentiedhere.Toupgradeanoptimal(R)]TJ /F5 11.955 Tf 12.47 0 Td[(1)-robust2-clubtoanoptimalR-robust2-cluboneneedstoaddonlyn)]TJ /F3 11.955 Tf 11.73 0 Td[(R)]TJ /F5 11.955 Tf 11.73 0 Td[(1edgestransformingonenon-hubnodeintoahubnode.Ingeneral,thispatternrequiresnR)]TJ /F6 7.97 Tf 13.81 5.48 Td[(R(R+1) 2edgestodesignanoptimalR-robust2-clubonnnodes.Theseconclusionsarealsodrawnfromthepatternthatweobserveinoptimalsolutionsforthesespecicinstances,withnoguaranteethatthesamepatternholdsforlargern.Inthenextsection,wealsoprovideaformalproofthatundercertainconditionsthispatternrepresentstheexactoptimalsolutionfortheR-robust2-clubnetworkdesignproblemforanyvalueofn.Althoughareabletoprovethatitisthecasewhenn3R+p 5R2)]TJ /F7 7.97 Tf 6.59 0 Td[(4R 2,thisconditiondoesnotseemtobeveryrestrictiveandisusuallysatisedforreal-worldlarge-scalenetworks(n>>R).NotethatR-robust2-clubshavebothedgeandnodestrongattacktoleranceproperties,whichcanalsobeobservedfromtheseillustrations. Finally,Figure 3-5 showsthesolutionsforoptimalR-robust2-clubenhancementofanexistingnetworkforR=1,2,3.Forillustrativepurposes,werandomlygeneratedasmallpower-law(scale-free)1networkwith=0.5,andn=20.Aninitialnetwork(Figure 3-5A )has53edges.ThenwesolvedproposedoptimizationproblemsforoptimalR-robust2-clubenhancementofanexistingnetworkforR=1,2,3.Toenhancethisnetworktobea2-clubaddingtheminimumnumberofedges,oneneedstoaddonly 1Apowerlaw(scale-free)networkwithaparameterisanetworkwherethenumberofnodeswithadegreekisproportionaltok)]TJ /F13 7.97 Tf 6.58 0 Td[(. 61

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7edges(Figure 3-5B ,theaddededgesareinblack).Foranoptimal2-robust2-clubenhancement,theminimumnumberofedgestobeaddedis15(Figure 3-5C ),sobyaddingjust8moreedges(comparedtothe2-clubenhancement)weareablenotonlytoensurethattheenhancednetworkhasasmalldiameter,butalsotoguaranteethattheenhancednetworkhasastrongattacktoleranceproperty.Foranoptimal3-robust2-clubenhancement,theminimumnumberofedgestobeaddedis25(Figure 3-5D ).Itisworthpointingoutthattherequirednumberofnewedgesisrathersmallcomparedtotheoriginalnumberofedgesinthisnetwork.Notethattheoriginalnetworknotonlydoesnothaveanyattacktoleranceproperties,butitisnotevena2-club. 3.5ExactAnalyticalSolutionsforOptimal2-club/2-coreandR-robust2-clubNetworkDesignProblems Inthissection,wepresentformalproofsofsomeobservationsthatwementionedintheprevioussection.Itwaspointedoutthatthesolutionsobtainedforoptimal2-club/2-coreandR-robust2-clubnetworkdesignproblemshavewell-denedpatterns(networkcongurations)inthecaseswhenalledgeshavethesameconstructioncosts(whichobviouslyimpliesthattheminimizingthetotalconstructioncostisequivalenttominimizingthetotalnumberofedges).Thetheoremsbelowshowthatthesepatternsprovideexactanalyticaloptimalsolutionsfortheaforementionedproblems. Thersttheoremestablishesthelowerboundonthenumberofedgesinany2-club/2-core.Itturnsoutthatany2-club/2-coreshouldhaveatleast(3n)]TJ /F5 11.955 Tf 12.18 0 Td[(3)=2edges.Theensuingcorollaryconrmsthattheobservedpatternsofoptimal2-club/2-coresrequireexactlyd(3n)]TJ /F5 11.955 Tf 11.96 0 Td[(3)=2eedges. Theorem3.1. LetG(V,E)bea2-club/2-core(n=jGj),thenjEj(3n)]TJ /F5 11.955 Tf 11.95 0 Td[(3)=2. Proof. SeeAppendix A 62

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Corollary4. LetG(V,E)beagraphwiththedegreesequencevectorw=(w1,...,wn),wherewi=8>><>>:n)]TJ /F5 11.955 Tf 11.95 0 Td[(1,i=1,2,i=2,...,n)]TJ /F5 11.955 Tf 11.96 0 Td[(1,wn=8>><>>:2,ifnisodd,3,ifniseven. thenG(V,E)istheoptimal2-club/2-coreintermsofthenumberofedges. Proof. First,weshouldcheckthatG(V,E)isa2-cluband2-core.Itisquiteobvioussincetherstnodeisconnectedtoanyothernode,hence,itisa2-club.Andsinceadegreeofanyothernodeisatleast2,then,itisa2-core. Second,weneedtoshowthatjEj=3n)]TJ /F5 11.955 Tf 11.95 0 Td[(3 2. ThatimmediatelyfollowsfromthedenitionofG(V,E),jEj=1 2nXi=1wi=n)]TJ /F5 11.955 Tf 11.95 0 Td[(1+2(n)]TJ /F5 11.955 Tf 11.95 0 Td[(2)+2 2=3n)]TJ /F5 11.955 Tf 11.96 0 Td[(3 2. LevelR)]TJ /F5 11.955 Tf 12.78 0 Td[(1strongattacktolerancepropertyof2-clubs,thatis,maintainingthe2-clubstructureafterR)]TJ /F5 11.955 Tf 12.38 0 Td[(1component(edgeand/ornodes)failures,mightseemveryrestrictive.WehaveshownabovethatR-robust2-clubsprovidetheseattacktoleranceproperties(inthecaseof2-clubswithstrongedgeattacktolerance,R-robust2-clubsareinfacttheonlynetworkcongurationsthatpossessthisproperty).HereweshowthatonecanconstructanR-robust2-clubonnnodesusingonlyRn)]TJ /F5 11.955 Tf 12.62 0 Td[((R(R+1)=2)(orO(Rn))edges,whichcomparesveryfavorablywithothertypesofrobustnetworkcongurations,suchascliquesanddegree/density-basedcliquerelaxationsthatallrequireO(n2)edges. TodesignanR-robust2-clubwithO(Rn)edges,weneedtohaveRnodes(hubs)inG(V,E)withadegree(n)]TJ /F5 11.955 Tf 12.79 0 Td[(1).Theremainingn)]TJ /F3 11.955 Tf 12.79 0 Td[(Rnodesautomaticallyhavea 63

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degreeR.Thefollowingtheoremstatesthattheconstructednetworkistheoptimalintermsofthenumberofconstructededges.Therefore,italsoshowshowtoconstructthemostcost-efcientR-robust2-clubonagivennumberofnodes.TheguresillustratingtheseR-robust2-clubcongurationsarediscussedintheprevioussection. Theorem3.2. LetG(V,E)beanR-robust2-clubwithn3R+p 5R2)]TJ /F7 7.97 Tf 6.59 0 Td[(4R 2(n=jGj),thenjEjRn)]TJ /F5 11.955 Tf 11.95 0 Td[((R(R+1)=2). Proof. SeeAppendix A Corollary5. LetG(V,E)beagraphwiththedegreesequencevectorw=(w1,...,wn),wherewi=8>><>>:R,i=1,...,n)]TJ /F3 11.955 Tf 11.95 0 Td[(Rn)]TJ /F5 11.955 Tf 11.96 0 Td[(1,i=n)]TJ /F3 11.955 Tf 11.96 0 Td[(R+1,...,n andn3R,thenG(V,E)istheoptimalR-robust2-clubintermsofthenumberofedges. Proof. First,weshouldcheckthatG(V,E)isanR-robust2-club.ThisstructurecanbeviewedasagraphinwhichRnodesareconnectedwithanyothernodewithanedge,andtherearenomoreadditionaledges.So,Rnodeshavethemaximumpossibledegreeofn)]TJ /F5 11.955 Tf 12.08 0 Td[(1,andn)]TJ /F3 11.955 Tf 12.08 0 Td[(RnodeshavedegreeR.Bydenition,G(V,E)isobviouslyanR-robust2-club. Second,weneedtoshowthatjEj=Rn)]TJ /F3 11.955 Tf 13.15 8.09 Td[(R(R+1) 2. ThisimmediatelyfollowsfromthedenitionofG(V,E),jEj=R(n)]TJ /F5 11.955 Tf 11.95 0 Td[(1)+(n)]TJ /F3 11.955 Tf 11.96 0 Td[(R)R 2=Rn)]TJ /F3 11.955 Tf 13.15 8.09 Td[(R+R2 2=Rn)]TJ /F3 11.955 Tf 13.16 8.09 Td[(R(R+1) 2. 64

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AnalremarkthatimmediatelyfollowsfromtheseresultsisthefactthatanoptimalR-robustk-club(fork>2)onnnodescanbeconstructedusingatmostRn)]TJ /F6 7.97 Tf 13.68 5.47 Td[(R(R+1) 2(O(Rn))edges.Althoughtheexactoptimalnumberofedgesinthesestructuresforeachspecick>2maynotnecessarilybederivedanalytically(althoughtheseissuesareworthinvestigatinginmoredetail),theresultsofthischaptershowthatR-robustk-clubnetworkcongurationscansimultaneouslyprovidelowdiameter,strongattacktolerancepropertieswithrespecttobothnodeandedgeattacks,andconstructioncostefciency. A2-connectedgraph B2-robust2-club Figure3-1. Illustrativeexampleforn=4A)2-connectedgraph;B)2-robust2-club. Figure3-2. Astronglyedgeattacktolerant4-clubwhichisnota2-robust4-club 65

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A B Figure3-3. Optimal2-club/2-corenetworkdesignforA)n=10,andB)n=11nodes. A B C D Figure3-4. OptimaldesignofA)2-club,B)2-robust2-club,C)3-robust2-club,andD)4-robust2-clubforn=10nodes. 66

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A B C D Figure3-5. A)Power-lawnetworkwith=0.5,n=20optimallyenhancedtoB)2-club,C)2-robust2-club,andD)3-robust2-club 67

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CHAPTER4DENSITY-BASEDCLIQUERELAXATIONSANDTHEIRASYMPTOTICBEHAVIORINRANDOMGRAPHS Thischapterconsidersdensity-basedcliquerelaxations,alsoknownasquasi-cliques,or-cliques.Inthiscontext,a-cliqueisasubsetofverticesSVofasimpleundirectedgraphG(V,E)suchthattheedgedensity(i.e.,theratioofthenumberofedgesintheclusterG(S)tothemaximumpossiblenumberofedges)intheinducedsubgraphG(S)isatleast.Weinvestigateasymptoticbehaviorof-cliquesintheuniformrandomgraphsanddevelopalinearmixed-integerproblemforidentifyingthelargest-cliqueinanynetwork. Randomgraphswerestudiedfrommanydifferentaspects,includingthewell-knownstudiespresentedin[ 39 ],[ 40 ],[ 41 ].Concerningtheasymptoticbehaviorofthemaximumcliquesizeinrandomgraphs,Matulain[ 43 ]showednumericalevidencethatthemaximumcliquesizehasastrongpeakaround2lnn=ln(1=p).GrimmettandMcDiarmid[ 42 ]provedthestrongerresultstatingthatasn!1themaximumcliquesizeisequalto2lnn=ln(1=p)+O(lnlnn)withprobabilityone. Inthischapterweshowthatthemaximum-cliquesize,asn!1,belongstotheinterval[2lnn=ln(1=p),2lnn=ln(1=p)]withprobabilityone,where1 p= p1)]TJ /F4 11.955 Tf 11.95 0 Td[( 1)]TJ /F3 11.955 Tf 11.96 0 Td[(p1)]TJ /F13 7.97 Tf 6.59 0 Td[( Wealsodemonstratethatthereexistsanabruptjump(rst-orderphasetransition)oftheorderofmagnitudeofthemaximumquasi-clique. 4.1DerivationofAsymptoticBehaviorProperties ConsiderarandomgraphG(n,p)withnvertices,andprobabilitypoftheexistenceofeachedge.Weareinterestedintheproblemofmaximum-cliquesizeinthatgraph.Particularly,weareinterestedintheasymptoticbehaviorofthemaximum-cliquesizewhenn!1. 68

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DeneNkastherandomvariableequaltothenumberof-cliquesofsizekinG(n,p).Note,thatthereare)]TJ /F6 7.97 Tf 5.61 -4.38 Td[(nkdifferentsubgraphsofsizekinthisgraph,andletjbeasubgraphnumber(j=1,...,)]TJ /F6 7.97 Tf 5.61 -4.38 Td[(nk).LetIjbeanindicatorrandomvariablesuchthatIj=1ifasubgraphjofsizekisa-cliqueandIj=0otherwise.ThenNk=(nk)Xj=1Ij. (4) Theunconditionalprobabilitiesthatanysubgraphofsizekisa-clique(i.e.,PfIj=1g)areequal.Then,theexpectednumberof-cliquesofsizekinG(n,p)isgivenby E[Nk]=nkPfasubgraphofsizekinGisa-cliqueg=nk(k2)Xm=d(k2)e)]TJ /F6 7.97 Tf 5.48 -4.38 Td[(k2mpm(1)]TJ /F3 11.955 Tf 11.95 0 Td[(p)(k2))]TJ /F6 7.97 Tf 6.59 0 Td[(m=nk1)]TJ /F5 11.955 Tf 11.96 0 Td[(Bin\004)]TJ /F6 7.97 Tf 5.48 -4.38 Td[(k2;)]TJ /F6 7.97 Tf 5.48 -4.38 Td[(k2,p,(4) whereBin(k,n;p)isthec.d.fofthebinomialdistribution.ThefollowingfactconcerningE[Nk]playsacentralroleinthesequel. Proposition4.1. Ifp<,theintegerk=knthatsatisesE[Nkn]=1 (4) isgivenbykn=2 ln p1)]TJ /F13 7.97 Tf 6.59 0 Td[( 1)]TJ /F6 7.97 Tf 6.59 0 Td[(p1)]TJ /F13 7.97 Tf 6.58 0 Td[(lnn+O(lnlnn),n1. (4) Proof. InestablishingTheorem 4.1 werelyonthefollowingresultdueto[ 44 ]. Theorem4.1([ 44 ]). Letp2(0,1)bexed,andpforsome1.Denex=)]TJ /F6 7.97 Tf 6.59 0 Td[(p ,where=p p(1)]TJ /F3 11.955 Tf 11.96 0 Td[(p).ThenXi=ipi(1)]TJ /F3 11.955 Tf 11.95 0 Td[(p))]TJ /F6 7.97 Tf 6.58 0 Td[(i=)]TJ /F5 11.955 Tf 11.95 0 Td[(1)]TJ /F5 11.955 Tf 11.95 0 Td[(1p)]TJ /F7 7.97 Tf 6.59 0 Td[(1(1)]TJ /F3 11.955 Tf 11.95 0 Td[(p))]TJ /F13 7.97 Tf 6.59 0 Td[(1)]TJ /F5 11.955 Tf 11.95 0 Td[((x) (x)exp(,,p)= 69

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where0(,,p)minfp =8,x)]TJ /F7 7.97 Tf 6.59 0 Td[(1g and(x)and(x)arethecumulativeandprobabilitydensityfunctionsofthestandardnormaldistribution,respectively. ProofofProposition 4.1 :Setting=)]TJ /F6 7.97 Tf 5.48 -4.38 Td[(k2,=de>p wehave=p p(1)]TJ /F3 11.955 Tf 11.95 0 Td[(p),x=)]TJ /F3 11.955 Tf 11.95 0 Td[(p p p(1)]TJ /F3 11.955 Tf 11.95 0 Td[(p)1=2 Thus,exp(,,p)==expfO()]TJ /F7 7.97 Tf 6.59 0 Td[(1)g=1+O()]TJ /F7 7.97 Tf 6.58 0 Td[(1),n!1 Fromthefactthatxincreaseswithn,itfollowsthat1)]TJ /F5 11.955 Tf 11.95 0 Td[((x) (x)=1 x+O(x)]TJ /F7 7.97 Tf 6.58 0 Td[(3)=p p(1)]TJ /F3 11.955 Tf 11.95 0 Td[(p) )]TJ /F3 11.955 Tf 11.96 0 Td[(p)]TJ /F7 7.97 Tf 6.58 0 Td[(1=2)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(1+O()]TJ /F7 7.97 Tf 6.59 0 Td[(1),n!1 wherethewell-knownexpansion1)]TJ /F5 11.955 Tf 11.96 0 Td[((t)=1 p 2e)]TJ /F6 7.97 Tf 6.58 0 Td[(t2=2)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(t)]TJ /F7 7.97 Tf 6.59 0 Td[(1+O(t)]TJ /F7 7.97 Tf 6.59 0 Td[(3),t!1 wasused.Finally,forthebinomialcoefcientwehave)]TJ /F5 11.955 Tf 11.95 0 Td[(1)]TJ /F5 11.955 Tf 11.95 0 Td[(1=\() \()\((1)]TJ /F4 11.955 Tf 11.95 0 Td[()+1) whereweusedaspecialcaseofthewell-knownGammafunction\(n)=(n)]TJ /F5 11.955 Tf 11.98 0 Td[(1)!.UsingStirling'sexpansionfor\(z),\(z)=r 2 zzze)]TJ /F6 7.97 Tf 6.58 0 Td[(z)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(1+O(z)]TJ /F7 7.97 Tf 6.59 0 Td[(1),z!1 70

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weobtain)]TJ /F5 11.955 Tf 11.95 0 Td[(1)]TJ /F5 11.955 Tf 11.95 0 Td[(1=1 p 2r 1)]TJ /F4 11.955 Tf 11.96 0 Td[((1)]TJ /F4 11.955 Tf 11.95 0 Td[()1)]TJ /F13 7.97 Tf 6.59 0 Td[()]TJ /F13 7.97 Tf 6.58 0 Td[()]TJ /F7 7.97 Tf 6.58 0 Td[(1=2)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(1+O()]TJ /F7 7.97 Tf 6.59 0 Td[(1) Thus,nally,thetailofthebinomialdistributioncanbeestimatedasXi=ipi(1)]TJ /F3 11.955 Tf 11.96 0 Td[(p))]TJ /F6 7.97 Tf 6.59 0 Td[(i=1 p 21)]TJ /F3 11.955 Tf 11.96 0 Td[(p )]TJ /F3 11.955 Tf 11.95 0 Td[(pr 1)]TJ /F4 11.955 Tf 11.95 0 Td[()]TJ /F7 7.97 Tf 6.58 0 Td[(1=2"p 1)]TJ /F3 11.955 Tf 11.96 0 Td[(p 1)]TJ /F4 11.955 Tf 11.95 0 Td[(1)]TJ /F13 7.97 Tf 6.59 0 Td[(#)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(1+O()]TJ /F7 7.97 Tf 6.58 0 Td[(1), where=)]TJ /F6 7.97 Tf 5.47 -4.38 Td[(k2.Consequently,Equation( 4 )canasymptoticallybewrittenasnk1 p 21)]TJ /F3 11.955 Tf 11.95 0 Td[(p )]TJ /F3 11.955 Tf 11.95 0 Td[(pr 1)]TJ /F4 11.955 Tf 11.95 0 Td[(k2)]TJ /F7 7.97 Tf 6.59 0 Td[(1=2"p 1)]TJ /F3 11.955 Tf 11.95 0 Td[(p 1)]TJ /F4 11.955 Tf 11.95 0 Td[(1)]TJ /F13 7.97 Tf 6.58 0 Td[(#(k2)1+O)]TJ /F6 7.97 Tf 5.48 -4.38 Td[(k2)]TJ /F7 7.97 Tf 6.59 0 Td[(1=1 Takingthelogarithmofbothsides,wehave)]TJ /F8 11.955 Tf 11.29 9.69 Td[()]TJ /F3 11.955 Tf 5.48 -9.69 Td[(k+1 2lnk+k+klnn+lnC1)]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 2lnk2+k2lnC2=Omaxk2 n,1 k whereC1=1 21)]TJ /F3 11.955 Tf 11.96 0 Td[(p )]TJ /F3 11.955 Tf 11.95 0 Td[(pr 1)]TJ /F4 11.955 Tf 11.95 0 Td[(,C2=p 1)]TJ /F3 11.955 Tf 11.96 0 Td[(p 1)]TJ /F4 11.955 Tf 11.96 0 Td[(1)]TJ /F13 7.97 Tf 6.59 0 Td[( Takingintoaccountthat1 2lnk2=lnk)]TJ /F5 11.955 Tf 11.96 0 Td[(lnp 2+O(k)]TJ /F7 7.97 Tf 6.59 0 Td[(1) thelastequationcanbewrittenask2 2lnC2)]TJ /F3 11.955 Tf 11.95 0 Td[(klnk+k(lnn+1)]TJ /F7 7.97 Tf 13.15 4.71 Td[(1 2lnC2))]TJ /F7 7.97 Tf 13.15 4.71 Td[(3 2lnk+lnp 2C1=Omaxk2 n,1 k Toobtainthemaintermoftheasymptoticalapproximationofthesolutionofthisequation,letusrestateitintheformk2(1 2lnC2+lnn k)]TJ /F5 11.955 Tf 13.15 8.08 Td[(lnk k+1)]TJ /F7 7.97 Tf 13.15 4.71 Td[(1 2lnC2 k)]TJ /F5 11.955 Tf 13.15 8.08 Td[(3lnk 2k2+lnp 2C1 k2)=Omaxk2 n,1 k Undertheaboveassumptionthatk=o(n)itcanbefurtherrewrittenas1 2lnC2+lnn k+olnn k=o(1) 71

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whencewehavethatk=2lnn lnC)]TJ /F7 7.97 Tf 6.59 0 Td[(12+(n),where(n)=o(lnn) Todeterminetheorderoftheterm(n),werewritetheequationask1 2lnC2+lnn)]TJ /F5 11.955 Tf 11.95 0 Td[(lnk=O(1) Writingdowntheexpressionfork=knintheformkn=2lnn lnC)]TJ /F7 7.97 Tf 6.59 0 Td[(12+(n) andsubstitutingitinthelastequation,weobtainthat(n)=O(lnlnn),whichfurnishesthestatementoftheProposition. Proposition4.2. For0
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iskn=o(n). Proof. FortheproofweusetheupperboundonthebinomialcoefcientfollowingfromtheStirling'sapproximation:nken kk (4) FortheupperboundonthebinomialdistributionweuseChernoff'sboundforthetailofthebinomialdistribution:nXi=mnipi(1)]TJ /F3 11.955 Tf 11.96 0 Td[(p)n)]TJ /F6 7.97 Tf 6.59 0 Td[(in)]TJ /F3 11.955 Tf 11.95 0 Td[(np n)]TJ /F3 11.955 Tf 11.95 0 Td[(mn)]TJ /F6 7.97 Tf 6.58 0 Td[(mnp mm, (4) wheremnp.Inourcasen=)]TJ /F6 7.97 Tf 5.48 -4.38 Td[(k2,m=)]TJ /F6 7.97 Tf 5.48 -4.38 Td[(k2(forsimplicity,weusem=)]TJ /F6 7.97 Tf 5.48 -4.38 Td[(k2),andsincep<,thenmnp;thus,therequirementonmisvalid.Thus,(k2)Xm=d(k2)e)]TJ /F6 7.97 Tf 5.48 -4.38 Td[(k2mpm(1)]TJ /F3 11.955 Tf 11.95 0 Td[(p)(k2))]TJ /F6 7.97 Tf 6.58 0 Td[(m )]TJ /F6 7.97 Tf 5.48 -4.37 Td[(k2)]TJ /F8 11.955 Tf 11.96 9.68 Td[()]TJ /F6 7.97 Tf 5.48 -4.37 Td[(k2p )]TJ /F6 7.97 Tf 5.48 -4.38 Td[(k2)]TJ /F4 11.955 Tf 11.96 0 Td[()]TJ /F6 7.97 Tf 5.48 -4.38 Td[(k2!(k2))]TJ /F13 7.97 Tf 6.59 0 Td[((k2) )]TJ /F6 7.97 Tf 5.48 -4.37 Td[(k2p )]TJ /F6 7.97 Tf 5.48 -4.38 Td[(k2!(k2)= = 1)]TJ /F3 11.955 Tf 11.96 0 Td[(p 1)]TJ /F4 11.955 Tf 11.96 0 Td[(1)]TJ /F13 7.97 Tf 6.59 0 Td[(p !(k2)(4) RecallthatEquation( 4 )canbewrittenasnk(k2)Xm=d(k2)e)]TJ /F6 7.97 Tf 5.48 -4.38 Td[(k2mpm(1)]TJ /F3 11.955 Tf 11.96 0 Td[(p)(k2))]TJ /F6 7.97 Tf 6.59 0 Td[(m=1, (4) Usingtheupperboundsin( 4 )and( 4 ),wecanwrite E[Nk]en kk 1)]TJ /F3 11.955 Tf 11.95 0 Td[(p 1)]TJ /F4 11.955 Tf 11.95 0 Td[(1)]TJ /F13 7.97 Tf 6.59 0 Td[(p !(k2) 73

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TakingthelogarithmofthisupperboundasinProposition 4.1 wecanconcludethatthereexistssuchconstantCthatifkClnn,thenen kk 1)]TJ /F3 11.955 Tf 11.96 0 Td[(p 1)]TJ /F4 11.955 Tf 11.96 0 Td[(1)]TJ /F13 7.97 Tf 6.59 0 Td[(p !(k2)<1. (4) Therefore,thesolutionk=kntotheequationE[Nk]=1cannotbegreaterthanClnn;thus,kn=o(n). NextpropositionestablishesthefactthatthenumberknwhichsolvestheequationE[Nk]=1playsacrucialroleindeterminingthesizeofthemaximumquasi-cliqueintheuniformrandomgraphwhenn!1.Letlnbethesizeofthemaximum-cliqueinauniformrandomgraphG(n,p).Sinceweconsiderthemaximumsizeinarandomgraph,lnisalsoarandomvalue,meaningthatforanyparticularrealizationofG(n,p),thevalueoflndeterminesthesizeofmaximum-cliqueinthatrealization. Proposition4.3. Ifp<,thenPfln>kng!0,whenn!1 (4) whereknisthesolutiontotheequationE[Nk]=1 (4) whichbyProposition 4.1 isgivenbykn=2 ln p1)]TJ /F13 7.97 Tf 6.59 0 Td[( 1)]TJ /F6 7.97 Tf 6.59 0 Td[(p1)]TJ /F13 7.97 Tf 6.58 0 Td[(lnn+O(lnlnn),n1. (4) Proof. First,weshowthatforany-cliqueG(V,E)ofsizem,andforanys
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toconsiders=m)]TJ /F5 11.955 Tf 11.96 0 Td[(1.SinceG(V,E)isa-clique,thenjEjm2. Therefore,thereexistsanodeiwithdeg(i)(m)]TJ /F5 11.955 Tf 12.54 0 Td[(1).Ifweconsiderthesubgraphwithoutthatnode,thenasubgraphG(Vni,Ei)a-clique,sincejEijm2)]TJ /F4 11.955 Tf 11.96 0 Td[((m)]TJ /F5 11.955 Tf 11.96 0 Td[(1)=m)]TJ /F5 11.955 Tf 11.96 0 Td[(12 Then,usingthisfact,wecanconcludethatinauniformrandomgraphG(n,p)ln>mifandonlyifNm+11.Thus,Pfln>kng=PfNkn+11g. NotealsothatPfNkn+11g=1Xi=1PfNkn+1=ig1Xi=1iPfNkn+1=ig=EhNkn+1i. Therefore,Pfln>kngEhNkn+1i. Hence,toprovethisproposition,weonlyneedtoshowthatEhNkn+1i!0,n!1.FromtheproofofProposition 4.1 weknowthatE[Nk]=nk1 p 21)]TJ /F3 11.955 Tf 11.95 0 Td[(p )]TJ /F3 11.955 Tf 11.95 0 Td[(pr 1)]TJ /F4 11.955 Tf 11.95 0 Td[(k2)]TJ /F7 7.97 Tf 6.59 0 Td[(1=2"p 1)]TJ /F3 11.955 Tf 11.95 0 Td[(p 1)]TJ /F4 11.955 Tf 11.95 0 Td[(1)]TJ /F13 7.97 Tf 6.58 0 Td[(#(k2)1+O)]TJ /F6 7.97 Tf 5.48 -4.37 Td[(k2)]TJ /F7 7.97 Tf 6.59 0 Td[(1, andfork=knnkn1 p 21)]TJ /F3 11.955 Tf 11.96 0 Td[(p )]TJ /F3 11.955 Tf 11.95 0 Td[(pr 1)]TJ /F4 11.955 Tf 11.95 0 Td[(kn2)]TJ /F7 7.97 Tf 6.59 0 Td[(1=2"p 1)]TJ /F3 11.955 Tf 11.95 0 Td[(p 1)]TJ /F4 11.955 Tf 11.95 0 Td[(1)]TJ /F13 7.97 Tf 6.59 0 Td[(#(kn2)1+O)]TJ /F6 7.97 Tf 5.48 -4.38 Td[(kn2)]TJ /F7 7.97 Tf 6.59 0 Td[(1=1. (4) 75

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Then,EhNkn+1i=nkn+11 p 21)]TJ /F3 11.955 Tf 11.96 0 Td[(p )]TJ /F3 11.955 Tf 11.95 0 Td[(pr 1)]TJ /F4 11.955 Tf 11.95 0 Td[(kn+12)]TJ /F7 7.97 Tf 6.58 0 Td[(1=2"p 1)]TJ /F3 11.955 Tf 11.96 0 Td[(p 1)]TJ /F4 11.955 Tf 11.95 0 Td[(1)]TJ /F13 7.97 Tf 6.59 0 Td[(#(kn+12)1+O)]TJ /F6 7.97 Tf 5.48 -4.38 Td[(kn+12)]TJ /F7 7.97 Tf 6.58 0 Td[(1n)]TJ /F3 11.955 Tf 11.96 0 Td[(kn kn+1nkn1 p 21)]TJ /F3 11.955 Tf 11.96 0 Td[(p )]TJ /F3 11.955 Tf 11.95 0 Td[(pr 1)]TJ /F4 11.955 Tf 11.95 0 Td[(kn2)]TJ /F7 7.97 Tf 6.59 0 Td[(1=2"p 1)]TJ /F3 11.955 Tf 11.95 0 Td[(p 1)]TJ /F4 11.955 Tf 11.95 0 Td[(1)]TJ /F13 7.97 Tf 6.59 0 Td[(#(kn2)+kn1+O)]TJ /F6 7.97 Tf 5.48 -4.37 Td[(kn2)]TJ /F7 7.97 Tf 6.59 0 Td[(1, andusingEquation( 4 ),wehave EhNkn+1in)]TJ /F3 11.955 Tf 11.95 0 Td[(kn kn+1"p 1)]TJ /F3 11.955 Tf 11.95 0 Td[(p 1)]TJ /F4 11.955 Tf 11.96 0 Td[(1)]TJ /F13 7.97 Tf 6.58 0 Td[(#kn1+O)]TJ /F6 7.97 Tf 5.48 -4.38 Td[(kn2)]TJ /F7 7.97 Tf 6.59 0 Td[(1, (4) Havingthatkn=2 ln p1)]TJ /F13 7.97 Tf 6.58 0 Td[( 1)]TJ /F6 7.97 Tf 6.59 0 Td[(p1)]TJ /F13 7.97 Tf 6.59 0 Td[(lnn+O(lnlnn), Inequality( 4 )canbewrittenasEhNkn+1in kn1 n2(1+O(1))=1 nkn(1+O(1)) Clearly,EhNkn+1i!0whenn!1,whatendstheproofofthisproposition. Thenextcorollaryshowsthatwhenngoestoinnity,themaximumsizeof-cliqueintheuniformrandomgraphisalmostsurelyabovethecertainvalueoftheorderofln(n).ItusesthewellknownfactestablishedbyGrimmettandMcDiarmid[ 42 ].Theyprovedthatthesizeofmaximumcliqueintheuniformrandomgraphisasymptoticallyalmostsurelyequalto2ln(n)=ln(p)+O(lnlnn).Notethatl1nistherandomvariablewhichisequaltothesizeofthemaximumclique(=1)inauniformrandomgraphG(n,p).Notealsothatlim!1)]TJ /F7 7.97 Tf 6.59 0 Td[(0kn=2ln(n)=ln(p), 76

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thenwecandenek1n=2ln(n)=ln(p)+O(lnlnn),andtheaforementionedstatementofGrimmettandMcDiarmid[ 42 ]canbewrittenas Pfk1nl1nk1n)]TJ /F5 11.955 Tf 11.95 0 Td[(1g!1,whenn!1 (4) Corollary6. Ifp<,thenPfk1n)]TJ /F5 11.955 Tf 11.95 0 Td[(1lnkng!1,whenn!1 (4) whereknisthesolutiontotheequationE[Nk]=1 (4) whichbyProposition 4.1 isgivenbykn=2 ln p1)]TJ /F13 7.97 Tf 6.59 0 Td[( 1)]TJ /F6 7.97 Tf 6.59 0 Td[(p1)]TJ /F13 7.97 Tf 6.58 0 Td[(lnn+O(lnlnn),n1. (4) Proof. ItfollowsimmediatelyfromEquation( 4 ),Proposition 4.3 ,andthefactthatthemaximumsizeof-cliqueforany<1isalwaysgreaterthanthesizeofthemaximumcliqueinthatgraph,i.ePfl1nlng=1,<1. Thus,weestablishedthefactthatforanyxed>ptheasymptoticbehaviorofthemaximum-cliquesizeistheorderoflnn.Intuitively,whenpthewholegraphG(n,p)becomes-clique,sothesizeofthemaximum-cliqueisthetheorderofn.Therefore,thenaturalquestionarisinghereiswhathappenswhenpisxedandapproachesp.Weshowthatthereisarstorderphasetransitionintheasymptoticbehavioroftheorderofmagnitudeofthemaximum-cliqueinthepoint=p. 77

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Particularly,weprovethatwithprobability1lim!p+0limn!1ln n=0, butlim!p)]TJ /F7 7.97 Tf 6.59 0 Td[(0limn!1ln n=1, whatshowsthatthereisajumpintheorderofmagnitudeofthemaximum-cliqueinthepoint=p. Proposition4.4. Iflpisthemaximumsizeof-cliqueintheuniformrandomgraphG(n,p)withxedp,thenwithprobability1lim!p+0limn!1ln n=0, (4) butlim!p)]TJ /F7 7.97 Tf 6.59 0 Td[(0limn!1ln n=1, (4) Proof. TherstlimitingcasefollowsfromProposition 4.3 ,sinceweprovedthatforanyxed>pwithprobability1limn!1ln n=0 ToproveEquation( 4 )letXijbetheBernoullirandomvariablewhichisequalto1ifthereexistsanedge(i,j)intheuniformrandomgraphG(n,p).Then,Pfln=ng=P8<:X(i,j)Xijn29=;=P8><>:P(i,j)Xij )]TJ /F6 7.97 Tf 5.48 -4.38 Td[(n29>=>; Fromtheweaklawoflargenumbersitfollowsthatforanyxed">0,P8><>:P(i,j)Xij )]TJ /F6 7.97 Tf 5.48 -4.38 Td[(n2)]TJ /F3 11.955 Tf 11.95 0 Td[(p<"9>=>;!1,n!1. 78

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Then,let"=p)]TJ /F4 11.955 Tf 11.96 0 Td[(,andP8><>:P(i,j)Xij )]TJ /F6 7.97 Tf 5.48 -4.38 Td[(n29>=>;=P8><>:p)]TJ /F8 11.955 Tf 14.32 27.56 Td[(P(i,j)Xij )]TJ /F6 7.97 Tf 5.48 -4.38 Td[(n2p)]TJ /F4 11.955 Tf 11.95 0 Td[(9>=>;=P8><>:p)]TJ /F8 11.955 Tf 14.32 27.56 Td[(P(i,j)Xij )]TJ /F6 7.97 Tf 5.48 -4.38 Td[(n2"9>=>;P8><>:p)]TJ /F8 11.955 Tf 14.32 27.55 Td[(P(i,j)Xij )]TJ /F6 7.97 Tf 5.48 -4.38 Td[(n2"9>=>;!1,n!1. Therefore,Pfln=ng!1,n!1 whatendstheproofoftheproposition. 4.2LinearMixed-IntegerFormulationsoftheMaximum-cliqueProblem Inthissectionwedeveloplinearmixed-integerformulationsofthemaximum-cliqueproblem.Toourknowledge,thelastformulationisthemostcompactlinearformulation.ItrequiresonlyO(n)entities. ConsideragraphG=(V,E)withnnodes,adjacencymatrixA,andconsideraproblemofndingamaximum-cliqueinthisgraph.SupposewepicksomesubgraphGs.Wewanttocheckwhetherthissubgraphisa-cliqueornot.Forthatreasondenex=(x1,...,xn)asa0-1vectorwithxi=1ifnodeibelongstoGs,orzerootherwise. ThesubgraphGsisa-cliqueifitcontainsatleastjGsj(jGsj)]TJ /F5 11.955 Tf 19.64 0 Td[(1)=2edges,or,equivalently,intermsofvectorx:1 2nXi=1xi nXi=1xi)]TJ /F5 11.955 Tf 11.95 0 Td[(1!=1 2 nXi,j=1xjxi)]TJ /F6 7.97 Tf 18.31 14.95 Td[(nXi=1xi!==1 2 nXi,j=1,i6=jxjxi+nXi=1x2i)]TJ /F6 7.97 Tf 18.3 14.95 Td[(nXi=1xi!=1 2nXi,j=1,i6=jxjxi sincex2i=xi. 79

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ThenumberofedgesinthesubgraphGscanbecalculatedas1 2xtAx=nXi,j=1,i6=jaijxjxi. Thereforetheproblemofndingthemaximum-cliqueinthegraphGcanbeformulatedasfollows:maxnXi=1xi subjecttonXi,j=1,i6=jaijxjxinXi,j=1,i6=jxjxi. Thisistheproblemwithalinearobjectiveandonequadraticconstraint.Weintroducethenewvariablestomakethisproblemlinear.Denewij=xixjforeverypairofnodes(i,j).Weneedonlyn(n)]TJ /F5 11.955 Tf 12.56 0 Td[(1)=2)]TJ /F3 11.955 Tf 12.56 0 Td[(nsuchvariablessincewij=wji.Theconstraintwij=xixjisequivalenttowijxi,wijxj,wijxi+xj)]TJ /F5 11.955 Tf 11.96 0 Td[(1,. Now,wemayformulatetheproblemasalinearproblem maxnXi=1xi subjecttonXi,j=1,i
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Thisproblemisalinear0-1problem,anditcontainsn(n)]TJ /F5 11.955 Tf 12.96 0 Td[(1)=2variablesand3 2n(n)]TJ /F5 11.955 Tf 11.47 0 Td[(1)+1constraints.Wemayrewriteitinmorecompactformsinceifxi=0,thenallwij=0,j=i+1,..,n.Insteadof( 4 )and( 4 )wemaywritenXj=i+1wijnxi,i=1,...,n)]TJ /F5 11.955 Tf 11.95 0 Td[(1, whatreducesthenumberofconstraintsto1 2n(n)]TJ /F5 11.955 Tf 11.95 0 Td[(1)+n. Next,weconsideranalternativelinearization.RecallthatoriginallywehadonlyoneconstraintnXi,j=1,i6=j(aij)]TJ /F4 11.955 Tf 11.95 0 Td[()xjxi0, whichwemayrewriteasnXi=1xi xi+nXj=1(aij)]TJ /F4 11.955 Tf 11.95 0 Td[()xj!0. Letusdenetheavariablewiforeveryiasfollowswi=xi xi+nXj=1(aij)]TJ /F4 11.955 Tf 11.96 0 Td[()xj!. Thisquadraticequalityisequivalenttofourlinearinequalitieswinxi,wi)]TJ /F3 11.955 Tf 21.92 0 Td[(nxi,wixi+nXj=1(aij)]TJ /F4 11.955 Tf 11.95 0 Td[()xj)]TJ /F5 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(xi)n,wixi+nXj=1(aij)]TJ /F4 11.955 Tf 11.95 0 Td[()xj+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xi)n,xi2f0,1g,wij2R,i,j=1,...,n. Therefore,theproblemofndingamaximum-cliquecanberepresentedasthefollowinglinearformulationwith2nvariables(n0-1variablesandncontinuousvariables)and4n+1constraints: 81

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maxnXi=1xi(4) subjecttonXi=1wi0,winxi,wi)]TJ /F3 11.955 Tf 21.92 0 Td[(nxi,wixi+nXj=1(aij)]TJ /F4 11.955 Tf 11.95 0 Td[()xj)]TJ /F5 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(xi)n,wixi+nXj=1(aij)]TJ /F4 11.955 Tf 11.95 0 Td[()xj+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xi)n,xi2f0,1g,wij2R,i,j=1,...,n. 4.3ComputationalExperiments Inthissectionwepresentcomputationalexperimentsperformedtoaddresstwoissues.First,wetrytoanalyzewhethertheboundsthatwederivedinSection 4.1 arereliableforrelativelysmalln.Second,theexistenceofajumpintheorderofmagnitudeofthemaximum-cliquesizeatthepointp=isprovedinthelimitingcase(n!1);therefore,itisofinteresttoconductcomputationalexperimentsinvestigatingthebehaviorofthemaximum-cliquearoundthepointp=forrelativelylargexedvaluesofn. Weprovedthefactthatforanyxedandptheprobabilitythatthemaximum-cliquesizebelongstotheinterval k1n,kn=2642lnn ln1 p;2lnn ln p1)]TJ /F13 7.97 Tf 6.59 0 Td[( 1)]TJ /F6 7.97 Tf 6.58 0 Td[(p1)]TJ /F13 7.97 Tf 6.59 0 Td[(375(4) tendsto1asnbecomesverylarge.Forsimplicity,wedonotincludetheresidualtermO(lnlnn)inthebounds.Sinceintherealworldeveryonedealsonlywithnitenetworks,itwouldbeveryusefultoknowiftheseboundsareapplicableforrelatively 82

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smalln.Intherstsetofcomputationalexperimentswegeneratedseveralinstancesofuniformrandomgraphsforn=100andprangingfrom0.05to0.15.Wegenerated100scenariosofG(100,p)foreveryp.Then,weusedthemixed-integerproblemformulationin( 4 )tondthemaximum-cliquesinthegeneratedscenariosfor=0.9,and=0.85(sinceshouldberathercloseto1inrobustnetworkclusters).WeusedFICOTMXpressOptimizationSuite7.1[ 22 ]tosolvetheproposedproblems.TheaveragemaximumsizesoverallscenariosisreportedinTable 4-1 andTable 4-2 inthelastcolumns.ItshouldbenotedthatforthissetofparametersInterval( 4 )isnotverywideandtheaveragesizeofthemaximum-cliquealmostallthetimebelongstothisinterval. Inthesecondsetofcomputationalexperimentswetriedtoanalyzethebehavioroftherelativemaximum-cliquesizeforaxedanddifferentvaluesofpandn.Weused=70%andsetthegridof200differentvaluesofpas(p1,...,p100)=(0.006,0.012,...,0.600),and(p101,...,p200)=(0.601,0.602,...,0.700).Notethatthesecondgridisdensersincepbecomesclosertoandthisregionshouldbeinvestigatedthoroughly.Foreachvalueofn=500,1000,5000,1000,20000andpifromthedenedgridwegeneratedinstancesofuniformrandomgraphsG(n,pi).Then,wetriedtondthemaximum-cliqueineachgeneratedinstance.Unfortunately,wecouldnotndtheexactsolutionsusingtheproposedformulations,sincetheseproblemsarecomputationallychallengingfortheconsideredvaluesofn,p,and.Therefore,weusedGRASPheuristics[ 1 ],whichperformsquitewellinmassivegraphs.Figure 4-1 reportstheresults.Onthehorizontalaxisweplotaparameterpofthegeneratedinstances.Ontheverticalaxisweplottherelativesizeofthemaximum-cliquedetectedbyGRASPalgorithmwithrespecttothesizeofthegraph.Eachlineonthegurerepresentshowtherelativemaximum-cliquesizechangesaspincreasesforeveryxedn. Intheprevioussectionweprovedthefactthatforanyxedp
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andforanyxedp>withprobability1limn!1ln n=1. Therefore,intuitively,asnincreases,therelativesizeofthemaximum-cliqueshoulddecreaseandtendtozeroforanyxedp<.Clearly,theobtainedresultsarequitesupportivetothat.Also,asnincreases,thechangesintherelativemaximum-cliquesizearoundthepointp=becomemoreabrupt. Figure4-1. Relativesizeofthemaximum-cliquesintheuniformrandomgraphsfor=70%,n=500,1000,5000,1000,20000,and0
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Table4-2. Averagemaximum-cliquesizesintheuniformrandomgraphsforn=100,and=85%.Thereare100scenariosgeneratedforeveryp.Thelowerandupperboundsarecalculatedfrom( 4 ). pLowerBound(k1n)UpperBound(kn)ScenarioAverage 0.053.074.323.090.063.274.663.190.073.464.983.430.083.655.303.820.093.825.624.130.104.005.944.430.114.176.264.680.124.346.584.970.134.516.915.120.144.687.255.320.154.857.595.65 85

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CHAPTER5IMPLIEDCOPULACDOPRICINGMODEL:ENTROPYAPPROACH Themarketofcreditriskderivativeswasboomingbeforetherecentnancialcrisis.CollateralizedDebtObligations(CDOs)accountedforasignicantfractionofthismarket.TheappealofCDOswasinhighprotmargins.CDOsofferedreturnsthatweresometimes2-3%higherthancorporatebondswiththesamecreditrating.ACDOisbasedonsocalledcredittranching,wherethelossesoftheportfolioofbonds,loansorothersecuritiesarerepackaged.InourworkweconsidersyntheticCDOsinwhichtheunderlyingcreditexposuresaretakenwithCreditDefaultSwaps(CDSs)ratherthanwithphysicalassets.Typically,inaCDO,fourtranchesaredesignatedassenior,mezzanine,subordinate,andequity.Lossesareappliedtothelaterclassesofdebtbeforeearlierones.Fromtheunderlyingpoolofinstruments,arangeofproductsarecreatedrangingfromtheveryriskyequitydebttotherelativelyrisklessseniordebt.Eachtrancheisspeciedbyitsattachmentanddetachmentpointsasthepercentagesofthetotalcollateral.Thelowertrancheboundaryiscalledtheattachmentpoint,whiletheuppertrancheboundaryiscalledthedetachmentpoint.TheCDOtranchelossoccurswhenthecumulativecollaterallossexceedsthetrancheattachmentpoint. Thetranchespreadisdenedasafractionofthetotalcollateral.Theamountofmoneythattheoriginatingbankshouldpayperyear(usually,paymentsaremadequarterly)tohavethistrancheinsuredisthespreadtimesthetranchesize.ThepricingofCDOcontractsisadifcultquantitativeproblemfacedbycreditriskmarkets.Themainissueisuncertaintyaboutobligorsdefaultrisk.Thischapterconsidersaso-calledimpliedcopulaCDOpricingmodelforcalibratingobligorhazardrates.Theideaofthismodelisthat,conditionalondifferentmarketstates,theobligorshavedifferenthazardrates.Forexample,ifthemarketgoesupthentheobligormayhavealowerriskofdefault(lowhazardrate),orifthemarketgoesdownthenitismorelikelyfortheobligortodefaultduringthecontractperiod(highhazardrate). 86

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Tondtheprobabilitydistributionofhazardrates,HullandWhite[ 27 ]suggestedtheso-calledimpliedcopulamodel.ThisisnotaspeciccopulalikeGaussian,Student-t,ordouble-t.Itiscalledimpliedbecauseitcanbededucedfrommarketquotes.TheCDOtranchequotesareusedforcalibration.Weconsideredthesimplestversionoftheimpliedcopulaapproachinwhichitisassumedthatallcompaniesbeingmodeledhavethesamehazardratesandthesamerecoveryrates(homogeneouscase).Dependingonthemarketscenariothesehazardratesaredifferent.Tosatisfymarketquotesweneedasearchforprobabilitiestoapplytoindividualhazardrates.TheHullandWhite[ 27 ]modelminimizesthesumofdeviationsfromno-arbitrageequationsandasmoothingterm.Themotivationinthedeviationtermcomesfromtheequalitybetweenthemid-priceoftheCDOtrancheandtheexpectedpayoffonthistranche(no-arbitrageconstraintinrisk-neutralsetting).ThisequalitymaynotbefeasibleforsomeCDOpricequotes.Thesmoothingtermisintroducedtoreducethenoiseinthedistribution.Weobserved,however,thattheoptimalsolutionisquitesensitivetothesmoothingtermcoefcient. Thischapterproposesanalternativeentropyapproachtotheimpliedcopulamodel.Wefoundthedistributionbymaximizingtheentropywithno-arbitrageconstraintsbasedonbidandaskpricesofCDOtranches.Inournumericalexperimentstheseconstraintswerefeasibleandwedidnotneedtointroducethedeviationfromno-arbitrageconstraints.Toreducethenoiseinthedistributionweintroducedanewclassofdistributions,calledCCCdistributions.Thisisawideclassofdistributionfunctionscontainingthenormal,gamma,andtheFdistributions.Bydenition,forthecontinuousdistributionsinthisclass,thePDFisconvexfromthebeginningtosomepoint,thenitisconcavetosomefurtherpoint,andthenitisagainconvextotheend.WecalledthisclassofdistributionstheCCCdistributions(CCCistheabbreviationforconvex/concave/convex).Fordiscretedistributionswegeneralizedthispropertytothe 87

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pointswherethediscretedistributionisdened.Byadiscretedistributionwemeanafunctionwhichassignssomeprobabilitytoeachofnitelymanyhazardrates. ThechapterpresentsacasestudyimplementingtheHullandWhitemodelandourentropyapproaches.WehavedemonstratedtheapproacheswiththeDecember,2006iTraxxtranchequotes.ThedatawereobtainedfromtheArnsdorfandHalperin[ 29 ]paper.Wedecidedtousetheirdatasincethereisinformationaboutbidandaskquotes.Wealsodemonstrateresultsforthemorerecentdatawherethemarketwasinunstablecondition.TodothecasestudyweusedthePortfolioSafeguard[ 31 ]package(MATLABEnvironment)byAmericanOptimalDecisions,Inc.Thecasestudyshowsthattheentropyapproachhasastableperformance,whiletheHullandWhitemodelissensitivetothesmoothingcoefcientandthenumberofhazardratesonthegrid. 5.1ConventionalCopulaandtheImpliedCopula ThissectionsummarizestheimpliedcopulaapproachproposedbyHullandWhite.ForafullmodeldescriptionareadermayrefertotheHullandWhite[ 27 ]paper.Aone-factorGaussiancopulamodel,rstintroducedbyLi[ 32 ],hasbecameanindustrystandard.Itmodelsdefaultintensitiesasaweightedsumofamarketfactorandanidiosyncraticterm,arm-dependentcomponent.Themodelprovidesacorrelationstructurebetweendefaultintensitiesofdifferentobligors. DenedefaultintensitiesXi(1in)by: Xi=iV+q 1)]TJ /F4 11.955 Tf 11.96 0 Td[(2iWi,(5) whereVisamarketfactorandWiisanidiosyncraticterm(rm-dependentcomponent).LetQi(t)bethecumulativedistributionof(unconditional)timetodefaultofcompanyiandletFi(t)bethecumulativedistributionofXi.DefaultintensityisthenmappedtodefaulttimeiasFi(Xi)=Qi(i). AconvenientwayofdeningQiisthroughacompanyhazardrate.Thelatterhasaninterpretationofdefaultintensityifthedefaultismodeledasthersteventin 88

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anonhomogeneousPoissonprocess.Thehazardratei()isrelatedtoQi(t)inthefollowingway: i()=1 1)]TJ /F3 11.955 Tf 11.96 0 Td[(Qi()dQi() d.(5) Hazardratesarepopularincreditriskapplicationsduetoeaseofimplementation,convenientanalyticexpressionsandclearphysicalinterpretation. Wedeneagrid1,...,Iofpossiblehazardrates.ThehazardratecanbeviewedastheseverityofthecreditenvironmentoverthelifeoftheCDOcontract.Weassumethatineachscenariothehazardrateisconstantandthesameforallobligors.AsinHullandWhite[ 28 ],wesetthelowesthazardratesuchthatthereisalmostnochancetodefault(1=10)]TJ /F7 7.97 Tf 6.59 0 Td[(8),andthehighesthazardratesuchthatalmostallcompaniesdefaultimmediately(I=100).Theintermediatehazardratesarechosensothatthelnkareequallyspaced.Wepresentresultsforthenumberofhazardratesonthegridfrom100to1,000.Wetrytondoutiftheincreasingnumberofscenariosofhazardratesleadstosomelimitingdistribution.Thispropertyisexpectedfromawelldenedmodelwheretheprecisionincreasinglyimprovestheperformanceofthemodel.Foraspecicvalueofthemarketfactor,defaultsofeachcompanyorobligationareindependentanddescribedbytheirconditionalhazardrates.Thesehazardratesaresimultaneouslyhigherorlower.HullandWhiteproposedaso-callimpliedcopulamodelprescribingthesameunconditionalhazardratetoeachcompanyandthenmovedallhazardratessimultaneously(or,moreprecisely,proportionally)sothatthecollateralhazardtakesonpre-denedvalues1,...,I.Theyalsoproposedanextensiontothismodelwheretheyassumeacorrelationbetweenobligors[ 27 ].Thescenariosforhazardrateihaveprobabilitiespitooccur. TottheprobabilitydistributionforhazardratestothemarketweconsiderCDOpricedata.Weusethe5-yearquotesforiTraxxindextranchesonDecember,2006.Weusethedatafrom[ 29 ].Wefoundthatpaperusefulsinceitprovidesthebidand 89

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askquotes.Bysamplingdefaultscenarioscorrespondingtoeachlevelofi,thenetpayoff(thedifferencebetweenexpectedpresentvalueofpremiumlegpaymentsanddefaultlegpayments)ofeachtranchejcanbedetermined,conditionalonthehazardratescenarioi.Denotethispayoffbyaij.Notethatthisnetpayoffiscalculatedwiththemid-quotesforthespreadsforeverytranche.Later,wewilldescribehowthebidandaskquotescanbeusedinno-arbitrageconsideration.Aprobabilitypiisassignedtoitoformaprobabilitydistributionofhazardrates.No-arbitrageconsiderationsinarisk-neutralsettingassumethattheexpectednetpayoffofeachCDOtrancheisequaltozero1 IXi=1aijpi=0j=1,...,J.(5) ThenumericalexperimentswiththemarketdatashowthatinsomecasesEquation( 5 )isinfeasible.InsuchcasesweneedtondadistributionapproximatelysolvingEquation( 5 ).Somecriterionhastobedenedtochooseadistributiontheclosesttoafeasibleone.HullandWhite[ 27 ]proposedsolvingthefollowingoptimizationproblemtondasuitableprobabilitydistribution: ProblemAminp(D(p)+S(p)) subjectto probabilitydistributionconstraints IXi=1pi=1,(5) 1Tranchepayoffs(withbothpaymentlegsincluded)havetobezerounderno-arbitrageassumptions.Thetranchespreadhastobeestablishedatsuchalevelthattheexpectedpayoffsthroughthepremiumlegarepreciselyequaltotheexpecteddefaultlosses,inotherwords,sothatthepremiumleghasthesamepresentvalueasthedefaultleg. 90

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pi0,i=1,...,I.(5) whereD(p)isadeviationterm D(p)=JXj=1 IXi=1piaij!2,(5) andS(p)isasmoothingterm S(p)=cI)]TJ /F7 7.97 Tf 6.58 0 Td[(1Xi=2pi+1+pi)]TJ /F7 7.97 Tf 6.58 0 Td[(1)]TJ /F5 11.955 Tf 11.96 0 Td[(2pi 0.5(di+1)]TJ /F3 11.955 Tf 11.95 0 Td[(di)]TJ /F7 7.97 Tf 6.58 0 Td[(1)2.(5) Thedeviationtermpenalizeslargedeviationsfromzeroofthenetexpectedpayoffofeverytranche.Thesmoothingtermenforcesthateverythreeconsecutivepointsonthehazardratedistributionareapproximatelyonthesameline.Thesmoothingtermislargerforlargerdifferencesfromthestraightline. Thesmoothingtermintroducesdistortionintoresultingdistribution,butareasonablelevelofdistortionmaybebetterthanaraggeddistributionshape.Thesmoothingeffectappearstodecreasewiththeincreaseinthenumberofatomsinthedistribution.Also,thecoefcientchastobechosenbytrialanderror. LetusconsiderforinstancethecasepresentedinthepaperofHullandWhite[ 27 ].Weranthecasestudytoanalyzeresults.First,weusedthedatafromFigure 5-1 tosimulatetheexpectedcashowsoneverytranchefordifferenthazardrates.ThenwesolvedtheoptimizationProblemA.Figures 5-2 and 5-3 showgraphsofoptimaldistributionsobtainedfordifferentnumbersofpointsanddifferentvaluesofthesmoothingtermcoefcientc. Itseemsthatifwendagoodsmoothingcoefcientcforaparticularnumberofatomsinthedistributions,itwillnotworkthesamewayifwechangethenumberofatoms.Therefore,thesmoothingcoefcientcshouldbechosenindividuallyforevery 91

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numberofpointsonthehazardrategrid.Moreover,wecannotndareasonablejusticationforwhyonesmoothingcoefcientisbetterthenanyotherone. Thenextsectionproposesanalternativeapproachwhichwecalltheentropyapproachforndingthebestprobabilitydistribution.Withthisapproachwetriedtoexcludefromthemodelarbitraryparameters,suchasthesmoothingcoefcient. 5.2ImpliedCopula:EntropyApproach Thissectionproposesanentropyapproachtotheimpliedcopulamodel,discussesthereasonswhyandwhensuchanapproachisuseful,andprovidesaheuristicalgorithmtondthebestprobabilitydistributionforhazardrates.HullandWhite[ 27 ]minimizethesumofsquareddeviationsoftranchepayoffsfromperfectt( 5 )andthesmoothingterm( 5 ).Intheirrecentpaper[ 28 ]thereisanotherapproachtondasuitableprobabilitydistribution.Theauthorsassumethatthedistributionisalog-tandcalibrateitsparameterstotthemarketquotes.WeproposeanalternativemaximumentropyprincipleandsuggestndingthedistributionintheclassofCCCdistributions(whichwillbedescribedlater). TheMaximumEntropyPrinciple(rstintroducedbyShannon,seealsoGolan[ 33 ])ispopularininformationtheory.Thisprincipleisactivelyusedinnancialapplications;seeforinstanceMillerandLiu[ 34 ],ChuandSatchell[ 35 ].TheessenceoftheMaximumEntropyPrincipleisthatwithsomegiveninformationaboutthedistribution(speciedthroughequationsandconstraints)wemaximizetheentropyandselectthemostunknowndistribution.Thereforewearetryingtondthemostunknowndistributioncontainingonlyavailableinformationaboutthedistribution. InthisrespectwewanttopointoutthattheinformationwhichisusedinHullandWhitemodelisnotcomplete.Thenon-arbitrageequationsareusedformid-spreadsfortranches.Thatmaybeareasonwhytheconstraintsmaynothaveafeasiblesolution.Wearguethatadditionalinformationisavailableinthebidandaskprices. 92

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Insteadofmidprices,weusebidandaskprices.Denotebya ijand aijtheexpectednetpayoffoftranchejconditionalonhazardrateiforaskandbidprices,respectively.Then,theno-arbitrageconstraintsareasfollows:foraskpricestheexpectednetpayoff(PIi=1a ijpi)ofeachCDOtrancheisnonpositiveandforbidprices(PIi=1 aijpi)isnonnegative. WemaximizeShannonentropyH(p)=)]TJ /F8 11.955 Tf 11.29 8.96 Td[(PIi=1pilnpisubjecttotheseno-arbitrageconstraints.Inotherwords,weproposetosolvethefollowingproblem: ProblemBminp)]TJ /F3 11.955 Tf 9.3 0 Td[(H(p) subjectto no-arbitrageconstraints IXi=1a ijpi0,j=1,...,J,(5) IXi=1 aijpi0,j=1,...,J,(5) probabilitydistributionconstraints IXi=1pi=1,(5) pi0,i=1,...,I.(5) WewanttoemphasizethatthesetofprobabilitydistributionssatisfyingConstraints( 5 ),( 5 )islargerthanthesetsatisfyingConstraints( 5 ).Therefore,Constraints( 5 ),( 5 )mayhaveafeasiblesetandwemaynotneedtointroducethedeviationtermD(p)totheobjective. HullandWhiteaddasmoothingtermtotheobjectivefunctionin( 5 ).Asweindicatedearlierintheprevioussection,theoptimalsolutionisverysensitiveto 93

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thechoiceofcin( 5 )andtothenumberofpointsIonthegrid(Figures3and4).Furthermore,itseemsthatwithanincreasingnumberofpointsIintheoptimizationProblemA,theoptimalsolutiondoesnotstabilize.Somekindofstabilizationcanbeseenforc=10)]TJ /F7 7.97 Tf 6.59 0 Td[(5attherightbottomgraphinFigure4.But,again,itisunclearwhyc=10)]TJ /F7 7.97 Tf 6.59 0 Td[(5shouldbeused. Herewewanttoquicklymentionwhatwedidnextandhowwecameuptotheconclusiontointroducethenewclassoffunctions.WesolvedProblemBfordifferentnumbersofpointsalso.Wefoundthattheshapeoftheoptimalsolutioneventuallystabilized.Beyondthenumberofpointsbeingequalto500,theoptimalsolutionshaveasimilarshape.Butwesawalsothatsomenoisewaspresentintheoptimaldistribution.TocopewiththatwewilllaterdenetheCCCclassofdiscreteprobabilitydistributions. WestartwiththegeneraldenitionofCCCclassoffunctions(notonlydistributions).Wesaythatafunctionbelongstothisclassifitisconvexontheleftuptosomepoint,thenconcaveuptoafurtherpoint,andthenagainconvexontheright.CCCistheabbreviationforconvex/concave/convex.Figure 5-1 showsanexampleofCCCfunction. Bydenition,afunctionf:R!Risconvexifforanyx1,x2,:2[0,1]thefollowinginequalityholds:f(x1)+(1)]TJ /F4 11.955 Tf 11.95 0 Td[()f(x2)f(x1+(1)]TJ /F4 11.955 Tf 11.95 0 Td[()x2) Letx3=x1+(1)]TJ /F4 11.955 Tf 12.72 0 Td[()x2;then(x2)]TJ /F3 11.955 Tf 12.73 0 Td[(x1)=x2)]TJ /F3 11.955 Tf 12.73 0 Td[(x3.Therefore,thereisonetoonecorrespondencebetweenandx3andtheconvexitypropertycanberewrittenasfollows: foranyx1,x2,x3suchthatx1x3x2followinginequalityholds:(x2)]TJ /F3 11.955 Tf 11.95 0 Td[(x3)f(x1)+(x3)]TJ /F3 11.955 Tf 11.96 0 Td[(x1)f(x2)(x2)]TJ /F3 11.955 Tf 11.96 0 Td[(x1)f(x3). Withthisobservationwecangeneralizetheconcavity/convexitypropertytoanysetXRnotnecessarilyconvex,closed,etc.Wesaythatf:X!RisconvexonXiffor 94

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anyx1,x2,x32X:(x2)]TJ /F3 11.955 Tf 11.95 0 Td[(x3)f(x1)+(x3)]TJ /F3 11.955 Tf 11.96 0 Td[(x1)f(x2)(x2)]TJ /F3 11.955 Tf 11.95 0 Td[(x1)f(x3) BelowistheformaldenitionoftheCCCclassoffunctionsingeneralcase. Denition4(generalcase). Letf:X!R.Thenf(x)belongstoCCCclassifandonlyifthereexistwl,wr2Rsuchthatthefollowinginequalitieshold: wlwr, (x2)]TJ /F3 11.955 Tf 11.96 0 Td[(x3)f(x1)+(x3)]TJ /F3 11.955 Tf 11.95 0 Td[(x1)f(x2)(x2)]TJ /F3 11.955 Tf 11.96 0 Td[(x1)f(x3),forallx1x3x22(,wl]\X, (x2)]TJ /F3 11.955 Tf 11.96 0 Td[(x3)f(x1)+(x3)]TJ /F3 11.955 Tf 11.95 0 Td[(x1)f(x2)(x2)]TJ /F3 11.955 Tf 11.96 0 Td[(x1)f(x3),forallx1x3x22[wl,wr]\X, (x2)]TJ /F3 11.955 Tf 11.94 0 Td[(x3)f(x1)+(x3)]TJ /F3 11.955 Tf 11.94 0 Td[(x1)f(x2)(x2)]TJ /F3 11.955 Tf 11.95 0 Td[(x1)f(x3),forallx1x3x22[wr,+1)\X. First,wedenetheCCCclassofcontinuousdistributions. Denition5(continuouscase). Letf:R!Rbeacontinuousdensityfunctionofsomecontinuousdistribution.Thenf(x)belongstoCCCclassofdistributionsiff(x)isaCCCfunction. Inourmodelwedealwithdiscretedistributions.LetusdenetheCCCclassofdiscretedistributions. Denition6(discretecase). Letf:fd1,...,dIg![0,1]beaprobabilitymeasurefunctiononasequenceofpointsd1,...,dI:d1
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1wlwrI, (di+1)]TJ /F3 11.955 Tf 11.95 0 Td[(di)f(di)]TJ /F7 7.97 Tf 6.59 0 Td[(1)+(di)]TJ /F3 11.955 Tf 11.95 0 Td[(di)]TJ /F7 7.97 Tf 6.59 0 Td[(1)f(di+1)(di)]TJ /F7 7.97 Tf 6.58 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(di+1)f(di),foralli:1
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ThedrawbackisthatthereisnoeconomicargumentallowingustoclaimthattherealhazardratedistributionbelongstotheCCCclass.Itiseasytoimaginesituationswhensomefutureeventisexpectedtoseriouslyaffectthehazardrates,andinvestorsaredividedintotwocampswithverydifferentexpectationsofhazardrates.Still,theadvantageisthatanynoiseisbeingeffectivelylteredout. TosolveProblemBintheCCCclassofdistributionsweuseProposition 5.2 tointroduceCCCconstraints.Wewanttomentionagainthatallln(i)areequallyspacedontheinterval[ln(10)]TJ /F7 7.97 Tf 6.58 0 Td[(8),ln(100)]inthissetting. TheCCCconstraintsincludeconstraintsontheleftslope,rightslopeandhump: Convexityoftheleftslope: pi)]TJ /F7 7.97 Tf 6.58 0 Td[(1+pi+1 2pi,i=2,...,wl)]TJ /F5 11.955 Tf 11.95 0 Td[(1,(5) Concavityofthehump: pi)]TJ /F7 7.97 Tf 6.59 0 Td[(1+pi+1 2pi,i=wl+1,...,wr)]TJ /F5 11.955 Tf 11.95 0 Td[(1,(5) Convexityoftherightslope: pi)]TJ /F7 7.97 Tf 6.59 0 Td[(1+pi+1 2pi,i=wr+1,...,I)]TJ /F5 11.955 Tf 11.96 0 Td[(1,(5) Thepointsw1,w2mayvaryfordifferentdiscretedistributions,thereforeweincorporatethemintotheoptimizationproblemasvariables.ByaddingtheCCCconstraintstoProblemBwehavethefollowingoptimizationproblem: ProblemCminwl,wr,p)]TJ /F3 11.955 Tf 9.3 0 Td[(H(p) subjectto no-arbitrageconstraints 97

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IXi=1a ijpi0,(5) IXi=1 aijpi0,(5) CCCconstraints: constraintsoninectionpoints 1wlwrI,(5)convexityoftheleftslope pi)]TJ /F7 7.97 Tf 6.58 0 Td[(1+pi+1 2pi,i=2,...,wl)]TJ /F5 11.955 Tf 11.95 0 Td[(1,(5)concavityofthehump pi)]TJ /F7 7.97 Tf 6.59 0 Td[(1+pi+1 2pi,i=wl+1,...,wr)]TJ /F5 11.955 Tf 11.95 0 Td[(1,(5)convexityoftherightslope pi)]TJ /F7 7.97 Tf 6.59 0 Td[(1+pi+1 2pi,i=wr+1,...,I)]TJ /F5 11.955 Tf 11.96 0 Td[(1,(5) probabilitydistributionconstraints IXi=1pi=1,(5) pi0,i=1,...,I.(5) Letuslookatasubproblemofthisproblem.Toformulatethisproblemsupposethatwexedwl,wr. 98

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ProblemC(wl,wr)minp)]TJ /F3 11.955 Tf 9.3 0 Td[(H(p) subjectto no-arbitrageconstraints IXi=1a ijpi0,(5) IXi=1 aijpi0,(5) CCCconstraints: convexityoftheleftslope pi)]TJ /F7 7.97 Tf 6.58 0 Td[(1+pi+1 2pi,i=2,...,wl)]TJ /F5 11.955 Tf 11.95 0 Td[(1,(5)concavityofthehump pi)]TJ /F7 7.97 Tf 6.59 0 Td[(1+pi+1 2pi,i=wl+1,...,wr)]TJ /F5 11.955 Tf 11.95 0 Td[(1,(5)convexityoftherightslope pi)]TJ /F7 7.97 Tf 6.59 0 Td[(1+pi+1 2pi,i=wr+1,...,I)]TJ /F5 11.955 Tf 11.96 0 Td[(1,(5) probabilitydistributionconstraints IXi=1pi=1,(5) pi0,i=1,...,I.(5) Clearly,tosolveProblemCweneedtosolveProblemC(wl,wr)forallpossiblepairsofintegerswl,wrsuchthat1wlwrI,andthenchoosetheminimumamong 99

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thesesolutions.Theoretically,theminimumshouldexist,butitmaynotbeunique.Inthiscase,wecanpickanysolutionwiththisminimum.Thenumberofsubproblems(ProblemC(wl,wr))tosolveistheorderofn2.RecallthatoriginallyweproposedtosolveProblemB,butsinceinourexperimentsitssolutionshaveanoise,wesuggestedtondthesolutiontoProblemBintheCCCclassoffunctions(ProblemC).WeprovideaheuristicalgorithmforsolvingProblemC.WesolveatrstProblemBandthenasequenceofProblemC(wl,wr)'sfordifferentpairsof(wl,wr).WedonotprovethatthisalgorithmprovidesanexactsolutionforProblemC. Hereistheformaldescriptionofalgorithm.Explanationsareprovidedaftertheformaldescription. Algorithm: Step0.Initialoptimalsolution. SolveProblemBanddenoteitssolutionobtainedforoptimizationproblembyp. Initializewl=wr=argmaxfpi:i=1,...,Ig2,k=0,H0=1. Step1.SolveProblemC(wl,wr) Setk=k+1,exit ag=0. SolveProblemC(wl,wr)andobtaintheoptimalsolutionpkandHk=H(pk). Step2.Shiftingwrtotheright Ifwr1thensetwl=wl)]TJ /F5 11.955 Tf 11.96 0 Td[(1. 2Ifthemaximumisnotunique,thealgorithmshouldbeperformedforeashpointinthesetargmaxfpi:i=1,...,Ig,andthenthesolutionwiththeleastobjectivevalueshouldbechoisen. 100

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Ifwl=1thenstopthealgorithm,andpk)]TJ /F7 7.97 Tf 6.58 0 Td[(1isanapproximationoftheoptimalsolution. Step4.SolveProblemC(wl,wr)(thesameasStep1) Setk=k+1. SolveProblemC(wl,wr)andobtaintheoptimalsolutionpkandHk=H(pk). Step5.Shiftingwltotheleft Ifwl>1andHkHk)]TJ /F7 7.97 Tf 6.59 0 Td[(1thensetwl=wl)]TJ /F5 11.955 Tf 11.95 0 Td[(1,exit ag=1,andgotoStep4. Ifexit ag=1,thengotoStep1. If(wl=1orHk>Hk)]TJ /F7 7.97 Tf 6.58 0 Td[(1)andexit ag=0,thenstopthealgorithm,andpk)]TJ /F7 7.97 Tf 6.58 0 Td[(1isanapproximationoftheoptimalpoint. Theideaofthisalgorithmisthatwestep-by-stepchangeinectionpointswl,wrandsolveProblemC(wl,wr).InStep0wesolveProblemBandobtainanoptimalsolutionp.Thenwesetwl=wr=argmaxfpi:i=1,...,Ig.Inotherwords,wendthemaximumcomponentofoptimalvectorpandmakewl,wrequaltoitsindex.InStep1wesolveProblemC(wl,wr)withthesewl,wrandobtaintheoptimalpointanditsobjectivevalue.Then,weshiftwrtotherightifitispossible,makingwr=wr+1.AfterthatwegotoStep1andagainsolveProblemC(wl,wr)toobtaintheoptimalpointanditsobjectivevalue.ThenwecomparethisobjectivevaluewiththepreviousoneobtainedinStep1(HkandHk)]TJ /F7 7.97 Tf 6.59 0 Td[(1).Thisprocedurestopswhenthenewobjectivevalueisgreaterthenthepreviousone(Hk>Hk)]TJ /F7 7.97 Tf 6.59 0 Td[(1),orwr=I.InSteps3to5werunthesameprocedure,butnowweshiftwltotheleft.Theprocedurealsostopswhenthenewobjectivevalueislargerthenthepreviousone(Hk>Hk)]TJ /F7 7.97 Tf 6.58 0 Td[(1),orwl=1.Ifduringthesteps1through4thelessobjectivevalueisfoundbyshiftingwrorwl,thenthesestepsareneededtobeperformedagain.Inotherwords,weshiftthepointswrandwltoreachlocaloptimality.Finally,thealgorithmreturnspk)]TJ /F7 7.97 Tf 6.58 0 Td[(1whichisconsideredasanoptimalpoint.Wedonotprovethatthisalgorithmprovidesanoptimalsolutionto 101

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ProblemC.WhatweobserveinourcasestudyisthatthisalgorithmworksfasterthansolvingProblemCandprovidesareasonablesolution. 5.3CaseStudy WeusedPortfolioSafeguard[ 31 ]inMATLABenvironmenttodothecasestudy.ForthecasestudywehaveconsiderediTraxxindexwithdifferentmaturities. First,weused5-yeariTraxxtranchequotestosimulatetheexpectedcashowmatrices.Forbidprices,midpricesandaskpriceswesimulateddifferentmatrices( aij)j=1,...,Ji=1,...,I,(aij)j=1,...,Ji=1,...,I,)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(a ijj=1,...,Ji=1,...,IforI=100,200,...,1,000.ThenumberoftranchesintheiTraxxindexissix,soJ=6. Forparticulari,jwesimulatedthetimestodefaultof125companiesintheiTraxxindexandthecorrespondingtranchecashows10,000timesandthantooktheaverage.Aswementionedearlier,thetimetodefaultofeachcompanyisexponentiallydistributedwithparameterj.Forsimulationweusedtheminimumhazardrate1=10)]TJ /F7 7.97 Tf 6.59 0 Td[(8,themaximumhazardrateI=100,andthedistancesbetweenln(i)areequal.Weassumedthatthetranchepaymentsaremadequarterly,therecoveryrateincaseofdefaultsubjecttoto40%andtheannualriskfreerateis4%.ThereadermayrefertotheHullandWhite[ 27 ]tondmoredetailsonthesimulationprocedure.Thisisaquitecommontechniqueandwedonotfocusonit. WesolvedProblemAforI=100,300,500and1,000points.ItshouldbenoticedthatintheirrecentpaperHullandWhitetestedanotherapproach[ 28 ].Theyusedoptimizationproceduretondalog-tdistributionttingthedistributionsofhazardrates.InthischapterwecomparedourapproachtotheapproachbyHullandWhite[ 27 ].WeusedsixdifferentsmoothingtermcoefcientsinProblemAtocompareresults.ThegraphsarepresentedinFigures 5-2 and 5-3 .Thedistributionfunctionsinthegraphsarenottheactualsolutionvectors.Wescaledthemsothattheareasunderthegraphareequalandthehorizontalaxisrepresentsln().Thesegraphscanbeviewedasimplied 102

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densitiesofhazardratedistributions.TheresultsarequitesensitivetotheparameterscandI. WithourapproachwerantheproposedheuristicalgorithmdescribedattheendofSection2for100,200,300,500,800and1,000points.TheentropymaximizationproblemcanbeeasilysolvedwithPSGinMATLABenvironmentbycallingPSGriskprogoptimizationsubroutine.Weonlyneedtoputthematrixofconstraintsandentropyrasparametersforthissubroutine.3Figure 5-5 showssixhazardratedistributiongraphsforthesixdifferentvaluesofImentionedaboveandhowthenaldistribution~p1differsfromtheintermediate~p0whichistheoptimalsolutionforProblemB.WefoundthatimposingCCCfunctionconstraintshasnotchangedsignicantlytheshapeofimplieddensityfunctions.Someirregularities(whichwecallnoise)werestreamlined. Figure 5-4 comparesthenaldistributions~p1fordifferentnumbersofhazardrates(I=100,200,300,500,800,and1,000)onthegrid.Wealsoscaledthemsothattheareasbelowthegraphsareequal.Thelastthreegraphsarealmostidentical,whichseemsquitenatural.WedidnotobservethesimilarstabilityintheHullandWhitemodelevenwithaxedsmoothingcoefcientc. WeappliedourapproachtotheiTraxxtrancheswithdifferentcontractperiods:5years,7yearsand10years.Itshouldshowwhethertheimpliedcopulamodelcanbeusedwiththehomogeneityassumption,i.ethathazardrateofthecompanystaysthesameduringthewholecontractperiod.Ifthisapproachisreasonable,weshouldobtainasimilardistributionofhazardratesfordifferentcontractperiods.WesimulatedthematricesofexpectedcashowsusingthepricesfromTable 5-1 forI=100for5,7and 3WeconductedthecasestudyonalaptopwithprocessorIntelCore2CPU@2GHz.TheoptimizationtimeforProblemBvariedfrom0.01sec.forI=100to0.06sec.forI=1,000,forProblemCitvariedfrom0.36sec.forI=100to600sec.forI=1,000. 103

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10-yearcontracts.Then,weusedthesematricestosolveProblemBandproposedheuristics.Figure 5-6 representsthesolutionsscaledthesimilarwayandwithln()onthehorizontalaxis.Thegraphsarequitesimilarandshowlittledependenceofthelengthofthecontractperiod. Wewanttopointoutthattheanalyzeddatawerethemarketquotesfor5,7,10-yeariTraxxonDecember20,2006.Atthattimethecreditderivativesmarketwasourishingandexpandingveryfast.Wealsotestedthismodelforthedatatakenfortherecenttimeswhenthemarketwasveryunstable. First,weusedthedataforthemarketquotesforthe5-yeariTraxxonfourdifferentdates:10/31/07,12/31/07,6/30/08and9/30/08.Thedataavailabletouscontainsonlytheclosingprices.Togetthebidandaskpricesweusedtypicalbid-askspreadsforthattimesvaryingfrom2%to7%dependingonthetranche.Then,usingthesimulationtechniquedescribedinthebeginningofthissection,wesimulatedexpectedcashowmatricesforthebidandaskpriceswiththenumberofhazardrategridpointsI=100.TheimplieddensityfunctionswereobtainedbysolvingProblemB.Figure 5-7 showscorrespondinggraphs.Thegraphsshowtheevolutionofthehazardratedistributionfunctionoverthetime.WewanttomentionthatProblemCisinfeasiblewiththeassumedbid-askspreads. Second,wepickedthetwolatestdatesforwhichwehavethepriceinformationforthemarketquotesfor5,7,10-yeariTraxx.Theexpectedcashowmatricesweresimulatedthesameway.Figure 5-8 showsthehazardratedistributionfunctionsforthiscase. Finally,wewanttomentionthattheobtainedhazardratedistributionscanbeusedforthepricingofvariouscreditriskinstruments. 104

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Table5-1. Marketquotesfor5,7,10-yeariTraxxonDecember20,2006.Quotesforthe0to3%tranchearethepercentoftheprincipalthatmustbepaidupfrontinadditionto500basispointsperyear.Quotesforothertranchesandtheindexareinbasispoints.ThedatawasobtainedfromArnsdorfandHalperin[ 29 ]. MaturityLowStrike(%)HighStrike(%)Bid(%)Ask(%) 20-Dec-110311.7512.0020-Dec-113653.7555.2520-Dec-116914.0015.5020-Dec-119125.756.7520-Dec-1112222.132.8820-Dec-11221000.801.3020-Dec-11010024.7525.2520-Dec-130326.8827.1320-Dec-1336130.00132.0020-Dec-136936.7538.2520-Dec-1391216.2518.0020-Dec-1312225.506.5020-Dec-13221002.402.9020-Dec-13010033.5034.5020-Dec-160341.8842.1320-Dec-1636348.00353.0020-Dec-166993.0095.0020-Dec-1691240.0042.0020-Dec-16122213.2514.2520-Dec-16221004.354.8520-Dec-16010044.5045.50 Figure5-1. ExampleofaCCCdistribution,wl=30andwr=50 105

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Figure5-2. Distributionsofthecollateralhazardrate,asimpliedin5-yeariTraxxtranchespreads.ThepricesfromTable 5-1 areused.TheimpliedcopulaofHullandWhiteapproachisused.ThedistributionswerefoundassolutionstoProblemAfornumbersofvariables100and300,anddifferentsmoothingtermcoefcientsc. 106

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Figure5-3. Distributionsofthecollateralhazardrate,asimpliedin5-yeariTraxxtranchespreads.ThepricesfromTable 5-1 areused.ThedistributionswerefoundassolutionstoProblemAfornumbersofvariables500and1000,anddifferentsmoothingtermcoefcientsc. 107

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Figure5-4. Distributionsofthecollateralhazardrate,asimpliedin5-yeariTraxxtranchespreads.ThepricesfromTable 5-1 areused.ThedistributionswerefoundassolutionstoProblemB,andtheproposedheuristicalgorithmtondasolutionintheCCCclassfor100,200,300,500,800,1000decisionvariables. 108

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Figure5-5. Distributionsofthecollateralhazardrate,asimpliedin5-yeariTraxxtranchespreads.ThepricesfromTable 5-1 areused.ThedistributionswerefoundintheCCCclassusingtheproposedheuristicalgorithmfor100,200,300,500,800,1000decisionvariables. 109

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Figure5-6. Distributionsofthecollateralhazardrate,asimpliedin5,7and10-yeariTraxxtranchespreads.ThepricesfromTable 5-1 areused.ThedistributionswerefoundassolutionstoProblemB(upperchart)andintheCCCclass(lowerchart)usingtheproposedheuristicalgorithmfor100decisionvariables. 110

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Figure5-7. Distributionsofthecollateralhazardrate,asimpliedin5-yeariTraxxtranchespreadsindifferentdates.ThedistributionswerefoundassolutionstoProblemBfor100decisionvariables. 111

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Figure5-8. Distributionsofthecollateralhazardrate,asimpliedin5,7and10-yeariTraxxtranchespreadsintwodifferentdates.ThedistributionswerefoundassolutionstoProblemBfor100decisionvariables. 112

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CHAPTER6OPTIMALSTRUCTURINGOFCDOCONTRACTS AnoptimalstructuringtechniquesmayhelptoincreaseprotabilityofCDOsandothersimilarderivatives.InatypicalCDOcontracttheattachmentanddetachmentpointsforeachtranchearethesameforthewholecontractperiod.Therefore,thebank-originatorshouldmakethesamepaymentseveryperiod(iftrancheisnotdefaulted).Thischapterconsidersstep-upCDOswheretheattachment/detachmentpointsmayvaryduringthelifeoftheCDO(typicallyincreaseeachtimeperiod).Aspecicriskexposurecanbebuiltineachtimeperiodbeforethecontractmaturity.ThischapteralsopresentsanoptimizationapproachfordeterminingtheattachmentpointsinaCDOcontractandselectingasetofinstrumentsintheCDOpool.Weconsiderseveralproblemstatements. Withtherstproblemstatementwechangeonlyattachment/detachmentpointsinaCDO:thegoalistominimizepaymentsforthecreditriskprotection(premiumleg)whilemaintainingspeciccreditratingsoftranches.Inthiscasethepoolofinstrumentsandincomespreadsaresupposedtobexed.Weconsideredseveralvariantsoftheproblemstatementwithvariousassumptionsandsimplications.WiththesecondproblemstatementweputarestrictiononthetotalincomespreadpaymentsandsimultaneouslyoptimizethesetofinstrumentsinaCDOpoolandtheattachement/detachmentpoints,maintainingcreditratingsoftranches. Thecasestudysolvestheproblemunderdifferentcreditratingandotherconstraints.Itisbasedonthetimetodefaultscenariosforobligors(instruments)generatedbyStandard&PoorsCDOEvaluatorR[ 30 ]forsomeexampledata.WeusedPortfolioSafeguardpackage[ 31 ]tobuildoptimalCDOs.Theresultsshowthatusingastep-upCDOversusastandardCDOallowsabankoriginatortosaveabout25%-35%ofoutgoingtranchespreadpayments. 113

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NextsectionprovidesabriefdescriptionofCDOsanddiscussesgeneralideasinvolvedinCDOstructuring.Then,wedescribeoptimizationmodels,provideformaloptimizationproblemstatementsandoptimalityconditions,andpresentthecasestudy. 6.1Step-upCDOBackground ThissectionprovidesabriefdescriptionofCDOsandideasinvolvedinCDOstructuring.CDOisacomplexcreditriskderivativeproduct.CashCDOincludesapool(portfolio)ofassetswithsomefuturecashowssuchasbonds,loans,orothersecurities.Thispoolisusedtocreateanewsetofxedincomesecurities.Everyconstructedsecurityhasitsownrisk-returnproleandpassessomefractionofcreditrisktootherinvestors.Aninvestormaybuyconstructedsecuritiesaccordingtorisk-returnpreferences. Thischapterconsidersaso-calledsyntheticCDOs.ACDOconsistsofaportfolioofCreditDefaultSwaps(CDS).ACDSisacreditriskderivativewithabondasanunderlyingasset.Itcanbeviewedasaninsuranceagainstpossiblebondlossesduetocreditdefaultevents.TheCDSbuyerpaysacertaincashow(CDSspread)duringthelifeofthebond.Ifthisbondincurscreditdefaultlosses,theCDSbuyeriscompensatedforthatlosses.Typically,thehighertheratingoftheunderlyingbond,thesmallerthespreadoftheCDS.ItshouldbenotedthattheCDSbuyerdoesnotneedtoholdtheunderlyingbondinitsportfolio. AsyntheticCDOconsistsofapoolofCDSs.TheCDOreceivespayments(CDSspreads)fromeachCDSandcoverscreditrisklossesincaseofdefault.Therefore,thisportfoliocoverspossiblelossesuptothetotalcollateralamount.CDOoriginatorrepackagespossiblecreditrisklossestocredittranches.Typically,fourtranchesaredesignatedassenior,mezzanine,subordinate,andequity.Lossesareappliedtothelaterclassesofdebtbeforeearlierones.Therefore,fromthebasketofCDSs,arangeofproductsarecreatedrangingfromaveryriskyequitydebttoarelativelyrisklessseniordebt. 114

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Thetranchespreadpaymentisdenedasafractionofthecollateralinthistranche.ToinsurecreditlossesinatrancheCDOshould(payperyear)spreadtimesthetranchesize.Usually,paymentsaremadeonquarterlybasis.Thespreadofatrancheismostlydeterminedbyitscreditratingwhichisbasedondefaultprobabilityofthistranche. Figure 6-1 showsthestructureofCDOcashows.Thebank-originatorsellstheCDSs.Thenthebankrepackageslossesandbuysaninsurance(creditprotection)foreachtranche.IfthesumofspreadsofCDSsinaCDOpoolisgreaterthanthesumoftranchespreads,theCDOoriginatorlocksinanarbitrage.ThenextsectiondiscussesoptimizationmodelsforminimizingthesumoftranchespreadsconditioningthatthepoolofCDSsisxed.FurtherthesemodelsareextendedtothecasewhenbankoriginatorsimultaneouslychoosesCDSstothepoolandadjustsattachment/detachmentpointsoftranches. EachtrancheinaCDOcontractcangetit'sownrating,e.g.,AAA,AA,A,BBB,inStandardandPoor's(S&P)classication.Atrancheratingcorrespondstoaprobabilityofdefaultestimatedbyacreditagency.Forexample,atranchehasAAAS&Pratingiftheprobabilitythatthelosswillexceedtheattachmentpointduringthecontractperiodislessthan0.12%.InastandardCDOcontracttheattachment/detachmentpointsforeachtranchearethesameforthewholecontractperiod.Therefore,thebank-originatorshouldmakethesamepaymentseveryperiodtohaveCDOtranchesinsured.Thelossesarecumulatedovertime;therefore,theprobabilitythatalosswillhitatrancheattachmentpointintherstperiodismuchsmallerthantheprobabilitythatalosswillhititinthelastperiod.Weshowthatbychangingtrancheattachmentpoints,wecanmaintaintranchescreditratingsanddecreasethecumulativeamountofspreadpaymentsfromthebankoriginator. Therefore,weconsideraCDOwithincreasingovertimeattachment/detachmentpoints(Figure 6-2 ).SuchaCDOcreatesadesirableriskexposureineachtimeperiod. 115

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6.2OptimizationModels ThissectionpresentsseveraloptimizationmodelsforCDOstructuring,i.e.,theselectionofCDOunderlyinginstrumentsandattachment/detachmentpoints.Theobjectiveistomaximizeprotsforthebank-originator. 6.2.1OptimizationofAttachement/DetachmentPoints(withFixedPoolofAssets) First,weconsiderastructuringproblemforaCDOwithaxedpoolofassets.Weselectoptimalattachment/detachmentpointsfortranches.ConsideraCDOwithapoolofIdifferentequallyweightedCDSs,acontractperiodT,andxednumberoftranchesM.NotethatthereareonlyM)]TJ /F5 11.955 Tf 10.5 0 Td[(1attachment/detachmentpointstobedetermined,sincetheattachmentpointforthersttrancheisxedanditisequaltozero.Let sm=tranchemspread; Lt=cumulativecollaterallossbyperiodt+1; xtm=attachmentpointoftrancheminperiodt. Henceforth,wealwaysassumethatxt1=0,andxtM+1=1forallt=1,...,T.Allthevaluesabovearemeasuredinthefractionofthetotalcollateral.TheCDOisusuallystructuredsothateachtranche,amongotherpossiblefeatures,hasaparticularcreditrating.HereweassumethatthetranchespreadonthistypeofCDOsisfullydeterminedbyitscreditrating.Inotherwords,thevectorofspreads(s1,...,sM)isxedifweassureappropriateratingsfortranches.Noticethats1>s2>>sMsincethehigherthetranchenumberthehigheritscreditratingandtheloweritsspread.DuringtheCDOcontractperiodthelossesareaccumulatingandthebank-originatormakesthepaymentsonlyfortheremainingamountofthetotalcollateral.Thepaymentsareusuallymadequarterly.Forinstance,ifthesizeofthetrancheM(supersenior)is70%ofthesizeofthetotalcollateral,thenthebankoriginatorshouldpay(100%)]TJ /F3 11.955 Tf 12.04 0 Td[(max(30%,Lt))sMintheperiodttohavethistranche(oritsremainingpart)insured.Then,thetotal 116

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paymentsforalltranchesintheperiodtareMXm=1(xtm+1)]TJ /F3 11.955 Tf 11.96 0 Td[(max(xtm,Lt))+sm, wherethefunction()+isdenedasx+=8>><>>:x,x0,0,x<0. ThevectorL=(L1,...,LT)isarandomvector.InourcasestudyweuseStandard&PoorsCDOEvaluatorRtogeneratethetimetodefaultscenariosforobligors(instruments)andcalculatethevectorLs=(Ls1,...,LsT)foraparticularscenarios. Wewanttondtheattachmentpointsfxtmg1,...,T2,...,Minordertominimizethepresentvalueoftheexpectedspreadpaymentsoverallperiodsforalltranches.Weimposeconstraintsondefaultprobabilitiesoftranches(toassurecreditrating)andsomeconstraintsonattachmentpoints.Let pratingm=upperboundondefaultprobabilityoftranchemcorrespondingtoitscreditrating; pTm=defaultprobabilityoftranchem(i.e.,probabilitythatthecumulativecollaterallossexceedsthetrancheattachmentpointatleastonceinperiods1,...,T)calculatedfromthegeneratedscenarios; ptm=probabilityofdefaultoftranchemattheperiodthavingthatithasnotdefaultedbeforetheperiodt(singeperioddefaultprobability)calculatedfromthegeneratedscenarios; qtm=upperboundforthesingle-perioddefaultprobabilityptm; r=oneperiodinterestrate. Inthesedenitionsm=2,...,M,sincetheattachmentpointofthelowesttrancheisxed(xt1=0). Theoptimizationproblemisformulatedasfollows 117

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ProblemA minimizepresentvalueofexpectedspreadpayments minfxtmgt=1,...,Tm=2,...,MTXt=11 (1+r)tMXm=1E(xtm+1)]TJ /F3 11.955 Tf 11.96 0 Td[(max(xtm,Lt))+sm subjectto ratingconstraints pTmpratingm,m=2,...,M,(6) singleperioddefaultprobabilityconstraints ptmqtm,m=2,...,M;t=1,...,T,(6) attachmentpointconstraints xtm>xtm)]TJ /F7 7.97 Tf 6.58 0 Td[(1,m=3,...,M;t=1,...,T,(6) 0xtm1,m=2,...,M;t=1,...,T,(6) otherlinearconstraints. NotethatConstraint( 6 )maintainsthecreditratingsoftranches.Theotherconstraintsmightbedrivenbythebank-originatorrequirementsorsomeotherconsiderations.Forexample,thelowesttranche(equity)detachmentpointmaynotbeconstantovertimeandmaybemorethatorequalto3%.TheexpectedvaluesintheobjectivefunctionaretakenoverallsimulatedlossesLs.AtypicalCDOcontractwithconstantovertimeattachmentpointscanbedenedbylinearconstraints: xtm=xt)]TJ /F7 7.97 Tf 6.59 0 Td[(1m,m=3,...,M;t=1,...,T.(6) 118

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TosolveProblemAwederiveanequivalentrepresentationoftheobjectivefunction. Theorem6.1(EquivalentRepresentationofObjective). Letsm=sm)]TJ /F3 11.955 Tf 12.77 0 Td[(sm+1,m=1,...,M)]TJ /F5 11.955 Tf 11.95 0 Td[(1,andsM=sM,thenthefollowingequalityholdsTXt=11 (1+r)tMXm=1E(xtm+1)]TJ /F3 11.955 Tf 11.95 0 Td[(max(xtm,Lt))+sm=TXt=11 (1+r)tMXm=1smE(xtm+1)]TJ /F3 11.955 Tf 11.95 0 Td[(Lt)+. Proof. SeeAppendix B WithTheorem 6.1 wewriteanequivalentformulationforProblemA. ProblemA(EquivalentFormulation) minimizepresentvalueofexpectedspreadpayments minfxtmgt=1,...,Tm=2,...,MTXt=11 (1+r)tMXm=1smE(xtm+1)]TJ /F3 11.955 Tf 11.95 0 Td[(Lt)+ subjecttoConstraints( 6 )-( 6 )andotherlinearconstraints. 6.2.2SimultaneousOptimizationofCDOPoolandCreditTranching ThissectionconsiderstheproblemofselectingboththeassetsintheCDOpoolandtheattachmentpoints.WewanttomaximizeexpectedprotofCDO.WeassumethatIisthenumberofCDSsforselectingtotheCDOpool.TheCDSspoolcompositionisdenedbythevectory=(y1,...,yI),whereyiistheweightofinstrumenti(i.e.,CDSi).Werequirethateachyiisboundedbyvalueu.Letusdenethefollowingparameters: ci=incomespreadpaymentforinstrumenti; )]TJ /F4 11.955 Tf 9.3 0 Td[(ti=randomcumulativelossofinstrumentibyperiodt+11; 1TheCDSlossoccurswhentheobligorofanunderlyingassetdefaults.Thelossamountiscalculatedasthefollowingproduct(totalamountofcollateral)*(1-(recoveryrate)),whererecoveryrateisarandomnumber. 119

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Lt(,y)=)]TJ /F6 7.97 Tf 16.47 11.36 Td[(IPi=1tiyi=randomcumulativelossoftheportfoliobyperiodt+1; l=lowerboundonCDOspreadincomepayments; u=upperboundontheweightonanyinstrumentintheCDOpool. Hereistheformulationofoptimizationproblem. ProblemB minimizepresentvalueofexpectedspreadpayments minfy,fxtmgt=1,...,Tm=1,...,MgTXt=11 (1+r)tMXm=2smE(xtm+1)]TJ /F3 11.955 Tf 11.96 0 Td[(Lt(,y))+ subjectto ratingconstraints pTmpratingm,m=2,...,M,(6) incomespreadpaymentsconstraints IXi=1ciyil,(6) budgetconstraints IXi=1yi=1,(6) instrumentweightconstraints 0yiu,i=1,...,I,(6) singleperioddefaultprobabilityconstraints 120

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ptmqtm,m=2,...,M;t=1,...,T,(6) attachmentpointconstraints xtmxtm)]TJ /F7 7.97 Tf 6.59 0 Td[(1,m=3,...,M;t=1,...,T,(6) 0xtm1,m=2,...,M;t=1,...,T,(6) otherlinearconstraints. WiththeincomespreadConstraint( 6 )weboundfrombelowtheincomepaymentstoCDO.ThisconstraintisactiveandtheCDOincomingpaymentsaredenedbytheaveragespreadl.Withthexedincomingpaymentsweminimizetheexpectedpresentvalueoftheoutcomingpayments.WecansolvemanyinstancesofProblemBwithdifferentvaluesoftheaveragespreadlandselectthemostprotableCDO. 6.2.3Simplication1:ProblemDecompositionforLargeSizeProblems Foramoderatenumberofscenarios(e.g.,50,000)andamoderatenumberofinstruments(e.g.,200)theProblemAcanbeeasilysolvedwiththeproposedformulations.However,forsolvingProblemAofalargersize(e.g.,500,000scenarios),wedecomposetheproblemintoM)]TJ /F5 11.955 Tf 12.3 0 Td[(1separatesub-problems,i.e,wendtheoptimalattachmentpointsforeachtrancheseparatelyandthencombinethesolutions.Weshowthatthefollowinginequality(minimumofthesumisalwaysgreaterthanthesumofminimumsofitsparts)minfxtmgt=1,...,Tm=2,...,MTXt=11 (1+r)tMXm=1smE(xtm+1)]TJ /F3 11.955 Tf 11.95 0 Td[(Lt)+ MXm=2minfxtmgt=1,...,TTXt=11 (1+r)tsmE(xtm)]TJ /F3 11.955 Tf 11.95 0 Td[(Lt)++TXt=11 (1+r)tsM+1E1)]TJ /F3 11.955 Tf 11.95 0 Td[(Lt,(6) 121

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undercertainconditionsbecomesequality,whereweusethefactthatxtM+1=1.ThelefthandsideofthisinequalityistheobjectiveofProblemA.Thus,tosolveProblemAwecansolveM)]TJ /F5 11.955 Tf 11.96 0 Td[(1followingproblemsforeachm=2,...,M. ProblemA(m) minimizepresentvalueofexpectedsizeoftrancem minfxtmgt=1,...,TTXt=11 (1+r)tE(xtm)]TJ /F3 11.955 Tf 11.95 0 Td[(Lt)+(6) subjectto ratingconstraints pTmpratingm,(6) singleperioddefaultprobabilityconstraints ptmqtm,t=1,...,T,(6) boxconstraintsforattachmentpoint 0xtm1,t=1,...,T,(6) otherlinearconstraints. NotethatweomitthetermsminObjective( 6 )sinceitisaxednonnegativenumber,anditdoesnotimpacttheoptimalsolutionpoint.BelowistheformalproofthatProblemAcanbedecomposedtoProblemsA(m),m=2,...,M. Theorem6.2(Decomposition). IfoptimalsolutionsforProblemsA(m),m=2,...,Msatisfyinequalitiesxtmxtm)]TJ /F7 7.97 Tf 6.59 0 Td[(1,m=3,...,M,t=1,...,T 122

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thentheseoptimalsolutionstakentogetheristheoptimalsolutionofthecorrespondingProblemA. Proof. DenotetheoptimalobjectivevaluesforProblemsA(m)byAmandtheoptimalobjectivevalueforthecorrespondingProblemA(withthesameparametersanddata)by A.Wecanrewrite( 6 )as AMXm=2Amsm+TXt=11 (1+r)tsM+1E1)]TJ /F3 11.955 Tf 11.96 0 Td[(Lt. TheoptimalsolutionsofProblemsA(m)satisfy( 6 );hence,theseoptimalsolutionssatisfyallConstraints( 6 )-( 6 ).Therefore,theseoptimalsolutionstakentogetherformafeasiblepointofProblemA.Thus, AMXm=2Amsm+TXt=11 (1+r)tsM+1E1)]TJ /F3 11.955 Tf 11.96 0 Td[(Lt. Consequently, A=MXm=2Amsm+TXt=11 (1+r)tsM+1E1)]TJ /F3 11.955 Tf 11.96 0 Td[(Lt. andthetheoremisproved. 6.2.4Simplication2:MinimizationofLowerandUpperBounds ThissectionconsidersaproblemofminimizingalowerandupperboundsofanobjectivefunctioninProblemA(m).Itshowsthattheproblemsofminimizingalowerandupperboundsareequivalent.UsingthefactthatthecumulativelossesLtarealwaysnonnegative,wecanwritextmE(xtm)]TJ /F3 11.955 Tf 11.95 0 Td[(Lt)+Extm)]TJ /F3 11.955 Tf 11.96 0 Td[(Lt=xtm)]TJ /F3 11.955 Tf 11.96 0 Td[(ELt foranym=2,...,M;t=1,...,T. Thus,theobjectiveinProblemA(m)canbeboundedby TXt=1xtm (1+r)tTXt=11 (1+r)tE(xtm)]TJ /F3 11.955 Tf 11.95 0 Td[(Lt)+TXt=1xtm (1+r)t)]TJ /F6 7.97 Tf 16.99 14.94 Td[(TXt=1E[Lt] (1+r)t(6) 123

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SinceE[Lt]doesnotdependonxtm,thentheproblemsofminimizinganupperandlowerboundsin( 6 )areequivalentinthesensethattheygivethesameoptimalvectors(althoughoptimalobjectivevaluesaredifferent). Theobjectivefunctioncanbewrittenas minfxtmg,t=1,...,TTXt=11 (1+r)txtm(6) WecanoptimizethisobjectiveforeitheraxedpoolofassetswithConstraints( 6 )-( 6 ),ornotxedpoolofassetswithConstraints( 6 )-( 6 ).OurnumericalexperimentsshowthatthereisnosignicantdifferencebetweentheoptimalsolutionsofProblemsA(m)andtheoptimalsolutionsofanupperboundminimization( 6 ).SuchasmalldifferencecanbeexplainedbythefactthatthecumulativeCDOlossesLtareusuallyprettysmallwithrespecttoxtm.Thehigherthetranchenumberthecloserbecomes(xtm)]TJ /F3 11.955 Tf 12.1 0 Td[(Lt)+andxtm.Theadvantageofsuchasimplicationisthattheobjectivefunctionofthesimpliedproblemin( 6 )isalinearfunctionofxtmandrequiresmuchlesstimetosolve. Inadditiontoourmathematicalexpressionsweprovideanintuitiveexplanationthatleadstothesamesimplication.Itcomesfromthefactthatthehighertheratingofthetranche,theloweritsspread.Therefore,thelargerthesizeofthehighesttranche,thelessisthecostforthetotalinsurance.Thus,wemaystartndingthebestattachmentpointforthesuperseniortranchewhilemaintainingitscreditrating.Whentheattachmentpointforthesuperseniortrancheisfound,onecanproceedwiththenextlowertranche(supposeitissenior)bysolvingthesameoptimizationproblemforthenewrating.Everyattachmentpointthatisfoundservesasadetachmentpointforthelowertranche.Thus,theattachmentpointscanbeobtainedrecursivelyforalltranches.Optimizingthesingletrancheisanindependentproblemassoonashighertranchesarexed. Theobjectivecanbewrittenas 124

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minfxtmg,t=1,...,TTXt=11 (1+r)txtm,(6) whatisthesameas( 6 ). Theproblemformulationisverysimpleanditssolutionmaybeagoodstartingpointforthedeeperanalysis.Therefore,theuseofitmayprovideapreliminaryanalysisontheassetsonemightwanttoincludeintheportfolio. 6.3CaseStudy WesolvedoptimizationproblemswithPortfolioSafeguard[ 31 ].ThereareseveraldocumentedcasestudiesconcerningCDOstructuringinthestandardversionofthePSGpackage2.Thischapterreportsnumericalresultsforsomeoftheoptimizationproblemsdescribedintheprevioussection.AreadermayrefertothestandardPSGinstallationtondsomeothercasestudies. ConsideraCDOwithT=5years,andM=5(numberoftranches).Thetimesoftheadjustmentsinattachmentpointsaret1=1,t2=2,t3=3,t4=4,t5=5. Thespreadpaymentsareusuallymadequarterly.Forsimplicity,weassumethatduringtheperiodiallthespreadpaymentsaremadeinthemiddleoftheoneyearperiod,sothatwediscountpaymentswiththecoefcient1=(1+r)ti)]TJ /F7 7.97 Tf 6.58 0 Td[(0.5.Wesetinterestrater=7%.Thecreditratings(S&P)ofthetranchesareBBB,A,AA,AAA.TheseratingscorrespondtothemaximumdefaultprobabilitiespAAA=0.12%,pAA=0.36%,pA=0.71%,pBBB=2.81%. 2ThesecasestudiesandthedataareavailableintheRegularEditionofPSG,freetrialofPSGallowingdownloadingdataandresultsisavailableathttp://www.aorda.com/aod/psg.action 125

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Theattachmentpointofthelowesttrancheisxed(xt1=0),thereforethereisnoratingconstraintonit.ThenumberofassetsinthepoolisI=53.WeuseStandard&PoorsCDOEvaluatorRtogeneratethetimetodefaultscenariosfortheCDSs(instruments). 6.3.1Case1a InthiscaseweoptimizedProblemsA(m)withratingconstraintsforeachtranchemseparatelym=2,...,M: OptimizationProblem1a minimizepresentvalueofexpectedsizeoftrancem minfxtmgt=1,...,55Xt=11 (1+r)t)]TJ /F7 7.97 Tf 6.58 0 Td[(0.5E(xtm)]TJ /F3 11.955 Tf 11.96 0 Td[(Lt)+ subjectto ratingconstraints p5mpratingm,(6) constraintsonattachmentpoints 0xtm1,t=1,...,5.(6) Forthiscaseweuse10,000timetodefaultscenarios. 6.3.2Case1b Toinvestigatethedifferencebetweenthestep-upCDOwithchangingovertimeattachmentpointsandthestandardCDOwithconstantovertimeattachmentpointsweoptimizedthesameproblemwiththexedattachmentpointsoverallcontractperiod. OptimizationProblem1b minimizepresentvalueofexpectedsizeoftrancem 126

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minfxtmgt=1,...,55Xt=11 (1+r)t)]TJ /F7 7.97 Tf 6.58 0 Td[(0.5E(xtm)]TJ /F3 11.955 Tf 11.96 0 Td[(Lt)+ subjectto ratingconstraints p5mpratingm,(6) attachmentpointconstraints xtm=xt)]TJ /F7 7.97 Tf 6.58 0 Td[(1m,t=2,...,5,(6) 0xtm1,t=1,...,5.(6) 6.3.3Case2 Also,wewanttoseethedifferencebetweenProblemA(m)solutionandthesolutionofthesimpliedproblemin( 6 ).WeoptimizedthesimpliedobjectivefunctiondescribedintheprevioussectionwiththesamedataforthesamestandardCDOcontractforeachtranchem(m=2,...,M). OptimizationProblem2 minimizationofanupperboundofobjectivefunctionofProblemA(m)minfxtmg,t=1,...,5TXt=11 (1+r)t)]TJ /F7 7.97 Tf 6.58 0 Td[(0.5xtm subjectto ratingconstraints p5mpratingm,(6) attachmentpointconstraints 127

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0xtm1,t=1,...,5.(6) Figure 6-1 showsthetablewiththecomputationalresults.Clearly,thereisaprettysubstantialdifferencebetweentheoptimalpointsandtheoptimalobjectivesforProblem1aandProblem1b.Thedifferencebetweentheoptimalobjectivesisaround25%-35%,anditisthedifferencebetweentheaveragespreadpaymentsforeachtrancheinthestep-upCDOandthestandardCDO.Theresultsshowthatusingastep-upCDOallowsthebankoriginatortosaveaprettysubstantialamountofmoney. ThesolutionsforProblem1aandProblem2shownosignicantdifference.Thisobservationjustiestheproposedsimplicationin( 6 ). 6.3.4Case3 Finally,wesimultaneouslyoptimizedtheCDOportfolioandcredittranchingwiththesuggestedsimplication.Foreachtranchem(m=2,...,M)weoptimizedthefollowingproblem. OptimizationProblem3 minimizepresentvalueofexpectedsizeoftrancem minfxtm,t=1,...,5;y1,...,y53gTXt=11 (1+r)t)]TJ /F7 7.97 Tf 6.59 0 Td[(0.5xtm subjectto incomespreadpaymentsconstraints 53Xi=1ciyil,(6) instrumentweightconstraints 0yiu,i=1,...,53,(6) 128

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ratingconstraints p5mpratingm,(6) attachmentpointconstraints 0xtm1,t=1,...,5.(6) Weconsideredtwodifferentbudgetconstraints:l=0.93%andl=0.97%.Theupperbounduwassettou=2.5%.Figure 6-2 showsthetablewiththeresults.Thereisasignicantdifferencebetweenthesolutionswithdifferentincomespreadpaymentconstraints. Actually,anefcientfrontiershouldbecreatedbyvariationthespreadpaymentconstraintl.Then,theparameterlcanbeselectedbymaximizingthedifferencebetweenincomingspreadpaymentslandoutcomingspreadpaymentsdescribedbytheobjectivesofoptimizationproblems. Alternatively,itispossibletoformulateaproblemofmaximizingthedifferencebetweenincomingandoutcomingspreadpayments.However,theseissuesarebeyondthescopeofthischapter. 129

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Figure6-1. SyntheticCDOspreadsowstructure Figure6-2. CDOattachmentanddetachmentpointsstructure 130

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Table6-1. Attachmentpoints(indecimals)ofthe5-periodCDOcontractobtainedfromoptimizationProblem1a,Problem1bandProblem2foreverym. TrancheRatingPeriod1Period2Period3Period4Period5Objective Problem1a BBB0.14910.18370.22130.26040.29100.5797A0.17680.21470.26110.29470.33420.7261AA0.18770.22390.27220.31410.35020.7814AAA0.22580.25540.29420.33070.36710.8908 Problem1b BBB0.26390.26390.26390.26390.26390.7818A0.30430.30430.30430.30430.30430.9524AA0.32120.32120.32120.32120.32121.0240AAA0.35000.35000.35000.35000.35001.1460 Problem2 BBB0.15130.18160.22130.25980.29100.9170A0.17680.20990.26520.30110.32991.0650AA0.18770.22390.27220.31410.35021.1199AAA0.22900.25540.29230.33070.36711.2308 Table6-2. Attachmentpoints(indecimals)ofthe5-periodCDOcontractobtainedfromoptimizationProblem3foreverym(u=2.5%,l=0.93and0.97). TrancheRatingPeriod1Period2Period3Period4Period5 l=0.93BBB0.11840.15090.18960.22210.2581A0.13880.17220.21460.25450.2870AA0.13980.18290.22990.26650.2938AAA0.14600.19250.24450.28650.3238 l=0.97BBB0.16160.19910.24910.28660.3241A0.19160.24160.27830.32830.3616AA0.20000.25000.30000.34160.3666AAA0.20000.25000.32290.37290.4229 131

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CHAPTER7CONCLUSION 7.1DissertationContribution Theoreticalmodelingandoptimizationapproachesforensuringrobustnessincomplexsystemsisachallengingtaskthathasbeenextensivelyaddressedoverthepastdecade.Asignicantamountofworkhasbeendoneonstudyingstructuralpropertiesofnetworkedsystemsintermsoftheirconnectivity;however,theresearchonvariouscharacteristicsofcomplexnetworksisfarfromcomplete.Inthisdissertationwehaveconsideredcertaintypesofnetworkcongurations(cliquerelaxations)thatensureasufcientdegreeofrobustconnectivitybetweenthenodes. Wehavedevelopedlinearintegerprogrammingtechniquesthatallowonetondexactsolutionsofmaximumk-clubproblemssubstantiallymoreefcientlythanthepreviouslyknownformulations.Inparticular,theproposedformulationsareefcientforndingmaximumk-clubsforanyk>2,whichprovidesasignicantadvantagecomparedtotheformulationpreviouslydevelopedby[ 17 ]and[ 3 ].Inaddition,weintroducedthenewconceptofanR-robustk-clubanddevelopedthecorrespondingcompactformulationforthemaximumR-robustk-clubproblem.Moreover,inthespecialcaseofR-robust2-clubs,onecanguaranteeandtheoreticallyjustifyimportantrobustnesscharacteristicsofthesenetworkclusters,inparticular,theirerrorandattacktoleranceproperties. Wehavealsoinvestigatedweakandstrongattacktolerancepropertiesofknownnetworkcongurations(inparticular,cliquerelaxations)andspecicallyfocusedonnetworksofdiameter2(2-clubs).Maintainingbothoverallconnectivityandsmalldiameterafterattacksonmultiplenetworkcomponentshasbeenidentiedasanimportantnetworkcharacteristicinthesituationswhereallnodesinanetworkneedtocommunicateeitherdirectlyorthroughatmostoneintermediary.WehaveshownnotonlythatanR-robust2-clubisthenecessaryandsufcientstructurethatprovides 132

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thestrongedgeattacktoleranceproperty,butalsothatanoptimalR-robust2-clubcanbeidentiedanalyticallyincertainspecialcases,andthattheoptimalcongurationiscost-efcientforanyk2(atmostO(Rn)edges)comparedtocliquesandothercliquerelaxations.Wehavealsoprovednecessaryandsufcientconditionsforweakattacktolerancepropertiesofdiameter-2networksandidentiedoptimalnetworkcongurations(2-club/2-core)forthistypeofnetworkdesignproblems.AlthoughtheconsiderednetworkdesignandenhancementproblemsareNP-hard,theobtainedanalyticalsolutionscanpotentiallybeusedtodevelopefcientheuristicsfortheseproblems. Wealsoconsideredanothertypeofrobustnetworkclusters(quasi-cliques)thatisbasedonrelaxingtherequirementonedgedensity(i.e.,theratioofthenumberofedgesintheclustertothemaximumpossiblenumberofedges).Wehavedevelopedthemostcompactknownlinearmixed-integerformulationforthemaximumquasi-cliqueproblem,analyzedtheasymptoticbehavioroftheseclustersinuniformrandomgraphsandderivedexplicitanalyticalformulasthatcharacterizethesizeofthemaximumquasi-cliqueinauniformrandomgraphdependingonitsparameters.Wehavealsoprovedthatthereexistsanabruptjumpoftheorderofmagnitudeofthemaximumquasi-cliquesize,whichtoourknowledgeistherstmathematicallyrigorousproofofsuchbehavior. Besidesnetworkedsystems,wehaveutilizedrobustmodelingtechniquestodevelopoptimizationmodelsinthenancialarea.Themarketofcreditriskderivativeswasboomingbeforetherecentnancialcrisis.CollateralizedDebtObligations(CDOs)accountedforasignicantfractionofthismarket.TherecentrecessionshowedthatthepricingofCDOcontractsisadifcultquantitativeproblem.Themainissueisuncertaintyaboutobligorsdefaultrisk.Aso-calledimpliedcopulaCDOpricingmodelisconsideredforcalibratingobligordefaultrates.Inthisdissertationweproposedanalternativeentropyapproachtotheimpliedcopulamodel.Thecasestudyshowedthat 133

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theentropyapproachhasastableperformance.WealsoconsideredanoptimizationframeworkofstructuringCDOs.InadditiontostandardCDOswehavestudiedsocalledstep-upCDOs.Weshowedthatstep-upCDOscansaveabout25%-35%oftranchespreadpayments(i.e.,protabilityofCDOscanbeboostedbyabout25%-35%). 7.2FutureWork DirectionsoffurtherresearchincludethedevelopmentofmathematicalprogrammingformulationsandsolutionalgorithmsforthegeneralcaseofthemaximumR-robustk-clubproblemwithinternallynode-disjointpaths.Italsoshouldaddresscomputationalefciencyissuesfortheintroducedproblems,aswellaspossibleextensionsoftheproposedconcepts.Inparticular,analyticalconstructionofprovablyoptimalR-robustk-clubsforanyxeddiameterkisasubjectofspecialinterestforfutureresearch. ConsideringtheCDOpricingmodelsitwouldbeinterestingtoanalyzetheperformanceoftheintroducedentropyapproachinthenon-homogenouscase.Ithiscasenotonlydifferentdefaultratescanbeconsidered,butalsoacertaincorrelationstructurecanbeassumed.Severalpossibleextensionshavebeenproposedin[ 28 ].FurtherresearchshouldalsoaddresstheissueofhowproposedoptimalCDOstructuringmodelscanbeappliedtoothersimilarexoticproductstomitigatenancialrisks. 134

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APPENDIXAMINIMALNUMBEROFEDGESINATTACKTOLERANT2-CLUBS TheoremA.1. LetG(V,E)bea2-club/2-core(n=jGj),thenjEj(3n)]TJ /F5 11.955 Tf 11.96 0 Td[(3)=2. Proof. Letdeg(j)beadegreeofnodejinthegraphG(V,E).Pickanodeisothat8j2V,deg(j)deg(i). Inotherwords,nodeihasaminimumdegreeinG(V,E).Ifdeg(i)3,thenjEj3n=2>(3n)]TJ /F5 11.955 Tf 12.84 0 Td[(3)=2andthetheoremstatementisvalid.Therefore,weonlyneedtoconsiderthecasewheredeg(i)=2,sinceina2-coreanynodehasadegreeofleast2. Let(i)betheneighborhoodofnodei,formallydenedas(i)=fj2V:(i,j)2Eg. Sinceweknowthatdeg(i)=2,thenlet(i)=fs,tg.DeneS=(s)nfi,s,tg,andT=(t)nfi,s,tg.NotethatsinceG(V,E)isa2-club,thenV=fi,s,tg[T[S,andjT[Sj=n)]TJ /F5 11.955 Tf 11.96 0 Td[(3.WithoutlossofgeneralitywealwaysassumethatjTjjSj Consider2differentcases: T\S6=S,andjT\Sj=k,jSj>0; T\S=S,andjSj=k. Inthesetwocaseskmightbeequaltozero,indicatingthateithersetsTandShavenocommonnodes,orthesetSisempty. IntherstcaseletT0=TnS,andS0=SnT.BothsetsT0andS0arenonempty.BythedenitionsofT,Sasneighborsofnodest,sthereareatleastn)]TJ /F5 11.955 Tf 13.03 0 Td[(3)]TJ /F3 11.955 Tf 13.04 0 Td[(k(jT0[S0j=n)]TJ /F5 11.955 Tf 12.02 0 Td[(3)]TJ /F3 11.955 Tf 12.02 0 Td[(k)edgesgoingfromft,sgtoT0[S0.Plusthereare2kedgesgoingfromft,sgtoT\S.Totally,wehaven)]TJ /F5 11.955 Tf 11.76 0 Td[(3+kedgesgoingfromtoft,sgtoT\S.Also,weknowthatanynodeinT0shouldbeconnectedtoanynodeinS0directly,orthroughsomeintermediarynode.ThisintermediarynodecanonlybelongtoT[S,soanypathiscontainedinT[S.Thus,fromanynodek2T0thereisapathinT[Stoanyother 135

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nodeinS0,andfromanynodek2S0thereisapathinT[StoanyothernodeinT0.Therefore,thereareatleastjT0[S0j)]TJ /F5 11.955 Tf 16.58 0 Td[(1=n)]TJ /F5 11.955 Tf 11.28 0 Td[(4)]TJ /F3 11.955 Tf 11.28 0 Td[(kedgesinthesubgraphT[S.Then,totally,wehaveatleast2+(n)]TJ /F5 11.955 Tf 11.97 0 Td[(3+k)+(n)]TJ /F5 11.955 Tf 11.96 0 Td[(4)]TJ /F3 11.955 Tf 11.96 0 Td[(k)=2n)]TJ /F5 11.955 Tf 11.96 0 Td[(5edgesinG(V,E),whatisgreaterthan(3n)]TJ /F5 11.955 Tf 11.96 0 Td[(3)=2(forn6). InthesecondcasewehaveasetSasasubsetofT.LetalsoT0=TnS(jT0j=n)]TJ /F5 11.955 Tf 9.94 0 Td[(3)]TJ /F3 11.955 Tf 9.94 0 Td[(k)andxn)]TJ /F5 11.955 Tf 9.94 0 Td[(3)]TJ /F3 11.955 Tf 9.94 0 Td[(kedgesgoingfromnodettoanynodeinT0,and2kedgesgoingfromt,stoS.Bydenitionof2-club,thereshouldbeapathfromnodestoanynodeT0.Ifthereisnoedge(s,t),thenallthesepathsaregoingthroughS,whatrequiresk(n)]TJ /F3 11.955 Tf 10.22 0 Td[(k)]TJ /F5 11.955 Tf 10.22 0 Td[(3)edgesgoingfromStoT0.Then,totally,jEj2+(n)]TJ /F5 11.955 Tf 10.23 0 Td[(3)]TJ /F3 11.955 Tf 10.22 0 Td[(k)+2k+k(n)]TJ /F3 11.955 Tf 10.22 0 Td[(k)]TJ /F5 11.955 Tf 10.22 0 Td[(3)whatisgreaterthan(3n)]TJ /F5 11.955 Tf 11.95 0 Td[(3)=2foranyk>0(withk=0theedge(s,t)mustexist). Nowassumethattheedge(s,t)existsinthisgraphandrecallthatG(V,E)isa2-core,soeverynodeinT0shouldhaveadegreeofatleast2.Butwehaveonlycountedadegreeof1ofeachnodeinT0.Therefore,thereshouldbeatleast(n)]TJ /F5 11.955 Tf 12.24 0 Td[(3)]TJ /F3 11.955 Tf 12.24 0 Td[(k)=2edgesinT0.Thus,totally,wehave3+(n)]TJ /F5 11.955 Tf 12.24 0 Td[(3)]TJ /F3 11.955 Tf 12.24 0 Td[(k)+2k+(n)]TJ /F5 11.955 Tf 12.25 0 Td[(3)]TJ /F3 11.955 Tf 12.24 0 Td[(k)=2edges,whatisequalto(3n)]TJ /F5 11.955 Tf 11.96 0 Td[(3+k)=2. So,G(V,E)shouldhaveatleast(3n)]TJ /F5 11.955 Tf 12.84 0 Td[(3)=2edges.Thisendstheproofofthetheorem. TheoremA.2. LetG(V,E)beanR-robust2-clubwithn3R+p 5R2)]TJ /F7 7.97 Tf 6.58 0 Td[(4R 2(n=jGj),thenjEjRn)]TJ /F5 11.955 Tf 11.95 0 Td[((R(R+1)=2). Proof. First,notethatinanyR-robust2-clubadegreeofanynodeisatleastR.Letdeg(j)beadegreeofnodejinthegraphG(V,E).Pickanodeisothat8j2V,deg(j)deg(i). 136

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Inotherwords,nodeihasaminimumdegreeinG(V,E),sayR+k(k0).Let(i)beaneighborhoodofnodei,formallydenedas(i)=fj2V:(i,j)2Eg. Now,letusdivideallnodesintheG(V,E)intothreenon-overlappingsubsets:i,(i)andVn((i)[i).Notethatj(i)j=R+k,andjVn((i)[i)j=n)]TJ /F3 11.955 Tf 11.13 0 Td[(R)]TJ /F3 11.955 Tf 11.12 0 Td[(k)]TJ /F5 11.955 Tf 11.12 0 Td[(1.Thisallowsustondsomenon-overlappingsubsetsofedgesintheG(V,E)toestablishalowerboundonjEj.Theyaredescribedbelowitembyitemwiththeboundsonthenumberofedgestheyhave. ThereareR+kedgesbetweensubgraphsiand(i). BydenitionofR-robust2-club,foranynodej2(i)thereareatleastRdistinctpathsbetweennodesiandjoflengthatmost2.Obviously,allthesepathsbelongtosubgraphi[(i),sothedegreeofnodejinthesubgraphi[(i)isatleastR.Thereisanedge(i,j)2E,thenthedegreeofnodejinthesubgraph(i)isatleastR)]TJ /F5 11.955 Tf 11.95 0 Td[(1Thus,thesubgraph(i)hasatleast(R+k)(R)]TJ /F5 11.955 Tf 11.95 0 Td[(1)=2edges. BydenitionofR-robust2-club,foranynodej2Vn((i)[i)thereareatleastRdistinctpathsbetweennodesiandjoflengthatmost2.Sincethereisnoedgebetweennodesiandj((i,j)=2E),thenallthepathsfromnodeitojoflengthatmost2gothrough(i).Hence,thenumberofedgesbetweensubgraphsjand(i)isatleastR.Thus,thereareatleastR(n)]TJ /F3 11.955 Tf 12.15 0 Td[(R)]TJ /F3 11.955 Tf 12.15 0 Td[(k)]TJ /F5 11.955 Tf 12.15 0 Td[(1)edgesbetweensubgraphs(i)andVn((i)[i).AssumethatwexexactlyRnumberofedgesgoingfromjto(i),orR(n)]TJ /F3 11.955 Tf 11.95 0 Td[(R)]TJ /F3 11.955 Tf 11.96 0 Td[(k)]TJ /F5 11.955 Tf 11.96 0 Td[(1)edgesbetweensubgraphs(i)andVn((i)[i). Thedegreeofanynodej2Vn((i)[i)isatleastR+k.WehavealreadyxedRnumberofedgesgoingfromjto(i).Therefore,otherkedgeswithinthenodejcaneitherbewithinsubgraph(i),orVn((i)[i).Thus,thereshouldbeatleastk(n)]TJ /F3 11.955 Tf 11.96 0 Td[(R)]TJ /F3 11.955 Tf 11.95 0 Td[(k)]TJ /F5 11.955 Tf 11.95 0 Td[(1)=2edgesthatwedidnotcountbythistime. Thesubsetsofedgesinallfouritemsthataredescribedabovearenon-overlapping.Hence,jEjR+k+(R+k)(R)]TJ /F5 11.955 Tf 11.96 0 Td[(1) 2+(n)]TJ /F3 11.955 Tf 11.95 0 Td[(R)]TJ /F3 11.955 Tf 11.95 0 Td[(k)]TJ /F5 11.955 Tf 11.95 0 Td[(1)R+k(n)]TJ /F3 11.955 Tf 11.95 0 Td[(R)]TJ /F3 11.955 Tf 11.96 0 Td[(k)]TJ /F5 11.955 Tf 11.95 0 Td[(1) 2, 137

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whatafterdoingarithmeticoperationsbecomes jEjRn)]TJ /F3 11.955 Tf 11.96 0 Td[(R(R+1)=2+k(n)]TJ /F5 11.955 Tf 11.95 0 Td[(2R)]TJ /F3 11.955 Tf 11.96 0 Td[(k)=2.(A) Ifn2R+k,thenthelastpartintherighthandsideofInequality( A )isnon-negative;thus, jEjRn)]TJ /F3 11.955 Tf 11.96 0 Td[(R(R+1)=2,(A) andthetheoremisproved. Ifn2R+k,thenweconsideranotherboundonjEj.NotethatR+kistheminimumdegreeofanynodeinthegraphG(V,E).Then,thetotalnumberofedgesinthisgraphisboundedby jEj(R+k)n 2.(A) Sincen2R+k,thenR+kn)]TJ /F3 11.955 Tf 11.96 0 Td[(R.Therefore,Inequality( A )canberewrittenas jEj(n)]TJ /F3 11.955 Tf 11.95 0 Td[(R)n 2.(A) Toseehowthatrelatesto( A )wecanrewriteitas jEjRn)]TJ /F3 11.955 Tf 11.95 0 Td[(R(R+1)=2+n2)]TJ /F5 11.955 Tf 11.95 0 Td[(3Rn+R2+R 2.(A) Usingtheformulafortherootofthequadraticequation,wecanconcludethatwithn3R+p 5R2)]TJ /F5 11.955 Tf 11.95 0 Td[(4R 2 thelasttermintherighthandsideofInequality( A )isnon-negative;thus,( A )holds,andthetheoremhasbeenproved. 138

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FigureA-1. Illustrationofthenon-overlappingsetsofedgesandnodesusedintheproofofTheorem A.1 FigureA-2. Illustrationofthenon-overlappingsetsofedgesandnodesusedintheproofofTheorem A.2 139

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APPENDIXBEQUIVALENTREPRESENTATIONOFOBJECTIVEINTHEOPTIMALCDOCONTRACTSTRUCTURINGMODEL TheoremB.1(EquivalentRepresentationofObjective). Letsm=sm)]TJ /F3 11.955 Tf 12.73 0 Td[(sm+1,m=1,...,M)]TJ /F5 11.955 Tf 11.95 0 Td[(1,andsM=sM,thenthefollowingequalityholdsTXt=11 (1+r)tMXm=1E(xtm+1)]TJ /F3 11.955 Tf 11.95 0 Td[(max(xtm,Lt))+sm=TXt=11 (1+r)tMXm=1smE(xtm+1)]TJ /F3 11.955 Tf 11.95 0 Td[(Lt)+. Proof. LetFt(z)bethecumulativedistributionfunctionofthecumulativecollaterallossLt.Then,bydenitionoftheexpectationELt=+1ZzdFt(z). Since)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(xtm+1)]TJ /F3 11.955 Tf 11.95 0 Td[(max(xtm,Lt)+=8>>>>>><>>>>>>:xtm+1)]TJ /F3 11.955 Tf 11.96 0 Td[(xtm,Ltxtm+1, then,E(xtm+1)]TJ /F3 11.955 Tf 11.95 0 Td[(max(xtm,Lt))+=(xtm+1)]TJ /F3 11.955 Tf 9.69 0 Td[(xtm)Ft(xtm)+xtm+1Zxtm(xtm+1)]TJ /F3 11.955 Tf 9.69 0 Td[(z)dFt(z)+0(1)]TJ /F3 11.955 Tf 9.69 0 Td[(F(xtm+1))==(xtm+1)]TJ /F3 11.955 Tf 11.96 0 Td[(xtm)Ft(xtm)+xtm+1(Ft(xtm+1))]TJ /F3 11.955 Tf 11.95 0 Td[(Ft(xtm))+xtm+1Zxtm()]TJ /F3 11.955 Tf 9.3 0 Td[(z)dFt(z)= =xtm+1Ft(xtm+1))]TJ /F3 11.955 Tf 11.96 0 Td[(xtmFt(xtm))]TJ /F6 7.97 Tf 14.61 20.69 Td[(xtm+1ZxtmzdFt(z).(B) Withtheintegrationbypartsruleforfunctionsu(z)andv(z)bZau(z)dv(z)=u(b)v(b))]TJ /F3 11.955 Tf 11.95 0 Td[(u(a)v(a))]TJ /F6 7.97 Tf 18.3 18.67 Td[(bZav(z)du(z) 140

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fora=xtm,b=xtm+1,u(z)=z,v(z)=Ft(z),wehave xtm+1ZxtmzdFt(z)=xtm+1Ft(xtm+1))]TJ /F3 11.955 Tf 11.96 0 Td[(xtmFt(xtm))]TJ /F6 7.97 Tf 14.61 20.68 Td[(xtm+1ZxtmFt(z)dz.(B) Finally,( B )and( B )imply E(xtm+1)]TJ /F3 11.955 Tf 11.96 0 Td[(max(xtm,Lt))+=xtm+1ZxtmFt(z)dz(B) Then,withsm=MPi=msi,m=1,...,M,therstsumintheobjectivecanberewrittenasMXm=1E(xtm+1)]TJ /F3 11.955 Tf 11.95 0 Td[(max(xtm,Lt))+sm=MXm=1smxtm+1ZxtmFt(z)dz=MXm=1MXi=msixtm+1ZxtmFt(z)dz. Bychangingtheorderofthesummation,weget MXm=1MXi=msixtm+1ZxtmFt(z)dz=MXi=1iXm=1sixtm+1ZxtmFt(z)dz=MXi=1siiXm=1xtm+1ZxtmFt(z)dz.(B) From( B ),puttingxtm=0andknowingthatLtisalwaysnon-negativewehave xti+1Z0Ft(z)dz=E(xti+1)]TJ /F3 11.955 Tf 11.96 0 Td[(Lt)+.(B) Finally,with( B ),( B )wehavethatMXi=1sixti+1Z0Ft(z)dz=MXm=1smE(xtm+1)]TJ /F3 11.955 Tf 11.96 0 Td[(Lt)+=MXm=1E(xtm+1)]TJ /F3 11.955 Tf 11.96 0 Td[(max(xtm,Lt))+sm. 141

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REFERENCES [1] J.Abello,M.G.C.Resende,S.Sudarsky,Massivequasi-cliquedetection,in:LATIN2002:TheoreticalInformatics,LectureNotesinComputerScience,Springer,2002,pp.598. [2] R.Albert,H.Jeong,A.-L.Barabasi,Errorandattacktoleranceofcomplexnetworks,Nature406(6794)(2000)378. [3] B.Balasundaram,S.Butenko,S.Trukhanov,Novelapproachesforanalyzingbiologicalnetworks,JournalofCombinatorialOptimization10(2005)23. [4] B.Balasundaram,S.Butenko,I.Hicks,S.Sachdeva,Cliquerelaxationsinsocialnetworkanalysis:Themaximumk-plexproblem(2010). [5] F.Bendali,I.Diarrassouba,A.R.Mahjoub,J.Mailfert,Thekedge-disjoint3-hop-constrainedpathspolytope,DiscreteOptimization7(4)(2010)222. [6] I.M.Bomze,M.Budinich,P.M.Pardalos,M.Pelillo,Themaximumcliqueproblem,in:HandbookofCombinatorialOptimization,Vol.4,KluwerAcademicPublishers,1999,pp.1. [7] W.Chaovalitwongse,P.M.Pardalos,O.A.Prokopyev,Anewlinearizationtechniqueformulti-quadratic0-1programmingproblems,OperationsResearchLetters32(2004)517. [8] C.-L.Li,S.T.McCormick,D.Simchi-Levi,Ontheminimum-cardinality-bounded-diameterandthebounded-cardinality-minimum-diameteredgeadditionproblems,OperationsResearchLetters11(1992)303. [9] R.D.Luce,Connectivityandgeneralizedcliquesinsociometricgroupstructure,Psychometrika15(2)(1950)169. [10] R.J.Mokken,Cliques,clubsandclans,QualityandQuantity13(2)(1979)161. [11] S.B.Seidman,B.L.Foster,Agraph-theoreticgeneralizationofthecliqueconcept,JournalofMathematicalSociology6(1978)139. [12] U.Schumacher,Analgorithmforconstructionofak-connectedgraphwithminimumnumberofedgesandquasiminimaldiameter,Networks14(1984)63. [13] T.Soneoka,H.Nakada,M.Imase,Designofad-connecteddigraphwithaminimumnumberofedgesandaquasiminimaldiameter,DiscreteAppliedMathematics27(1990)255. [14] A.Veremyev,V.Boginski,Identifyinglargerobustnetworkclustersvianewcompactformulationsofmaximumk-clubproblems(2010). 142

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[15] A.Veremyev,V.Boginski,P.Krokhmal,D.E.Jeffcoat,Asymptoticbehaviorofcliquerelaxationsinrandomgraphs(2010). [16] W.P.Adams,R.J.Forrester,Linearformsofnonlinearexpressions:Newinsightsonoldideas,OperationsResearchLetters35(2007)510. [17] J.-M.Bourjolly,G.Laporte,G.Pesant,Anexactalgorithmforthemaximumk-clubprobleminanundirectedgraph,EuropeanJournalofOperationalResearch138(2002)21. [18] F.Chung,L.Lu,V.Vu,Thespectraofrandomgraphswithgivenexpecteddegrees,InternetMathematics1(2004)257. [19] F.Chung,B.Aiello,L.Lu.,Arandomgraphmodelforpowerlawgraphs,ExperimentalMath10(2000)53. [20] F.Glover,E.Woolsey,Convertingthe0-1polynomialprogrammingproblemtoa0-1linearprogram,OperationsResearch22(1974)180. [21] ILOGCPLEX, http://www.ilog.com/ (2009). [22] FicoTMxpressoptimizationsuite7.1, http://www.fico.com/en/Products/DMTools/Pages/FICO-Xpress-Optimization-Suite.aspx (2010). [23] Pajekversion1.26, http://vlado.fmf.uni-lj.si/pub/networks/pajek/ (2010). [24] O.A.Prokopyev,C.Meneses,C.A.S.Oliveira,P.M.Pardalos,Onmultiple-ratiohyperbolic0-1programmingproblems,PacicJournalofOptimization1(2)(2005)327. [25] T.-H.Wu,Anoteonaglobalapproachforgeneral0-1fractionalprogramming,EuropeanJournalofOperationalResearch101(1)(1997)220. [26] L.Andersen,J.Sidenius,Extensionstothegaussiancopula:Randomrecoveryandrandomfactorloadings,JournalofCreditRisk1(1)(2004)29. [27] J.Hull,A.White,Valuingcreditderivativesusinganimpliedcopulaapproach,JournalofDerivatives14(1)(2006)8. [28] J.Hull,A.White,Animprovedimpliedcopulamodelanditsapplicationtothevaluationofbespokecdotranches,Workingpaper(2008). [29] M.Arnsdorf,I.Halperin,Bslp:Markovianbivariatespread-lossmodelforportfoliocreditderivatives,Quantitativeresearch,JPMorgan(2009). [30] Standard&poor'scdoevaluatorR, http://www.standardandpoors.com/products-services/CDO-Evaluator/en/us (2007). [31] Portfoliosafeguard,version2.1, http://www.aorda.com/aod/welcome.action (2008). 143

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[32] D.X.Li,Ondefaultcorrelation:Acopulafunctionapproach,JournalofFixedIncome9(4)(2000)43. [33] A.Golan,Informationandentropyeconometrics:Editorsview,JournalofEconometrics107(2002)1. [34] D.Miller,W.H.Liu.,Ontherecoveryofjointdistributionsfromlimitedinformation,JournalofEconometrics107(2002)259. [35] B.Chu,S.Satchell,Ontherecoveryofthemostentropiccopulasfrompriorknowledgeofdependence,Workingpaper(2005). [36] A.Veremyev,V.Boginski,Provablyoptimaldesignandenhancementstrategiesforlow-diameternetworkswithguaranteedattacktoleranceproperties(2010). [37] A.Veremyev,S.Uryasev,Optimalstructuringofcdocontracts:Optimizationapproach(2010). [38] A.Veremyev,A.Nakonechnyi,S.Uryasev,T.R.Rockafellar,Impliedcopulacdopricingmodel:Entropyapproach(2009). [39] P.Erdos,A.Renyi,Ontheevolutionofrandomgraphs,PublicationoftheMathematicalInstituteoftheHungarianAcademyofSciences5(1960)17. [40] B.Bollobas,P.Erdos,Cliquesinrandomgraphs,MathematicalProceedingsofCambridgePhilosophicalSociety80(1976)419. [41] B.Bollobas,RandomGraphs,CambridgeUniversityPress,2001. [42] G.R.Grimmett,C.J.H.McDiarmid,Oncolouringrandomgraphs,MathematicalProceedingsofCambridgePhilosophicalSociety77(1976)313. [43] D.W.Matula,Onthecompletesubgraphsofarandomgraphs,CombinatoryMathematicsanditsApplications(1970)356. [44] B.D.McKay,Onlittlewood'sestimateforthebinomialdistribution,AdvancesinAppliedProbability21(2)(1989)475. 144

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BIOGRAPHICALSKETCH AlexanderVeremyevwasbornin1984inLeningradskaya,Russia.Hereceivedhisbachelor`sdegreeinmathematicsfromMoscowStateUniversityUniversityin2002.AlexanderVeremyevworkedparttimeasanancialanalystin2001forlargeRussianinsurancecompany,wherehehadanopportunitytoworkonoptimizationmodelstodeterminethepremiumsforcarinsurance.In2007,AlexanderVeremyevjoinedthegraduateprograminindustrialandsystemsengineeringdepartmentattheUniversityofFlorida.HereceivedhisMasterofSciencedegreeinindustrialandsystemsengineeringfromtheUniversityofFloridainthesummerof2009,andhereceivedhisPh.D.fromtheUniversityofFloridainthespringof2011.AlexanderVeremyevistheauthorofseveralscienticpapers.HewasalsoaTAinEngineeringEconomyandStochasticOptimizationclasses. 145