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A mathematical model of political borders

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Title:
A mathematical model of political borders
Creator:
Syme, James C. ( author )
Place of Publication:
Denver, CO
Publisher:
University of Colorado Denver
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Language:
English
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1 electronic file. : ;

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Subjects / Keywords:
Voronoi polygons ( lcsh )
Social sciences -- Statistical methods ( lcsh )
Boundaries ( lcsh )
Acquisition of territory ( lcsh )
Political geography ( lcsh )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
Cities and metropolitan areas act as the centers of national cultural, linguistic, governmental, and economic exchange, while also acting as portals for international immigration and foreign relations. I hypothesize that, due to relative cultural dispersion from city centers, preeminent world and primate cities can act as generating nodes for the geographic expanse of nation states. I present a 3 phase model of the geographic territorial expanse of nation states: (1) use bounded Voronoi tessellations generated with longitude/latitude coordinates of cities as central nodes to create regions of urban influence (RUIs); (2) generate countries using RUIs, an alliance matrix, and simulated border segments; and (3) use dynamics to update the alliance matrix and generate new states. A dynamical model is presented to demonstrate the ability of the model with the intent of encouraging social scientists to apply their own models.
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System requirements: Adobe Reader.
Thesis:
Applied mathematics
General Note:
Department of Mathematical and Statistical Sciences
Statement of Responsibility:
by James C. Syme.

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|University of Colorado Denver
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|Auraria Library
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890814985 ( OCLC )
ocn890814985

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AMATHEMATICALMODELOFPOLITICALBORDERS by JAMESC.SYME B.S.PureMathematics,UniversityofNewMexico,2011 B.A.PolticalScience,UniversityofNewMexico,2011 Athesissubmittedtothe UniversityofColoradoDenver inpartialfulllment oftherequirementsforthedegreeof MasterofScience AppliedMathematics 2013

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ThisthesisfortheMasterofSciencedegreeby JamesC.Syme hasbeenapprovedforthe DepartmentofMathematicalandStatisticalSciences by LorenCobb,Chair LynnBennethum WeldonLodwick November14,2013 ii

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Syme,JamesC.M.S.,AppliedMathematics AMathematicalModelofPoliticalBorders ThesisdirectedbyResearchProfessorLorenCobb ABSTRACT Citiesandmetropolitanareasactasthecentersofnationalcultural,linguistic,governmental,andeconomicexchange,whilealsoactingasportalsfor internationalimmigrationandforeignrelations.Ihypothesizethat,duetorelativeculturaldispersionfromcitycenters,preeminentworldandprimatecities canactasgeneratingnodesforthegeographicexpanseofnationstates.Ipresent a3phasemodelofthegeographicterritorialexpanseofnationstates:use boundedVoronoitessellationsgeneratedwithlongitude/latitudecoordinatesof citiesascentralnodestocreateregionsofurbaninuenceRUIs;generate countriesusingRUIs,analliancematrix,andsimulatedbordersegments;and usedynamicstoupdatethealliancematrixandgeneratenewstates.A dynamicalmodelispresentedtodemonstratetheabilityofthemodelwiththe intentofencouragingsocialscientiststoapplytheirownmodels. Theformandcontentofthisabstractareapproved.Irecommenditspublication. Approved:LorenCobb iii

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DEDICATION Mythesisisdedicatedtomyparents,withoutwhomIwouldnothavebeen abletocompletegraduateschool. iv

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ACKNOWLEDGMENT Iwouldliketothankmythesisadvisor,Dr.LorenCobb,forallhisworkin helpingtomecraftthismodelandresearch.IwouldalsoliketothankDr. GerhardJageroftheUniversityofTubingenforprovidingmewithhisdataset forlinguisticsimilaritybetweenlanguages. v

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CONTENTS Figures....................................viii 1.Introduction................................1 1.1HistoryofCitiesasCentersofStates.................2 1.2TheoriesofCitiesasCentersofStates................6 1.3LiteratureSearch............................9 1.4DescriptionofModelandSimulation.................11 2.StatisticalAnalysisofSegments.....................14 2.1BorderSegments............................14 2.2SamplingIntervals...........................21 3.AModeloftheTerritorialExpanseofNationStates..........25 3.1VoronoiTessellationsandRegionsofUrbanInuence........25 3.2BorderSimulation...........................30 3.3DynamicalModelofNationStatesUsingRUIs............37 4.Simulation.................................48 4.1DataandDescription..........................48 4.2SimulationResults...........................51 5.Conclusions................................69 5.1PotentialImprovements........................69 5.2FutureWork..............................73 References ...................................74 vi

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Appendix A.CitiesIncludedinSimulation.......................79 vii

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FIGURES Figure 2.1MapofEuropeshowingallnodesactingasendpointsforborder segments.................................15 2.2Scatterplotofnetborderlengthversusnetborderpusharea....19 2.3Regressionmodelofnetborderpushareaasafunctionofborder lengthwithmean95%condencebands................20 2.4RotationofthebordersegmentseparatingAlbaniaandMacedonia withanexampleofanintervalsamplinglengthshown.......22 2.5Histogramofallratiosofborderdeviations L .............23 2.6Histogramofallratiosofborderdeviations L andpdfoftexponentialdistribution...........................23 3.1Somesimulatedbordersegmentsgeneratedwithvaryingparameters a L thevarianceof z k theexpectednumberofsampleintervalsand n X .............................35 4.1MapofsimulationnodesincontinentalEurope............51 4.2MapofallgeneratedregionsofurbaninuenceRUIs........52 4.3FloatingcomponentsandrespectiveVoronoiregionsinSouthernEurope...................................53 4.4SimulatedVoronoistates N k forcontinentalEurope........53 4.5Simulatednations N 0 k forcontinentalEuropewithsimulatedbordersegments...............................54 viii

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4.6SimulatedVoronoistates N k attime t =1forcontinentalEurope.55 4.7SimulatedVoronoistates N k attime t =2forcontinentalEurope.56 4.8SimulatedVoronoistates N k attime t =3forcontinentalEurope.57 4.9SimulatedVoronoistates N k attime t =4forcontinentalEurope.57 4.10SimulatedVoronoistates N k attime t =5forcontinentalEurope.58 4.11SimulatedVoronoistates N k attime t =6forcontinentalEurope.59 4.12SimulatedVoronoistates N k attime t =9forcontinentalEurope.60 4.13SimulatedVoronoistates N k attime t =12forcontinentalEurope.61 4.14SimulatedVoronoistates N k attime t =14forcontinentalEurope.61 4.15SimulatedVoronoistates N k attime t =15forcontinentalEurope.62 4.16SimulatedVoronoistates N k attime t =16forcontinentalEurope.63 4.17SimulatedVoronoistates N k attime t =19forcontinentalEurope.63 4.18SimulatedVoronoistates N k attime t =30forcontinentalEurope.64 4.19SimulatedVoronoistates N k attime t =36forcontinentalEurope.64 4.20SimulatedVoronoistates N k attime t =37forcontinentalEurope.65 4.21SimulatedVoronoistates N k attime t =38forcontinentalEurope.66 4.22Simulatednations N 0 k attime t =39forcontinentalEuropewith simulatedbordersegments.......................67 ix

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1.Introduction Citiesarethecoreofnationalempires.Theyactasthecultural,linguistic, educational,governmental,andeconomichubsofsovereignstates,whilemajor airandseaportslocatedinthesecitiesactasportsofentryforanationintothe globaleconomy.Evenbeforeglobalization,citiesactedascentersforcultural exchangesandmarkets.Furthermore,citiesactasregionalcenters,projecting signicantinuencetosurroundingareas.Inthissense,cities,actingasnodes inaplane,canbeusedtogeneratenationstatesmathematicallyusingtessellations.Ipresenthereinamodelofnationstatesbasedontheuseoftessellations andstochasticbordersegmentsbetweenstates.Aframeworkforthedynamical evolutionofnationstatesisdiscussedandpresented,andamodelandsubsequentsimulation,usedtodemonstratethesedynamics,isalsopresented. Theincreaseinurbanizationratesfutherincreasestheimportanceofthecityto itsnation;citiesarehometoanincreasingmajorityofthepopulationinboth developedanddevelopingnations.AccordingtotheUnitedNations,asof2011, 52.1%oftheworldpopulationlivedinurbanlocations,with77.7%ofdeveloped regionsbeingurbanized.Europeisoneofthemostdevelopedregionsofthe world.AccordingtheUnitedNationsUNHumanDevelopmentIndexHDI, 6ofthetop10countrieswiththehighestHDIrankingsarelocatedinEurope, andtheregionofEuropeandCentralAsiahasthehighestaverageHDIofall regionsintheworld,withanaverageHDIof0.771[42]. 1

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ThetotalproportionofthepopulationofEuropelivinginurbanareasis72.9% [41].Asof2011,outofthe47countriesthattheUNclassiedasEuropean, therewereonlysixcountrieswithaproportionofthepopulationlivinginurban areasbelow50%:Liechtenstein.4%,ChannelIslands.2%,FaroeIslands .1%,RepublicofMoldova.7%,BosniaandHerzegovina.3%,and Slovenia.9%.NotethatLiechtensteinisasmall,doublyland-lockedmicrostatelocatedentirelyintheAlpsbetweenSwitzerlandandGermany,making full-scaleurbanizationadicultprospectgiventhegeography,whiletheChannelIslandsandFaroeIslandsaresmallislandnationswithverylowpopulations. ThentheRepublicofMoldova,BosniaandHerzegovina,andSloveniahaveUrbanpopulationfractionscloseto50%,whereMoldovaandBosniaandHerzegovinaareexpectedtohaveUrbanpopulationfractionsabove50%by2015, andSloveniaby2020[41].TheseconditionsmakeEuropetheidealcandidate todemonstratethemodelpresentedherein. 1.1HistoryofCitiesasCentersofStates Culturalsimilaritiesanddierencesactaspartofthebasisfornational growthcohesion,empiricalexpansion,andwars.Thisfundamentalassertionis welldevelopedinsocialscienceliterature.Forexample,SamuelHuntington's ClashofCivilizations isanimportantandcontroversialpieceontheextentand growthofnationstatesandempiresovertime,andhasbeenanalyzedextensively inliterature,withresponsesvaried;see[9],[23],and[49]formoreinformation ondieringviewpoints.ThefundamentalbasesforHuntington'scivilizational clashandnationalevolution,asdescribedbyHolmes[18],lieinreligiousand 2

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culturaldierences.ThehistoryofEuropeislitteredwithconictandconquest basedoncultural,religious,andethnicdierences,and,historically,urbancentershaveplayedacrucialroleintheexpansion,dissolution,andre-aggregation ofempiresandnationstatesinEurope.SeveralkeyexamplesinEuropeanhistorysupportthetheoryofcitiesasthecentersofnation-states. MuchofancientGreecewasdenedbytherealmofthecity-state,or polis inantiquity.CitystatessuchasAthensorSpartawerethewell-knowntypeof Greekcitywhichwithitsterritoryconstitutedanautonomousstateand,inthis respect,wasquitesimilartothecentresoftheItalianRenaissance....Eachof themhaditsownfreedom,itsindividualprideasanindependentrepublic.[32]" ThesepoleiswerethebirthplacesandcenterpointofGreekdemocracy,trade, andintellectualgrowth[32][31],and[are]notthesameasthecity-state,although[theyrely]onthephysicalandsocialframeworkofthecity-statefor protectionandshelter.[31]"Mostimportantly,thepoliswasthefoundation forGreeksociety.AsnotedbyKitto,[without]aclearconceptionwhatthe poliswas,andwhatitmeanttotheGreek,itisquiteimpossibletounderstand properlyGreekhistory,theGreekmind,ortheGreekachievement[22]." TheRomanRepublic,andlatertheRomanEmpire,wasdenedbyRome 1 whereRomewasthecenterofculture,art,economics,andgovernmentandthe city-statethatgavebirthtotherepublicandsubsequentempire.WhiletheRomanmodelofurbanization[was]...weakatenduring[21],"thedisseminationof 1 Untiltheendoftheempire,whenitre-centeredonConstantinopleandpartiallyevolved intotheByzantineempire. 3

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RomancultureandlawwassignicantlyaidedbytheRomanurbanstructure: ThedrivetoRoman-styleurbanizationwaswidespreadandcontemporary....ItshouldbeassumedthattheadministrativepracticesoftheRomanEmpirewerelargelyuniformfromprovinceto province....ThefacilitiesprovidedintheRomanizedtownthereforemarkedculturalchangesinthetribalaristocracies....Thenew leaders'actionsshowhoweectivelytheRomanculturalsystemhad beentransmittedtothem.Bystraightforwardpoliticalactsthey wereabletocreatetheframeworkbywhichwecanrecogniseaRomantown.Atrstsightitseemsthatthisprocessofculturalchange overwhelmedalltracesoftheearliervalueswhichhadoperatedinthe oppida 2 ....ThemostimportantchangewasthattheformofexpressionshowedtheacceptanceofasetofRomanvalues,whichextended beyondtheconstructionofgrandpublicbuildingstopayingforthe activitiesthatwentoninsidethem...[21]" MuchofthedisseminationofRomanculturetotheRomanEmpirewasaccomplishedthroughurbannetworks,providingmorehistoricalevidenceofurban localesasthecentersofculturalnetworksofanationorempireinEurope. TheexpansionoftheOttomanempireplayedoutheavilyincities.AccordingtotheIstanbulMetropolitanMunicipality,[with]thesiegeofIstanbul,the OttomansproceededtoestablishhegemonyovernumerousindependentTurkish 2 Theoppidaisdescribedbythe EncyclopediaoftheBarbarianWorld asaLatinwordfor adefendedsite,oftenwithurbancharacteristics,andso,byextension,simplya`town.'[3]" 4

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statesBeylikwithinAnatoliaAsiaMinor.Theresultofimperialconquest wastounifytheTurkishpopulationsinAnatolia.Inturn,othernon-Turkish, Muslimcommunitiesandprincipalitieswerebroughttogetherundertheaegisof OttomanleadershipsothattheOttomanBeylikwouldeventuallyexpandinto anEmpire.[27]"ThefallofIstanbulwasanecessarystepintheconstruction oftheOttomanempire. TheEuropeantheaterofWorldWarIIplayedoutheavilyinurbancenters. Brakman,Garretsen,andScrhamm[6]notethatthestrategicbombingofGermancitiesduringWorldWarIIhadasignicantimpactontheGermanwar economyandsubsequentcity-growth.ThefallofFrancewasincompleteuntil Parisbecameanopencity,eectivelysurrenderingthecountrytoNaziforces viaFrance'sprimatecity[33]. Brenner[7]argues,fromaneconomicstandpoint,[the]ongoingrearticulation ofstatesovereigntyisembodiednotonlyintheterritorialtransformationsassociatedwiththeconsolidationofEuropeanUnionEUgovernanceinstitutions, butequally,inawide-rangingrecalibrationofinterscalarrelationsamonginheritedtiersofstatepowerandintheassociated'splintering'ofregulatoryarrangementsamongdistinctiveurban-regionalinstitutionalcongurations,"and thusnationaleconomiesare,moreandmore,delegatingregulatorypowerand thuseconomicpowertourbanandmetropolitanregions.Brenneralsoargues thatfollowingthe1980s,anewmosaicofurbanandregionaldevelopmentcrystallizedthroughouttheEuropeancity-system....[and]establishedmetropolitan 5

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coressuchasLondon,Amsterdam,Paris,Frankfurt,Milan,andZurichwerebeingtransformedintostrategicnodalpointswithinglobalandEuropeannancial networks[7]."Thesemajorcentersactasthegatewaystoworldeconomiesand haveanincreasingdegreeofcontroloverregionaleconomicactivity. 1.2TheoriesofCitiesasCentersofStates Asidefromhistoricalevidenceforcitiesasthecentersofnationstates,there issignicanttheoreticalsupportforurbancentersasthebasesfor,andportals to,nationstatesineconomic,artisticandcultural,linguistic,ethnic,religious, governmental,andmilitaristicterms.Themodelpresentedhereinfocussesprimarilyon primatecities globalcities orworldcities,andotherurbancenters ofinuence.TheconceptofglobalcitiesisespousedbytheGlobalizationand WorldCitiesGaWCResearchNetwork,whereaglobalcityisanycitythat isconnectedsignicantlytointernationalgovernmentandeconomy,actingasa nationalportal.TheideaofaprimatecitywasrstdiscussedbyMarkJeerson, whodescribestheprimatecityinthe1939article TheLawofthePrimateCity [20]: WhyisaMexicanfromMazatlannowinMexicoCity?Because hewasdiscontentedineverywaywiththenarrowopportunitiesof thelittletown.Ifhewasdoingbadly,hethoughhecoulddobetterin thecapital.Ifhewasdoingwell,helearnedthattheeldopportunity islargerthere.ThemostfamousMexicansofthedaylivethere,and hewantedtoseethem,orhecanbuybetterorsellbetter.Perhaps hewillmakehisfortunethere....[Once]acityislargerthanany otherinitscountry,thismerefactgivesitanimpetustogrowthat 6

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cannotaectanyothercity,anditdrawsawayfromallofthemin characteraswellasinsize.Itisthebestmarketforallexceptional products.Itbecomesthe primatecity .[20]" Theconceptofthecentralityoftheseprimatecitiestotheirrespectivenations isusedasthebasisofthemodel.Furthermore,smallernationstendtohave primatecities,whilelargernationstendtohaveregionalprimatecities,orthose citiesthatdenesub-nationalpoliticalbodiesorculturalregions.Linskynotes, [Most]largecitiesareintegratedwiththerestoftheirnationor regionbythereciprocalexchangeofservices.Theretendstobeonly oneverylargecityinacountrywheredistinctprimacyoccurs.All `largecity'servicesrequiredbytheentirecountrywouldhavetobe providedfromtheprinciplecenter.Itisclearthattheservicingtask ofthatcitywouldbecomparativelygreaterinaterritoriallyextensivecountrybecauseoftransportationandcommunicationproblems thanitwouldbeinacountrythatwasareallycompact.[25]" Linskyndsthathighurbanprimacyoccursmostfrequentlyincountrieswith smallarealextentofdensepopulation.Thisndingsupportstheuseofconvextessellationstorepresenttheregionsofcultural,governmental,economic, educational,andinfrastructuralinuenceofcities,whereconvexityisusedas thebasisofthemodelofnationstates.Convexityallowsfortheprimatecities ormajorurbancentersofsmallernationstobeeasilydefensible,whilealso centrallylocatinginfrastructure,education,economiccenters,andsoonandso forth.Fornationswithoutprimatecities,regionalprimatecitiesandcitiesof 7

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culturalimportanceareincluded. CentralPlaceTheoryisageographictheorythatfurthersupportstheuseof citiesasnationalcenters.CentralPlaceTheoryisatenantofeconomicgeographydevelopedbyGermanscholarWalterChristallerthatdescribesspatial growthanddistributionofcitiesandcitysizeaccordingtosomefundamental economicideals.AsdescribedbyGetis,thechieffunctionorcharacteristic ofatown,Christallersaid,istobethecenterofaregion.Settlementswhich areprevalentlycentersofregionshecalledcentralplaces[15]."Theworkof Christallerandtheideaofcitiesascentralplacesisexpandedbyseveralauthors;see[16]and[40]formoredescriptionsofcentralplacetheory.Citiesare vitaltonationaleconomies.Majorcorporationstendtobeheadquarteredor haveasignicantpresenceinlargescalecities.Murillonotesthatcitiesactas centersofinnovation[17],whereinnovationiscentraltotheenduringvitality ofnationsinaglobaleconomy.Majoruniversitiesandinstitutionsofhigher learningarelocatedinlarge-scaleandglobalcities.Additionally,nationaland international-levelgovernmentservice,operations,andpersonellarelocatedin majorcities.Itisclearthatmajorcitieshaveasignicantinuenceonthe trajectoryofanation. Thereisextensivetheoryaddressingtherelevanceofartisticandculturalinteractionandexchangethatoccursinurbancenterstonationalidentity.For example,iconicarchitectureandurbandesignlocatedincapitalandcentral citiescontributeheavilytourbanandnationalidentities[35][43].Valeargues, 8

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citybuildingandnationbuildingarelinked,especiallywhenthecityisamodel capital.Capitalcitydesignentersintotherelationshipbetweencitiesandnationalismatmultiplescales.Mostbroadly,designmattersattheinternational scaleofnation-staterelations....Citybuildingandnationbuildingarealsoconnectedatthenationalscale.Capitalcities,especiallythosedesignedtoserve newlyindependentcountriesthatarecomposedofmanysmallerunits,often mustserveasfulcrumsthatbalancecontendingsubnationalforcesbasedindiverseregionalcenters[43]." Finally,racialandethnicsegregationincitiescanactasamicrocosmforinternalstrifeandconict,especiallygiventhecosmopolitanismandtolerance forwhichcitiesaregenerallyknown[10].Keyurbanandmetropolitancenters ofnationstatesoertheidealsoftoleranceanddiversitywhilecultivatinga nationalidentity,governmentalandeconomicpower,andincreasingly,thevast majorityofthepopulationofanation,especiallyindevelopeddemocracies. Collectively,theideasbehindprimatecities,centralplacetheory,andthecrucialbasisofcitiestonationaleconomics,government,military,infrastructure, trade,andculture,allreinforcethemodelingofnationstatespresentedherein. 1.3LiteratureSearch Aliteraturesearchwasperformedwiththegoalofdeterminingapplicable, similar,andrelatedworkavailableintheliterature.Thereissomeworkavailableontheuseoftessellationstodenepoliticalboundaries,thoughmuchof 9

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itinvolvesthegenerationofpoliticaldistricts,suchascongressionaldistricts. Pereiraetal.[29]presentthreeindicestocompareterritorialpartitionsindistrictingproblems,usingtwoterritorialpartitionstoevaluatetheindices.Ricca etal.[30]presentanovelapproachtopoliticaldistrictingbasedonweighted Voronoitessellations;thisworkisthemostsimilartotheworkIhaveperformed, althoughitisdistinctlydierentinitsnaloutcomeanditsgoalcongressional districting. Secondly,thestudyofthepoliticalbreak-upandintegrationofnationsiswell establishedinliterature.AlesinaandSpolaore[2]presentaninterestinggametheoreticdynamicalmodelthathassimilargoalsasthedynamicalmodelpresentedinSection3.3.Asdescribedin[2],AlesinaandSpolaore[study]the relationshipbetweeninternationalconictandthesizedistributionofcountries inamodelinwhichbothpeacefulbargainingandnonpeacefulconfrontations arepossible....[they]alsostudytheroleofinternationallawandshowhow betterdenedinternational`propertyrights'mayleadtocountrybreakupand morenumerouslocalconicts.[2]"Thismodelhassimilarimplicationstothe dynamicalmodelpresentedherein,andcouldbeadaptedtooperateasthedynamicalengineforthemodel,oraspartofalargersystem.Bolton,Roland,and Spolaore[4]presentaneconomictheorytodescribethenatureofthedynamical changeinstates,asserting, athespreadofdemocracyleadstothecreationoftoomanynationstates;btheextentoffactormobilityplaysakeyroleindeterminingtheincentivestowardsseparationorintegration;ceconomic 10

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integrationthroughthedevelopmentofinternationaltrademayrequiregreaterpoliticalintegrationtocreateandeectivelymaintain alevelplayingeldininternationaltrade...[4] SirmonandLanepresentamodelofculturaldierencesandinternationalallianceperformancetoexplaintheambiguousndingsregardingtheinuenceof nationalculturedierencesonallianceperformance.[34]"Whilethemodelpresentedin[34]isusedforbusinessalliancesandthestructuresofmultinational corporations,someoftheconceptscanbemadeanalogoustopoliticalscience andpoliticalorganizations. 1.4DescriptionofModelandSimulation Inaccordancewiththepositionthatcitiesactasnationalcentersofidentity,militaryandgovernmentalstructure,andthusterritorialexpanse,Ipresent hereinamodelofnationstatesbasedoncities.Themodelhasthreeprimary components:thedivisionoftheplane,orborderregion,intotessellations usingVoronoitessellations,alsoknownasDirichlettessellationsorThessien polygons;themodelingofbordersegmentsbetweentessellations,whereregressionmodelsandstatisticalinferenceguidebordersegmentsimulationparameters;andaframeworkforadynamicalmodelalongwithamodelto demonstratethisframework.Inkeepingwiththetheoreticalbasis,andsome otherbasicnotions,twokeyassumptionsaremade: Equivalenceofnodeweights: InordertosimplifythecalculationofVoronoi tessellations,noweightingofregionsofinuenceisperformed.Thisassumptioncorrespondstoauniformrelativeinuenceofcitiesinregions. 11

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Priorityofculturaldiusionvialandoversea: Inconstructingthetessellationsoftheborderregiontodescribetherelativeterritorialinuence ofurbancenters,Iassumethatlandismoreconducivetoculturalandmilitarydispersionthansea,andthusoatingconnectedcomponentsofeach Voronoitessellationareattachedtothenearestcentralcitycomponent seeSection3.1. PopulationStability Inordertosimplifythemodel,populationwasheld constantforeachcity.Thisisasimplifyingassumptionwhichcouldeasily bechangedinsuccessivemodeliterations. FixedNodes Thenumberandpositionofnodesisxed.However,infuture work,themodelcouldbeadaptedtoallowforvaryingcriteriatodetermine theuseofcitiesasnodes,withpopulationdynamicsgoverningchanges withincities. UniformPolicingofBorderSegments Thisassumptionguidestheuseof thecatenarycurve.Iassumethat,acrossabordersegment,military presence,defenses,andphysicalmanifestationsofpoliticaltensionand borderdefensewillbeuniformacrossthatsegment.Theseextensionsof governmentpowercanbethoughtofanalogouslytoforce,andthusthe netforceoneachsegmentpushingthatsegmentinagivendirectionis uniform.Thecatenarycurverepresentssuchacurve. Thegoalofthemodelistopresentadynamicalframeworkfortheevolutionof bordersegmentsovertime;eventually,usingmodelsdevelopedbypoliticalscientists,militaryscientists,economists,historians,andsocialscientistsinother 12

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relevantelds,themodelshouldbeabletocloselymirrorthedynamicalchange ofnationstatesthathasbeenseeninEuropeoverthepastmillennium.In ordertoaccomplishthisreplication,anadaptableframeworkwithinthemodel ofnationstatesshouldbeabletoaccommodate{andbeeasilyadaptableto{ dynamicalmodelspresentedbypoliticalscientists,economists,militaryscientists,anthropologists,historians,andotherrelevantsocialscientists.Themodel presentedhereingivessuchaframework. InChapter2,IdescribetheanalysisofbordersegmentsinEuropethatwas performedinordertoguidethemodelofbordersegments.Chapter3isdevoted entirelytodescribingthemodelofthegeographicextentofnations.InSection 3.1,IdescribetheconstructionofVoronoiregionsofUrbaninuence.Then,in Section3.2,Ipresentthemodelofbordersegments,andinSection3.3Ipresent adynamicalmodelforthecollapseandre-aggregationofpoliticalnation-states basedsolelyonreligiousandlinguisticdata.InChapter4,IpresentasimulationofthemodeldescribedinChapter3.Chapter5discussesimplications, improvements,andfutureworkforthemodel. 13

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2.StatisticalAnalysisofSegments AstatisticalanalysisofthenatureofpoliticalbordersinEuropewasperformedinordertoaidinthesimulationofborders.Giventhenatureofborders, themodelofbordersegmentsiscomprisedofthreeprimaryelements:the directionofthepush;thenetpush{concaveorconvexcomponent{ofthe segmentfromonecountryintoanother;andastochasticcomponent.The statisticalanalysispresentedhereisusedasabasisforthemodelpresented herein.AllanalyseswereperformedusingMathematica,ahigh-capabilitycomputinglanguageandprogram.ThebordersegmentmodelcomponentisdiscussedinSection3.2. 2.1BorderSegments ExcludingallexclavessavefortheKaliningradOblastexclaveofRussia, thereare90bordersegmentsinEurope,notincludingafewdisputedsegments betweenKosovoanditsneighbors.ShowninFigure2.1isamapofallEuropean bordersegmentsanalyzed. 14

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Figure2.1: MapofEuropeshowingallnodesactingasendpointsforborder segments. Theanalysiswasdoneinordertoguidethedevelopmentofthemodel ofnationstates,morespecically,thebordersimulation.Toperformthe analysis,geographicinformationsystemGISpolygonshapelesoftheadministrativeboundariesofEuropewereobtainedfromtheEuropeanCommission,Eurostat/GISCO 1 [14].Eachpolygonintheshapeleisasetof orderedpointsin R 2 .UsingtheGISpolygondata,bordersegmentswere easilydenedasorderedmembersoftheintersectionofadjacentcountries C 1 = f c 1 ;:::;c n 1 g ;C 2 = f c 1 ;:::;c n 2 g c i j 2 R 2 ,where n 1 and n 2 arethe 1 Acknowledgementnotice: c EuroGeographics, c FAOUN, c TurkStatSource:EuropeanCommission c Eurostat/GISCO 15

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lengthsofthepolygons C 1 and C 2 ,respectively.Bordersegmentsweregeneratedasfollows: 1.Ifeither c 1 or c n 1 iscontainedin C 1 C 2 ,butnotboth,let p min =min p : c p 2 C 1 C 2 ,andlet p max =max p : c p 2 C 1 C 2 ;thentheborder S canbegivenas S = f c p min ;:::;c p max g ; 2.Ifboth c 1 and c n 1 arein C 1 C 2 ,thenwemaycountbackwardsfrom n 1 to p 0 ,where p 0 > 2theremustbeatleastonepointnotintheintersection istherstpointsuchthat c p 0 )]TJ/F16 7.9701 Tf 6.586 0 Td [(1 62 C 1 C 2 .Let n int = j C 1 C 2 j ;then S = f c m p o + k;n 1 j 0 k n int )]TJ/F15 11.9552 Tf 13.257 0 Td [(1 g ,where m p;n = p mod n if p mod n 6 =0and n otherwise. Foreachbordersegment,the netpusharea iscalculatedbyrstrotatingeach bordersegment.Thenetpushareaisdenedasthenetareaoftheborder segmentoneithersideofthelinesegmentbetweenthetwoendpointsofthe bordersegment.For S R 2 abordersegment,let n S = j S j andlet s 1 = x 1 ;y 1 and s n S = x n S ;y n S .If y 1
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0 = s 1 and n = s n S .Thenletthepolygon p k bethepolygoncreatedby takingallelementsfrom k )]TJ/F16 7.9701 Tf 6.586 0 Td [(1 to k from ^ S .Thenwedeterminetheareaofeach polygon p k andtotaltheareasofall k suchthat k mod2areequivalent,giving areas a + and a )]TJ/F15 11.9552 Tf 7.085 -4.339 Td [(,where a + isthetotalofallareasofpolygons p k locatedabove the x axisand a )]TJ/F15 11.9552 Tf 11.211 -4.339 Td [(isthetotalofallareasofthepolygons p k locatedbelowthe x axis.Then,foreachbordersegment,thenetpushareais a = a + )]TJ/F18 11.9552 Tf 11.955 0 Td [(a )]TJ/F15 11.9552 Tf 7.084 -4.339 Td [(. If a isnegativepositive,thenthecountrysharingthesegmentabovebelowthe x axispushesintothecountrythatsharesthesegmentbelowabove the x axis.Ifthecountryaboveissmallerthanthecountrybelowand a< 0,or ifthecountrybelowissmallerthanthecountryaboveand a> 0,thenwesay thatthesmallercountrypushesintothelarger.Thiscalculationwasperformed for88ofthe90bordersegmentsinEurope,withtheonlytwoexceptionsbeing theclosedbordersegmentsofVaticanCityandSanMarino,bothofwhichare enclavesofItaly.Outofthe88bordersegmentsanalyzed,at59thesmaller countrypushedintothelargerandatone{theborderbetweenGibraltarand Spain{thenetpushofthesmallerintothelargerwas0,meaningthatthe bordersegmentwasonlyalinesegment.Forthiscase,Ideferredtothesmaller country.Theproportionofsegmentswherethesmallercountrypushedintothe largerwasfoundtobe 60 88 0 : 6818;inmanycaseswherethiswasnottrue, geographicfeaturesgovernedthebehavioroftheborders. Initially,thehypothesiswasthatthenetpushofabordersegmentwasgovernedbytherelationshipbetweenthemilitarystrengthofcountriessharing 17

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asegmentinadditiontoculturalelementssuchaslanguageandreligion.To simplifythis,severalfactorsbetweennationssharingabordersegmentwere considered,includingmilitaryexpenditures,GDP,militarypopulation,number oflanguages,andnumberofreligions.Nosignicantrelationshipwasfoundbetweenanyoftheseparametersandthenetpushofabordersegment.However, thisanalysisdirectlysupportsthemodelofnationstatespresentedandwasthe foundationfortheideatodevelopthemodel.Smallernationstendtohave smallerpopulationsorlesscities,andfurthermore,thesenationstendtohave primatecitiesmoresothanseverallargecities.Inmostcases,theexistenceofa primatecityleadstoasinglecitybeingincludedintheinitialalliance.Inthese cases,theprimatecitiesrepresentthecenterofgovernment,culture,language, andhistoryfortheirrespectivecountries,andareconsideredonetheirprimary assetsfromadefensibilitystandpoint.Inaddition,theconvexshapeofacountrypresentsthemosteasilydefensibleposition.Sincethemodelisbasedo ofVoronoitessellations,whichareconvex,theconvexityofasingletessellation wouldimplythatthiscountrywouldpushintosurroundingcountries. Next,ananalysisofthepushingcomponentofeachbordersegmentwasperformed.Foreachbordersegment S ,denethenetdisplacementastheEuclidean distancebetween s 1 and s n S .Apatternemergedbetweenthedistancebetween endpointsofeachbordersegmentandthenetpushareaofthebordersegment fromonecountryintoanother.ShowninFigure2.2isascatterplotofthenet displacementofbordersegmentsversusthecalculatednetpusharea. 18

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Figure2.2: Scatterplotofnetborderlengthversusnetborderpusharea. ThePearson'sRcorrelationbetweenthenetdisplacementoftheborder andtheborderpushareaisapproximately0.7625,whichislarge,and,assuming normality,signicantatthe < 0 : 0001level.However,thiscorrelationisnot thebasisofthemodel;aregressionmodelis.Thefourregressionmodelsthat werettothedataareenumeratedbelow,where^ y istheregressionvalueofthe dependentvariablenetborderpusharea,and x istheindependentvariablenet displacementofbordersegment: 1.Alinearmodel,intheformof^ y = ax + b ; 2.Anexponentialmodel,intheformof^ y = ae bx + r ; 3.Aquadraticmodel,intheformof^ y = ax 2 + bx + c ;and 4.Acatenarymodel,intheformof^ y = a cosh )]TJ/F19 7.9701 Tf 6.675 -4.976 Td [(x )]TJ/F19 7.9701 Tf 6.587 0 Td [(b a )]TJ/F18 11.9552 Tf 11.955 0 Td [(a cosh )]TJ/F19 7.9701 Tf 7.112 -4.977 Td [(b a )]TJ/F18 11.9552 Tf 11.955 0 Td [(c 19

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Foreachmodel,theadjusted R 2 valuewascalculated,wherethelinearmodel hadthelowestvalueat0.5765;theexponentialmodelhadthenextsmallest, at0.7425;thecatenaryhadavalueof0.7545;andthequadraticmodelhada valueof0.7582.Whilethequadraticmodelhadaslightlyhigheradjusted R 2 valuethanthecatenarymodel,thecatenarycurvewastheinitialhypothesized convexcomponentofthesegmentmodel,andthusIretainedthecatenarycurve astheregressionmodelfortherelationshipbetweenbordernetdisplacement andpusharea.ShowninFigure2.3isthescatterplotofFigure2.2withthet regressionmodelwithcoecientsroundedtothenearest0.01 ^ y c =7 : 58cosh[0 : 14 x +0 : 65] )]TJ/F15 11.9552 Tf 11.955 0 Td [(7 : 18.2 and95%condencebandsofthepredictionofthemean.Themodelshownin equation.2supportsthehypothesisthatthepushingcomponentofaborder iswellrepresentedbyacatenarycurve. Figure2.3: Regressionmodelofnetborderpushareaasafunctionofborder lengthwithmean95%condencebands. 20

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2.2SamplingIntervals ThebordersegmentsimulationrestsonanapproximationofaWienerprocess,whichmustbesampleddiscretelytoproduceviableandsucientlysmooth bordersegments.Therefore,inaneorttoguidethesamplingintervalsofthe stochasticbordersimulation,theintervalsbetweendeviationsfromthelinesegmentbetweentwobordersegmentendpointsisconsidered.Forarotatedborder segment S ,letdenoteasamplingintervalbetweenendpoints;thenforany s i = x i ;y i ,let i = j x i +1 )]TJ/F18 11.9552 Tf 11.732 0 Td [(x i j .Theinterval i canalsobewrittenofasthe distancebetweentheprojectionsofeachbordersegmentontothevectorbetween endpoints;i.e.,where v = s n S )]TJ/F18 11.9552 Tf 11.446 0 Td [(s 1 thevectorbetweenendpointsofthesegment S, i =Proj v s i +1 )]TJ/F18 11.9552 Tf 12.488 0 Td [(s i ShowninFigure2.4isthebordersegment,whereall pointsin R 2 correspondtoEurostat/GISCOlongitude/latitudecoordinates[14], separatingAlbaniaandMacedoniawiththevariableshown. 21

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Figure2.4: RotationofthebordersegmentseparatingAlbaniaandMacedonia withanexampleofanintervalsamplinglengthshown. Forallbordersegmentsandall i ineachsegmentinEurope, L was calculated,where L = k s n S )]TJ/F18 11.9552 Tf 11.815 0 Td [(s 1 k isthedistancebetweenendpointsofaborder segment,alsoreferredtoasthe netdisplacement ofthebordersegment.Wecall thesetheratiosofbordersegmentsampleintervalstolength;then,asshown inFigure2.5,theseratios appear tobeexponentiallydistributed,wherefor eachGISsegment S withnetdisplacement L andsegmentsampleintervals i 1 i n s )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ; P n s )]TJ/F16 7.9701 Tf 6.586 0 Td [(1 i =1 i L =1. 22

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Figure2.5: Histogramofallratiosofborderdeviations L Figure2.6: Histogramofallratiosofborderdeviations L andpdfoft exponentialdistribution. Thepdf,whichwastusinganautomatedroutineintheutilityMathemat23

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ica,andtheCramerVon-Misesdistributiongoodnessofttest,wasfoundto haveparameter 81 : 6596,where E [ L ]= 1 impliesthattheexpectednumber ofsamplingintervalsperbordersegmentis .However,thenullhypothesisfor thetestisthatthedataareexponentiallydistributedaccordingto 81 : 6596; anear-zero p -valuefortheCramerVon-Misestestwascalculated,indicating thattheintervalswerenotexponentiallydistributed.However,giventherelativequalitativeclosenessoftheexponentialcurvetothedatashowninFigure 2.6,thecurvewasusedasaguidefortheinterval,i.e.,forthebordersegment model,itisthenassumedthatthesamplingintervalratiosareexponentiallydistributed.Thesamplingalgorithmisshownasalgorithm3.2.However,foreach bordersegment,arelationshipbetweenthenetdisplacementandnetpusharea ofeachbordersegmentand ,theparameterfortheexponentialdistribution, isdetermined.UsingthesameeurostatGISdata,amultivariateregressionwas performedtoestimatethenumberofintervalspersegment^ y s ,thedependent variableasafunctionofboththebordernetdisplacement x 1 andnetpush area x 2 .Thequadraticregressionmodelwasfoundtobewithcoecients roundedtothenearest0.01 ^ y s x 1 ;x 2 =56 : 69 x 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(6 : 85 x 2 1 +40 : 37 x 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(0 : 24 x 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(7 : 33 : .3 Theregressionequationshownin.3hasanadjustedR-squaredvalueof 0.649212.Thevalueof^ y s isusedtoestimatetheexpectednumberofsampling intervalsforeachsimulatedbordersegmentasdescribedinSection3.2. 24

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3.AModeloftheTerritorialExpanseofNationStates Themodelpresentedhereiniscomprisedofthreeprimarycomponents: 1.ThegenerationofboundedVoronoitessellationsusingselectcitiesasthe centralnodesoftheVoronoiTessellations,wherelatitude/longitudecoordinatesofcitiesandcontinentaloceanandlandbordersareused; 2.Generationofsimulatedbordersegmentstoreplacenon-stochasticborder segmentsbetweenalliedVoronoistates,wherethesimulationofborder segmentsisguidedbythestatisticalanalysisshowninChapter2;and 3.Aniterativedynamicalmodel,wherethechangeofalliancesattime t +1 isgeneratedusingculturaldatafromtimestep t ;thedynamicalmodel servesasanexampleofhowtoimplementdynamicalmotion{specically theaggregation,collapse,andre-aggregationofnationstates{inthe frameworkpresentedinthispaper. Notethatsomevariablesarerepeatedacrosssections;inthesecases,thevariable appliesonlytothesectionthatitisdescribedin. 3.1VoronoiTessellationsandRegionsofUrbanInuence AVoronoidiagrampartitionsaplaneinto n convexpolygonswith n centers c i ,whereallpointscontainedwithinpolygon i areareclosertothecenter c i than anyothercenter[44][45].ThesepolygonsareknownasVoronoitessellations, Dirichlettessellations,orThiessenPolygons.Let C = f c i j 1 i n g C R 2 25

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bethesetofcentralcities,ornodes,whereeachcity's x;y locationinthe planeisdeterminedbyitslongitudeandlatitude,respectively.The i th Voronoi tessellation V i oftheVoronoidiagram V = fV i g n i =1 isdenedas V i = f x 2 R 2 : k x )]TJ/F18 11.9552 Tf 11.956 0 Td [(c i kk x )]TJ/F18 11.9552 Tf 11.955 0 Td [(c j k8 j 6 = i; 1 j n g ; .1 wherethenormweuseisthestandard ` 2 normin R 2 ,ortheEuclideandistance. Notethatin R 2 ; S n i =1 V i = R 2 ,andthusatleastsomeofthe V i arepartially unbounded.Therefore,wemayboundthediagrambasedonthecontinental border.Let B bethesetrepresentingthecontinentalborder,where B isa closed,connectedsubsetof R 2 ,and C B allcitiesarecontainedonland withintheborder.Notethatrequiring B tobeconnectedimpliesthatnoislands orseparatedpartsofacontinente.g.,Scandinaviaisseparatedfromtherestof EuropeviaRussiaandtheBalticSea,andthusitisconsideredseparately.Let B 1 = f x j x;y 2 B g and B 2 = f y j x;y 2 B g betheprojectionsof B ontothe dimensionalsubspacesof R 2 .Toboundtheset,welet B besuchthat B = x;y j min u 2 B 1 x max u 2 B 1 ; min v 2 B 2 y max v 2 B 2 ; .2 where B closedimpliesthat B 1 and B 2 obtainmaximaandminima.Thenwe lettheboundedVoronoitessellations V i be V i = V i B ; .3 andthusall V i areclosed,convex,boundedsetswhere B S n i =1 V i and S n i =1 V i = B .Dene A = A ij asthenodeadjacencymatrix,whereforeach1 i;j n;A ij isdenedas A ij = 8 > < > : 1 j V i V j B j > 1 0else : .4 26

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Theadjacencymatrix A willbeusedtoconstructpotentialalliancesinthedynamicalmodelpresentedinSection3.3. Briey,recallfromSection1.4thatIassumethattheinuenceofacitylinguistically,religiously,ethnically,andmilitarilyoverlandisgreaterthaninuenceof thecityoversea,andthuseachnalregionofinuenceofeachcityisrequired tobeconnected.Thisimpliesthattheregionsofinuenceofcity i cannotbe writtensimplyas V i B .Therefore,let B i representthenalconnectedpolygon containing c i ,whichrepresentstheregionofurbaninuenceof c i .Wedenote thispolygonas B i since S n i =1 B i = B ,and B i isconstructedusingthefollowing fourprimarysteps. 1.Let @V i denotetheboundaryof V i .Then @V i = Bn V i V i ,since V i closed.Let V 0 i denotetheinteriorof V i ,andthus V i = @V i [ V 0 i 2.Foreach x 2 V 0 i T B ,let G 0 x bethecollectionofallconnectedsubsets of V 0 i containing x ;thenlet 0 x = T U 2 G 0 x U ,andthus 0 x isthemaximal connectedsetcontaining x intheinteriorof V 0 i .Furthermore,wehavethe resultthatforall x;y 2 V 0 i B ,either 0 x = 0 y or 0 x 0 y = ; .Therefore, wedenetheset E 0 i = f 0 x j9 y 2 V 0 i B : y = x g .5 asthesetofconnectedcomponentsof V 0 i B ,where T U 2 E 0 i U = V 0 i B and U 1 T U 2 = ;8 U 1 ;U 2 2 E 0 i ;U 1 6 = U 2 alldistinctcomponentsof E 0 i aredisjoint.Inthecaseofthismathematicalmodel,therearenitely manyconnectedcomponentsin E 0 i forall1 i n .Iconsiderthesetof 27

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connectedcomponentsoftheinteriorofeachconvextessellation V 0 i T B {ratherthantheconnectedcomponentsofthesetitself{inorderto ensurethatthecomponentsareconnectedontheinterior,andthatthere isnotsimplyabridge"betweencomponentslocatedontheboundary @V i Thisensuresthat,inthecontextofthemodel,thereisanon-trivialland connectiontofosterthedispersionoflanguage,religion,andethnicgroups fromcity c i .Foreach 0 x ,let x = 0 x betheclosureofthecomponent 0 x Then,denetheset E i = f = 0 j 0 2 E 0 i g .6 asthesetofclosuresofallconnectedcomponents 0 of V 0 i .Wecallthe connectedcomponent c i the coreurbancomponent of E i .Notethatthe elementsof E i arenotnecessarilydisjoint. 3.If j E i nf c i gj > 0,thenwecalltheelementsof E i nf c i g the oatingcomponents of V i B .Anexampleofoatingcomponentsinthesimulationis showninFigure4.3.Thesecomponentsarewithinthe Voronoiregion of inuenceofcity c i V i B ,butthereisnosignicantlandconnectionbetweentheoatingcomponentand c i locatedin V i B .Sincetheposition ofthispaperisthatculturalandmilitarydispersionoverlandisfargreater andmoreecientthanoversea,aoatingcomponentisinuencedbythe urbanregionclosesttoitwhichiseventuallyconnectedtoit.Therefore, since B isconnectedandtherearenitelymanyconnectedcomponentsof eachVoronoiregion,each 2 E i nf c i g mustbeconnectedtoatleastone other V j B oraoatingcomponentof E j forsome j .Theneveryoating componentmusteventuallybeconnectedtoacoreurbancomponent c i 28

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forsome i .Let E f bethesetofalloatingcomponents.Clearly j E f j is nite,sodenote n f = j E f j asthenumberofoatingcomponents.Then, foreach1 j n f ,lettheset j = f k : j j k j > 1 ; 1 k n f g including j = k .Thenwemaydenoteanitesequenceofsets f I j m g j m =0 where I j m = S k 2 I j m )]TJ/F17 5.9776 Tf 5.756 0 Td [(1 I j m )]TJ/F16 7.9701 Tf 6.586 0 Td [(1 [ k for0 j j ,where I j 0 = j foreach 1 j n f .Sincethesequenceisnite,thereisanalelementwhichwe maydenote I j = I j j .Then k 2 I j impliesthat k iseventuallyconnectedto j ;furthermore,forall1 j;k n f ;k 2 I j j 2 I k .Then let j = [ k 2 I j k ,andlet E r bethereducedsetofoatingcomponents suchthat E r isthesetofalluniqueelementsof f k j 1 k n f g ,andlet n r = j E r j ,so n r n f .Thentoformthepolygon B i ,weattachtoeach c i allreducedoatingcomponentsin 2 E r ,shouldtheyexist,suchthat k )]TJ/F18 11.9552 Tf 12.617 0 Td [(c i k =min 1 j n k )]TJ/F18 11.9552 Tf 12.617 0 Td [(c j k ,where isthecenterofmassofeach .Formally,foreach1 j n r theindexofreducedoatingpolygons, deneanassignmentfunction F : E r !f 1 ;:::;n g ,where F = i if k )]TJ/F18 11.9552 Tf 10.969 0 Td [(c i k =min 1 j n k )]TJ/F18 11.9552 Tf 10.968 0 Td [(c j k and i isunique;iftherearemultiplevalues suchthat k )]TJ/F18 11.9552 Tf 12.49 0 Td [(c i k =min 1 j n k )]TJ/F18 11.9552 Tf 12.489 0 Td [(c j k ,thenrandomlychooseoneof the i andassignitto F .Then B i = c i [f 2 E r : F = i g Andthustheregion B i isconstructed,wherecertainly S n i =1 B i = B .Iwillrefer to B i asthe regionsofurbaninuence RUIs.Next,considerthealliancematrix,whichvarieswith t; A t .Forinitialtime t =0,setthealliancesofregions associatedwitheachcityastheycurrentlystand;theRUIrepresentedbyacity c i isalliedwith c j if c i and c j areininthesamecountry.Forthepurposesof thesimulationpresentedinChapter4,onlyconnectedcountriesincontinental 29

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Europe 1 {excludingRussia{areconsidered. ThenodeadjacencymatrixandalliancesmatrixhavesomerelationshipsIdrop the t notationtoshowthefollowingrelationships: A ij =0implies A ij =0, and A ij =1implies A ij =1,where A ij =1impliesthatregion i andregion j areinthesamenationtogether.Then,foreach1 i n ,denetheset J i = f j jA ij =1 ; 1 j n g ;then J i isthesetofindicesofRUIsalliedwith RUI i .Thenlet N t bethenumberofcountries,where N t =Rank A t Let J t bethecollectionofallunique J i t ,andsothereare N t elements J k t 2J t .Theelementsof J t representthealliednations,ornationstates, theterritorialexpanseofnation k; 1 k N t ,withoutstochasticbordercomponents,isconstructedas N k = S i 2J k B i .Each N k isknownasa VoronoiState andistheinitialexpanseofterritory.ThesestatesarecalledVoronoistates astheboundariesofeachstatearetheboundariesofsomeboundedVoronoi tessellation.Togeneratethenalmapofsimulatednationstates,eachborder segmentsharedbetweenVoronoistatesisreplacedbyasimulatedbordersegment,asdescribedattheendofSection3.2.Finally,notethatforthediscrete dynamicalmodel,Iallow A t ;N t ; J t ; andconsequently N t tovarywith t seeSection3.3{givingthebasisandfordynamicalaggregation,collapse, andre-aggregationofstatesovertime. 3.2BorderSimulation Themodelofbordersegmentshastwoprimarycomponents:astochasticcomponentthatmaybesimulatedusingrealdataasaguide,andan 1 AsexplainedinSection4.1,theKaliningradOblastisincludedwhilethevastmajority Russiaisnotincludedinthesimulation. 30

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overarchingbordertrend,ortheconvexbridgethatwerepresentusingthecatenaryequation.Someparametersguidingthesecomponentswereanalyzedin Chapter2,andIprovidethemathematicalandtheoreticalbasisforthemodel here.Ibeginwithabriefdiscussionofboththestochasticcomponentandthe catenarycomponent,followedbythenalequationusedtosimularbordersegments.Someoftheparametersusedtogeneratebordersegmentsarecontrolled byrealdatapertinenttothetwocountriessharingthesegment. Sincebordersegmentsexhibitstochasticproperties,weincludeastochastic componentusinganiteapproximationoftheKarhunen-Loeveexpansionofa WienerProcess.ThestandardKarhunen-Loeveexpansionisgivenas[1][11] X t = 1 X k =1 z k p k e k t ; .7 where, 8 k 2 N z k N ; areindependentGaussianrandomvariableswith variance intheWienerProcessapproximation, =1, e k t are ` 2 [0 ; 1] orthogonal,and k aretheeigenvaluesoftheintegraloperation T X f t = R 1 0 K X t;s f s ds ,where K X t;s = E X t X s ; 1 s;t 1,isthecovariancefunction[1]. TheKarhunen-LoeveexpansionofaBrownianbridgeontheinterval[0 ; 1]is foundusing e k t = p 2sin kt and k = )]TJ/F16 7.9701 Tf 9.443 -4.977 Td [(1 k 2 [28].Usingthesevalues,we canrewrite.7as X t = p 2 1 X k =1 z k sin kt k : .8 31

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Notethat X = X =0.Forthemodel,wewishtoparametrizethe expansionfromtheinterval[0 ; 1]totheinterval[0 ;L ],where L isthedistance betweensegmentvertices.Therefore,were-parametrize t to L ,sothat 2 ;L ,where.8becomes X = p 2 1 X k =1 z k sin )]TJ/F18 11.9552 Tf 5.479 -9.684 Td [( k L k : .9 Sinceweareconsideringaniteapproximation,wereplacetheinnitesummationin.9withanitesummationto n X ,i.e., X = p 2 n X X k =1 z k k sin k L ; .10 where X isthensucientfortheapproximationofthestochasticcomponent ofpoliticalborders.Varying n X willallowthesmoothnessofthebordertovary; highvaluesof n X givemorenoise,wellsmallervaluesof n X willgivesmoother borders. Theconvexcomponentofbordersisthensimulatedusingthecatenaryequation.Theuseofthecatenaryequationissupportedbybothquantitativesee thestatisticalregressionandanalysisdescribedinChapter2andqualitative observationsofbordersegmentsinEurope.Furthermore,thecatenaryisacurve thatrepresentsuniformforcewithrespecttonetdisplacement;thisisadesirable trait,whereIassumethatpoliticaltension"andmilitaryforcespolicingborder segmentswillgenerallybeuniformacrosssegments.Thecatenaryisgivenin generalformby y t = a cosh t a ; .11 32

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where a determinesthescaleofthecatenary.Sincethere-parametrizedBrownianBridge X isdenedontheinterval[0 ;L ],thecatenarymustberecentered.Thecoshfunctionisanevenconvexfunction,i.e.,cosh t =cosh )]TJ/F18 11.9552 Tf 9.299 0 Td [(t andiscenteredat0,wherecosh=1=min t 2 R cosh t =min t 2 [0 ; 1] cosh t Re-centeringthecatenaryshownin.11to t = L 2 gives y t = a cosh t )]TJ/F20 5.9776 Tf 7.782 3.258 Td [(L 2 a Sincewewishthebordersegmenttohavethevalueof0at t =0and t = L ,wemustre-parametrize t as t = L ,so 2 [0 ;L ],andsubtract thevalueof y = y L =max t = L 2 [0 ;L ] y ,wherethismaximumisthen y = y L = a cosh L 2 a .Therefore,welet ^ y = a cosh a )]TJ/F18 11.9552 Tf 15.167 8.088 Td [(L 2 a )]TJ/F18 11.9552 Tf 11.955 0 Td [(a cosh L 2 a : .12 Inordertodeterminethedirectionalpushofthebordersegment,thecatenary componentmustbeabletoipbetweenbeingconvexsmallercountrypushes intolargerandconcavelargerpushesintosmaller.Let & 2f)]TJ/F15 11.9552 Tf 27.503 0 Td [(1 ; 0 ; 1 g ;then wemaymultiply^ y t by & todeterminewhetherornotthecatenarycomponent isconvex,concave,orzero.Ifthebordersegmentisestimatedtobeconvex, then & =1;ifthesegmentisconcave,then & = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1;andifthesegmentiseven betweenthetwocountriessharingthesegment,then & =0,leavingnocatenary component.Thentheentirecatenarycomponent t isfoundtobe = & ^ y = &a cosh a )]TJ/F18 11.9552 Tf 15.167 8.087 Td [(L 2 a )]TJ/F18 11.9552 Tf 11.955 0 Td [(&a cosh L 2 a : .13 Therefore,thebordersegmentsimulationequation ;a;& isfoundtoby adding.10and.13,i.e. ;a;& = X + ,where expandsto ;a;& = p 2 n X X k =1 z k k sin k L + &a cosh a )]TJ/F18 11.9552 Tf 15.167 8.087 Td [(L 2 a )]TJ/F18 11.9552 Tf 10.801 0 Td [(&a cosh L 2 a : .14 33

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However,if t;a;& istakentobecontinuous,thensimulatedbordersegments areinsucientlysmooth,havingfartoomuchnoisetoaccuratelyrepresenta potentialsegment.Therefore,discretesamplingofthesignalgeneratedby isperformedatallintervalpoints i intheorderedset= f i g n i =0 ,where n andall i aredeterminedviaalgorithm3.2, a p isthedesiredpushareaof thecatenarycomponentwhere,inthismodel,thenetpushareaoftheinitial territorialexpanseofnation k intoitsadjacentnationforwhichthesegmentis beingsimulatedisused, L isthenetdisplacementofthesegmentagain,in thismodelthisisthenetdisplacementofthebordersegmentintheboundaryof thesimulatedVoronoistate N k ,and^ y isasin.3.Thenthenalsimulated Algorithm3.1 Algorithmusedtodeterminesamplingintervalsforsimulated bordersegment. 0 =0; i =1; while i
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where f i g n i =0 isthenalsetrepresentingthesimulatedbordersegmentand canbeconsideredasaconnectedpathin R 2 .ShowninFigure3.1aresome simulatedbordersegmentswithsomevaryingvaluesof a and n X Figure3.1: Somesimulatedbordersegmentsgeneratedwithvaryingparameters a L thevarianceof z k theexpectednumberofsampleintervals and n X Theparameters a and & arecontrolledbythesimulatednationstates,where thenetpushandnetdisplacementofthecatenarycomponentofthesimulated 35

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bordersegmentareequivalenttothenetpushandnetdisplacementofthe state N k intoadjacentstate N m .Bydoingthis,thesimulatedbordersegment maintainsanetpushareaclosetothatoftheinitialsegmentbetweentheVoronoi states,withsomesmallstochasticvariations.Anumericalsolutionisusedto ndthecatenaryscaleparameter a forasimulatedbordersegment,where a is determinedbythenumericalsolutionto a p = R L 0 )]TJ/F18 11.9552 Tf 9.299 0 Td [(a cosh )]TJ/F19 7.9701 Tf 6.675 -4.977 Td [(x a )]TJ/F19 7.9701 Tf 14.637 4.707 Td [(L 2 a + a cosh )]TJ/F19 7.9701 Tf 8.162 -4.977 Td [(L 2 a = a L cosh )]TJ/F19 7.9701 Tf 8.162 -4.977 Td [(L 2 a )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 a sinh )]TJ/F19 7.9701 Tf 8.161 -4.977 Td [(L 2 a .16 Numericalaccuracyisnotcrucialforthismodel,soasolutionwithanaccuracy of10 )]TJ/F16 7.9701 Tf 6.586 0 Td [(4 waschosenforthesimulation,andthebisectionmethodwasusedto solvefor a Toproduceamapofthesimulatedbordersegmentsbetweentwoadjacent simulatedcountriesattime t; N k t and N m t ,considertheboundariesof twoarbitraryVoronoistates, @ N k t and @ N m t .Eachnon-emptyintersection @ N k t @ N m t isalinesegmentin R withadisplacementdistance betweenendpoints L andnetpusharea a p ;usingthesedata,abordersegmentcanbesimulated,androtatedintoplaceaccordingly.Givenparameters L and a p foreachborderbetweenVoronoistates,thecatenaryparameter a see[3.15]canbenumericallysolvedfortogivetheconvexcomponentofthe bordersegment.ThebordersegmentisthengeneratedforVoronoistateborder @ N k t @ N m t : j @ N k t @ N m t j > 1,wherethesimulatedsegment replacestheboundaryformedbytheVoronoistates.Thentheboundaryofeverycountryisupdatedtoincludethesimulatedbordersegment.Letthesenal 36

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simulatednationsbedenotedas N 0 k t ,wheretheinterioroftheboundaryofthe nationstatewithboundariesreplacedbysimulatedbordersegmentsrepresents theextentofthenation. 3.3DynamicalModelofNationStatesUsingRUIs Thedynamicalchangeofterritoriesovertimeshownhereinisaninteresting exampleofthecapabilitiesofthemodel.Itisareasonablemodel,thoughnot arigorousone.However,itprovidesthebasisforfuturedynamicalmodels andservesasanexampleofhowthismodelcaneasilybecomedynamicusing dierentculturalandnationaldata.Thegoalofthismodelistoencourage politicalscientists,historians,andotherstodevelopandincorporateinteresting dynamicstocontrolthismodel.Thetheoreticalbasisforthismodelborrows fromidentitypoliticsandideasbehindculturalevolution,althoughnoformal theoriesareinvolved.First,somefundamentalassumptionsaremade: HomogeneityofReligiousSpatialDistribution Iassumethatfortheinitialstateofthemodelsimulation,religiouspopulationsarehomogeneously distributedacrosscountries,andthereforeeachRUIhasthesameproportionofadherentstoagivenreligionaseachotherRUIcontainedinthe country.Thisassumptionisnecessarygiventhelackofdataonproportionalreligiouspopulationsbycity.Forsuccessivetimesteps,thehomogeneityofreligionisrestrictedonlytoeachRUI,as,dependingon successivealliances,theseproportionsmaychange. HomogeneityofSpatialDistributionofLanguage Iassumethatlanguage ishomogeneouslydistributedspatiallyacrosseachRUI,wheretheprimary 37

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and,ifapplicable,secondaryandtertiarylanguageproportionswereestimatedusingseveralsources.Thisassumptionismaintainedforeachtime step. StationaryPopulation Iassumethatpopulation,andmorespecically,the ratiosofpopulationsbetweenRUIs,doesnotchange.Thisisanon-realistic assumption,anditisusedonlytopreservefocusontheeectsoflanguage andreligionsonnationalextent.However,theassumptionissupportedby Brakmanetal.[6],whonotethat[the]relativesizeofindividualcities andtheresultingcitysizedistributionareremarkablystableovertimefor mostcountries.[6]" Thedynamicalmodelrestsoftheassumptionthatlinguisticandreligioussimilaritiesarekeyindicatorsfortheculturalcohesionthatbindsanationtogether. Inthismodel,thereligiousandlinguisticcharacteristicsofeachcityareassumed torepresenttheserespectivecharacteristicsfortheentireRUI.Notethatthis doesnotnecessarilyhavetobethecase,astheprevalenceofGISdataregardingthesedemographicpropertiesareavailable,butwereinaccessibleforthis research. Let n L bethenumberoflanguagesspokenintheregionbeingmodeledEurope,andlet n R bethenumberofreligionspracticedintheregion.Forthis simulation,Iuseareducednumberoflanguagesandreligionsfromthetotal, simplifyingcertainreligionsandlanguages.Foreachcentralcity i ,let L i 2 R n L beavectoroflanguages,wherethe j th entryinthevectoristheproportion ofthepopulationin c i thatspeakslanguage j .Notethat P n L j =1 L ij 1,since 38

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manyinhabitantsaremultilingual.Thenlet R i 2 R n R bethereligiousproportionvector,wherereligionisconsideredauniqueidentierforeachperson,i.e., P n R j =1 R ij =1forall i .Finally,let L bethe n n L matrixwithrowvectors L i andlet R bethe n n R matrixwithrowvectors R i Supposethatthelanguagesandreligionsandcultureingeneralforeachcity anditscorrespondingRUIaremostinuencedbythosecitiesclosesttoitas wellasbythecountryasawhole.InuencecomingfromthoseRUIsnearby comesfromlanguagecontact,work,travel,television,economicexchange,etc, whileinuencefromthecountrycomesfromclassrooms,roadsigns,forms,etc. Whileglobalizationandincreasedconnectivityhasdiversiedthegeographical rangeoftheseinuences,regionalandlocalinuencesarestillmoreprevalent thanglobalinuence. 2 Idenethesetof exchangenodes as E i = f j j B j B i 6 = ;_ B j 2 J i g : .17 TheexchangenodesaretheindicesofallotherRUIsthathaveeitheranondisjointintersectionwith B i orarealliedwith B j .ThenIproposethateach RUI i experiencesalevelofculturalexchangewiththosecitieslocatedintheset E i ,wherethatlevelofculturalexchangeisdependentonthedistancebetween c i and c j ; 8 j 2E i ,and P i and P j ,where P i isthepopulationofcity i .Note that i 2E i .Toproducedynamics,let L i t and R j t dependontimestep t t 2 N ,andthentheexchangeequationsaregivenas L i t +1= 1 W L X j 2E i L + y j L e )]TJ/F19 7.9701 Tf 6.586 0 Td [(d ij L P j L j t .18 2 Incorporatingtheseinuencesisdesirable,butsignicantlyincreasesthecomplexityof themodel,aseconomic,trade,andtourismindicatorswouldbeneededaswell. 39

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and R i t +1= 1 W R X j 2E i R + y j R e )]TJ/F19 7.9701 Tf 6.586 0 Td [(d ij R P j R j t ; .19 where: W L = P j 2E i L + y j L e )]TJ/F19 7.9701 Tf 6.586 0 Td [(d ij L P j isthesumofalllanguageweights; W R = P j 2E i R + y j R e )]TJ/F19 7.9701 Tf 6.586 0 Td [(d ij R P j isthesumofallreligionweights; L and R arethebaselineexchangeratesoflanguageandreligion; L and R aretheincreaseinexchangeratesoflanguageandreligionthatis realizedifthenodethat i isexchangingwithisin J i i.e.,iftheexchange nodeisalliedwithRUI i ; L and R aretheratesofcontactforlanguageandreligion,respectively; y j isabinaryvariablesuchthat y j =1if j 2 J i and y j =0if j 62 J i ;and d ij istheEuclideandistancebetween i and j ,ortheentriesin D = d ij ThemodelchangesproportionsofthepopulationofeachRUIthatspeaka certainlanguageandpracticeagivenreligionduetolanguageandreligiouscontact,wherethelevelandratesofcontactdependonthedistanceandpopulation betweeneachnode.Furthermore,inordertosimulateimmigrantslearninga newlanguage,ateachiterationbeforetheexchangeshownin.18occurs,if max 1 k n L L ik t < 1,then L i t isadjustedtosimulateimmigrantslearning theprimarylanguageofnode i .Let k 1 bethepositionin L i t ofthemaximum {thisrepresentstheprimarylanguageofRUI i {and k 2 bethepositionin L i t ofthesecondlargestvalue{thesecondarylanguageofRUI i {then,if 40

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L ik 1 )]TJ/F18 11.9552 Tf 12.622 0 Td [(L ik 2 > forsomethreshold ,replace L ik 1 withanincreasedamount. Mathematically, L ik 1 t L ik 1 t + )]TJ/F18 11.9552 Tf 11.955 0 Td [(L ik 1 t if L ik 1 )]TJ/F18 11.9552 Tf 11.955 0 Td [(L ik 2 >; .20 where '> 0isaproportionofthosespeakerswhodidn'tspeakthelanguage whentheyarrivedwhonowspeakit,and isathresholdthatdetermineswhat theprimarylanguageis.Theinclusionof isnecessaryasitallowsformovementtodetermine languageshift ,orthechangeintheprimarylanguagespoken byapopulation,shouldasecondarylanguagebecomecloseenoughinpropagationtotherstlanguage.Forthismodel, wassetat0.25.Thedynamical modelworksbydynamicallychangingthealliancematrix A t as t varies,where N t =Rank[ A t ]changeswith t aswell.Tochangethealliances,adistance metricmustbeusedtodeterminethedistancesbetweeneachvector.Let ij bethedistancebetweenlanguagevectors i and j ,andlet ij bethedistance betweentworeligiousvectors R i and R j .Firstconsiderthemetricusedforlanguages.Quantifyingsimilaritiesinlanguageisanimportantpartoflinguistics, andthusthereareavailablemetricsforthedistancebetweentwolanguages.For thissimulation,thelinguisticdistanceiscalculatedusingtheworkofJager[19], whereJagerusestheAutomatedSimilarityJudgementProgramdatabase[46] tocalculatethedistancebetweentwolanguages.Forspecicinformationonthe calculation,see[19].Let km denotethelanguagedistancebetweenlanguages k and m .Asdescribedin[19],0 1.Then,since P n L j =1 L ij 1,and since n L isconsideredrelativelylargeinthissimulations,the ` 2 normisnot used.Instead,considertheexpectedlanguagedistancebetweenthetopthree languagesspokenineachcityorRUI.Thetopthreeareusedtosimplifythe 41

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calculation,andsincemostlocalestendtohaveonlyoneortwoandinrarer casese.g.,Switzerland,Luxembourg,Belgium,threelanguagesspokenbya signicantproportionofthepopulation,thissucesforthemodel.Let ij t denotethelanguagedistancebetweenRUI i andRUI j attime t Let L i t = l ij 1 ;l ij 2 ;l ij 3 bethevectoroftheproportionsofthetopthreelanguagesspokeninRUI i attime t ,where1 j 1 ;j 2 ;j 3 n L and j 1 6 = j 2 6 = j 3 6 = j 1 Ifonlyoneortwolanguagesarespoken,thentheproportionsofthepopulationspeakingtheremaininglanguagesaresettozero.Theexpecteddistance E [ ij t ]iscalculatedasfollows.Let S = ff 1 g ; f 2 g ; f 3 g ; f 1 ; 2 g ; f 1 ; 3 g ; f 2 ; 3 g ; f 1 ; 2 ; 3 gg beanorderedset,andlet Q ik betheeventthataspeakerrandomlyselectedin RUI i speaksonlythelanguages l ij m suchthat m 2 S k ,the k th elementof S Thenthelanguagedistance ij betweenRUI i and j isfoundas ij = 8 > < > : 0 L i t = L j t E [ ij t ]else ; .21 whereweassumethat L i t = L j t impliesthetwoRUIsarelinguisticallythe samesothateachRUImaybeselfsimilar.Thevalueof E [ ij t ]iscalculated undertheassumptionthateachentry l ik of L i canbeconsideredastheprobabilitythatarandomlyselectedindividualfromthepopulation P i speakslanguage k .Thentheexpectedvaluecanberewrittenas ij t = 8 > < > : 0 L i t = L j t P n R k =1 P n R m =1 E [ ij t j Q ik Q jm ] P Q ik Q jm else : .22 42

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Notethat P Q ik iseasilycalculatedassumingtheindependenceofeachevent forsimplicity'ssakeforeach i ,and E [ ij t j Q ik Q jm ]= 1 j S k jj S m j X 2 S k X 2 S m .23 isalsoeasilycalculatedwhen Q ik and Q jm areassumedindependentforall i;j;k;m .Theexpectedvalueisusedinsteadofthe ` 2 normsincelanguageis notnecessarilyauniqueidentierlikereligion;somepeoplecanspeakmultiple languages.Furthermore,considertheexpecteddistancebetweentwospeakers, 1and2,wherespeaker1speakslanguage j 1 ,andspeakertwospeakslanguage j 1 and j 2 .Theexpecteddistancedoesnotconsiderthisdistancezero,even thoughbothspeakerscanspeak j 1 .Thisoutcomeisdesiredasitisassumed,in thiscase,thatspeaker2thushasadierentculturalbackgroundthanspeaker 1.Next,considerthereligiousdistance ij t .Since P n R j =1 R ij =1foreach i ,a normalizedEuclideandistanceissucienttouseforthereligiousdistance,i.e., ij t = 1 p 2 v u u t n R X k =1 R ik t )]TJ/F18 11.9552 Tf 11.955 0 Td [(R jk t 2 : .24 Whileusinganexpectedreligiousdistancewouldhavebeendesired,thereisno knownmeasurementforsimilaritybetweenreligionsanalogoustothatofJager's linguisticdistancemetric[19].Therefore,themeasurement,whichistheroot meansquareofthedierencesofproportionsofthepopulationpracticingthe religions,issucient.Forthismodel,religiousadherencewassignicantlysimpliedtosevenprimaryclasses:Catholicism,Hindu,Islam,Judaism,Orthodox Christianity,ProtestantChristianity,andOtherorNone.ThoseintheOther orNone"categorywereprimarilyagnosticoratheist,andanyminorityreligions 43

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wereconsideredtohavelittleeectontheoverarchingculture.FurtherdescriptionofthedatausedandsimplicationisavailableinSection4.1. Aftercalculatingalinguisticexchangesandreligiousexchanges,thedistances between ij t and ij t areusedtocalculatea cohesionscore forallalliednations J k ,1 k N .Thecohesionscoreisdevelopedasananalogytotension, where,if F isaforcevector,asystemisconsideredinequilibriumif P F = ~ 0; i.e.,thesumofallforcesinthesystemiszero,andthusthereisnonetforce. Forthecohesionscore H k t ofacountry k attime t ,let L k t R k t ,and c k t bethemeanlanguageandreligionvectorsofall L i t R i t ,and c i ,weighted bythepopulations P i i 2J k ,i.e., L k t = 1 P i 2J k P i X i 2J k P i L i t ;.25 R k t = 1 P i 2J k P i X i 2J k P i R i t ;.26 and c k t = 1 P i 2J k P i X i 2J k c i : .27 Then,foreachcountry k andeach i 2J k ,let F ik t = c i )]TJ/F15 11.9552 Tf 12.01 0 Td [( c k t beavectorin R 2 ,andlet q ik t =1 )]TJ/F24 11.9552 Tf 11.955 22.323 Td [(s w [ ik t ] 2 + w [ ik t ] 2 w + w .28 betheculturalsimilarityscorebetweennode i andthemeanpointsofcountry k at t ,where: ik t isthelanguagedistancebetween L i t and L k t ,calculatedusing.22; 44

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ik t isthereligiousdistancebetween R i t and R k t ,calculatedusing.24; and w and w aretherespectiveweightsofthelanguageandreligiondistances, asdeterminedbytheuserinthesimulationpresentedherein,respective weightsof1and1.5wereused. Theculturalcohesionisthenconsideredasthesumofthenetculturalforceof eachofcomponentRUIsinthecountry,i.e.,the culturalcohesion ofcountry k attime t iscalculatedas H J k t = X i 2J k t F ik t q ik t ; .29 where H J k t iscalculatedateachiterativetimestep t .Thenlet ^ H bethe cohesionthreshold ,orthevalueforwhichif H J k t < ^ H ,thecountryisconsideredtohaveweakculturalcohesion,implyingthat J k t collapsesintoits components,andthus J k t doesnotexistattime t +1.Instead,eachcomponentRUIbecomesanindependentstateattime t +1,wherethesecomponent RUIsmaychoosetore-allywithotheradjacentstatesinsubsequenttimesteps. Theuseofathresholdisnecessarysince,giventhemodel,theprobabilitythat P i 2J k t F ik t q ik t =0isnearzeroafterexchangesoccur.Ifcountry k collapsesattime t ,thenforeach i 2J k t ,row i ofthealliancematrix A t +1 becomes i ,ortheKroneckerdeltavectorfor i ,where ij =0if i 6 = j and ii =1 seealgorithm3.2. Forcountriesthatdonotcollapseattime t ,adecisionmustbemadewhether 45

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ornottoallywithanothercountry.Inthiscase,analgorithmmovesbetween alliancesandconsiderspotentialcohesionscores,startingwiththealliancewith thegreatestpopulation.Thealgorithmoperatesundertheassumptionthat countrieswithlargerpopulationswillexpandandannexsurroundingcountries withlowerpopulations.Thisalgorithmcouldeasilybechangedtooperatebased onGrossDomesticProductGDP,militaryspendingperGDP,orotherindicators,includingmultipleindicatorssimultaneously.Thealgorithm,whichis performedateachtimestep t ,isshownbelowasalgorithm3.2,where A ij t iselement ij of A t and A i t isthe i th rowof A t ; ij istheKroneckerDeltafunction,where ij =1if i = j and ij =0if i 6 = j ; ^ J k t isthesetofallindicesofallcountriesadjacenttocountry k ; H k m = H J k t [J m t isthe potentialcohesionscore ,calculatedusing .29overthe potentialalliance J k t [J m t ; 2 R isascalar,0 < 1,whichensuresthatexpandingcountriesdonot enterintoweakalliances; T j isthesetmembershipfunctionof T forelement j ,i.e., T j = 8 > < > : 1 j 2 T 0 j 62 T ;.30 Usingalgorithm3.2ateachiteration t givesdynamicalmotioninthenatureof states. 46

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Algorithm3.2 Thealgorithmusedtodeterminethealliancematrixfortime t +1. Set I = f 1 ; 2 ;:::;N t g ; while j I j > 0 do Select k 2 I : P i 2J k P i =max 1 k N t P i 2J k P i ; if H k t < ^ H then for i 2J k t do A ij t +1= ij ; endfor I I nf k g ; else Find ^ J k t N : m 2 ^ J k t jN k t N m t j > 1; for m 2 ^ J k t do H k m = H J k t [J m t endfor if max m 2 ^ J k t H k m > ^ H then Let m = m : H k m =max m 2 ^ J k t H k m ; Let T = J k t [J m t ; for i 2 T do A ij t +1= T j ; endfor I I nf k ;m g ; else for i 2J k t do A i t +1= A i t ; endfor I I nf k g ; endif endif endwhile 47

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4.Simulation ThemodelpresentedinChapter3wasimplementedforasimulationof continentalEuropeusing n =97citiesascentralnodesand t =100simulationsteps.Thissectiondescribesthedatausedinthesimulation,graphically showssomeofthestepsofthesimulationforclarity,andpresentsasetofmaps generatedusingthedynamicalmodelsimulation. 4.1DataandDescription Thecitieswerecompiledusingalistofcitiesthatwereofsignicantgovernmental,military,linguistic,andculturalimportancetotheirnationsandas standalonebodies.ThecityincludedGlobalandWorldCitiesResearchNetwork[36]globalcitiesfor2010;alistofprimatecities 1 Twentyfouradditional citieswereincludedtocompletethesimulationforoneofvereasons: 1. Importancetoitsrespectivecountry: Mostadditionalcitieswere includedforthisreason.ForexampleBrno,CzechRepublicwasincluded sincetheConstitutionalCourtoftheCzechRepublicislocatedinBrno. 2. Primatecityofanexclave: Kaliningrad,Russiawasincludedasitthe clearprimatecityoftheKaliningradOblastofRussia,aRussianexclave innortheasternEurope. 3. Largestcityinamicrostate: AndorraLaVella,Andorra;Vaduz, Liechtenstein;andGibraltar,Gibraltarwereincludedforthisreason. 1 SeeChapter ?? forabriefdiscussionofprimatecities;also,see[25],[20],and[26]for morein-depthdiscussions. 48

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4. Proxyforanothercity: Padova,Italywasincludedasaproxyfor Venice,Italytheyareconsideredaspartofthesamemetropolitanstatisticalareasincethelatitude/longitudecoordinatesforVenicelieoutside ofthelatitude/longitudecoordinatesoftheboundaryofEurope. 5. Geographical/culturalproxy: Afewcitieswereincludedtoactas culturalproxies,orcitieswhoseinclusionactedasaninitialstate t =0 boundarybetweentheintersectionofmanyculturesMetz,France,orgeographicproxies,signicantcitiesthatwereincludedtoactasgeographic boundariese.g.,Toulouse,FranceactedasaproxyforthePyreneesMountains;InnsbruckAustriaactedasaproxyfortheAlps,etc.. AfewmajorEuropeanCitieswerenotincluded.Forexample,Copenhagen, Denmark,wasnotincludedsinceitislocatedonanislandwithnootherprimatecities,violatingtheconnectednessoftheboundary B .Thecompletelist ofcitiesusedinthesimulationisavailableinappendixA. Languageandreligiondatawereacquiredfromseveralsources.Asdescribedin Section3.3,languagedistanceswereacquiredfromGerhardJageroftheUniversityofTubingen,wherethemethodforthecalculationoftheselanguage distancesisdescribedindepthin[19].Theinitiallanguagevectors L i were setusingamaximumofthreelanguagesspokenineachcity,i.e.,ifonlynolanguageisspokenbyasignicantminority > 10%ofthepopulation,thenonly theociallanguageofthecountrythatthecitywaslocatedinwasconsidered, andthislanguagewassettobespokenby100%ofthepopulation.Ifasecond languageofsignicancewasspoken,thislanguagewassettobeinitiallyspoken 49

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by30%ofthepopulation,andifathirdlanguage,orasecondlanguagespoken byasmallminorityofthepopulation,wasdeterminedtobespoken,theinitial proportionofthepopulationintheRUIspeakingthislanguagewassetto15%. Theseinitialproportionsweresetafterqualitativelyexaminingseveralcitiesfor whichdetailedproportionswereavailable;notethatnogooddataonthetrue proportionsoflanguagesspokenineachcitywereavailable.Thetopthreelanguagesweredeterminedusingavarietyofsources,includingEthnologue[24], theWorldAtlasofLanguageStructures[13],andtheCIAWorldFactbook[8]. ThenumberoflanguagesinEuropewasconsideredtobe36,andthus n L =36. Theinitialproportionsforreligiousadherenceineachcitywastakentobe thesameasthatofthenation.ThesedatawereobtainedfromtheCIAWorld Factbook[8],andallreligionsweresimpliedintosevendistinctclassesi.e., n R =7:CATHOLIC,HINDU,ISLAM,JUDAISM,OTHERNONE,ORTHODOX,andPROTESTANT. Theinitialcountriesincludedinthesimulationarethoselocatedincontinental Europe,orallcountriesshowninappendixA.Togeneratethemapsandperform thesimulation,thelatitudes,longitudes,andpopulationsofallcitiesincluded inthesimulationwerefoundusingWolframMathematica8.Thesourcesfor theselatitudesandlongitudesareshownin[47].Furthermore,themapofthe boundaryofcontinentalEuropewasgeneratedusingpolygondataofcountries incontinentalEuropefromMathematica.Thesesourcesareavailablein[48]. Thepolygondataformapsusedlongitudelatitudecoordinates,andsubsequent 50

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mapsweregeneratedusingtheAlbersequal-areaprojection.Theinitialalliance matrixwasgeneratedusingthealliancesofcitiesastheycurrentlystand,i.e., citieslocatedinthesamecountryareconsideredallied. 4.2SimulationResults Thesimulationproducedsomeinterestingmapsanddynamicalmotionbetweencountries.First,shownbelowinFigure4.1isamapofEuropewithall citiesincludedinthesimulationshownaspoints,whileshowninFigure4.2is thesetofallgeneratedRUIs. Figure4.1: MapofsimulationnodesincontinentalEurope. 51

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Figure4.2: MapofallgeneratedregionsofurbaninuenceRUIs. FollowingthecomputationoftheRUIs,alloatingcomponentshadtobe reattached.ShowninFigure4.3isanexampleofsomeoatingpolygonsthat hadtobereattachedaccordingtothefargreaterinuenceofalandconnectionbetweenlocationsonculturaldiusion.Inthegure,weseethatpartof thesoutheasterntipofItalyfellundertheinuenceoftheRUIsofPodgorica, MontenegroandTirana,Albania.Clearly,theinuenceofNaples,Italyisfar greateronthisregion,andsotheywereattachedtotheRUIassociatedwith Naples.Additionally,averysmallpartoftheRUIrepresentedbyThessaloniki hadtobeattachedtotheAthensRUI,andalargerpartoftheAthensRUIhad tobeattachedtotheThessalonikiRUI. 52

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Figure4.3: FloatingcomponentsandrespectiveVoronoiregionsinSouthern Europe. Figure4.4: SimulatedVoronoistates N k forcontinentalEurope. TheresultsofthedynamicalsimulationareshownbelowinFigures4.5 -4.22.Outof100timesteps,allianceschangedonly17times,anddidnot changeaftertime t =39,suggestingthatanequilibriummayhavebeenreached, 53

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althoughthiswasnotconrmed. Figure4.5: Simulatednations N 0 k forcontinentalEuropewithsimulated bordersegments. Thesimulatedcountriesat t =0areshownin4.5,whereall37countriesin initialstatesareincluded.Notethatthestochasticbordersegmentcomponents areshownonlyintheinitialandnalstagesofthesimulation,wheremapsof theintermediateiterationsonlyhaveVoronoistatesshown. 54

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Figure4.6: SimulatedVoronoistates N k attime t =1forcontinental Europe. After1timestep,signicantchangesinthealliancesoftheRUIsoccur.The numberofcountriesdeclinesfrom37to29:LiechtensteinisabsorbedbyAustria; SloveniaandCroatiabecomeonecountry;HungaryandSerbiajoin;Albaniaand Kosovojoin,clearlybolsteredbytheirsimilaritiesasstateswithamajorityofthe populationidentifyingthemselvesasMuslim;BosniaandMontenegrobecome onecountrygroupedtogether;RomaniaandMoldovacoalesce;Belarusand theUkrainebecomeonecountry;andBulgariaandMacedoniaallythemselves. NoticethatthevastmajorityofmotionoccurredinEasternEurope,aregion wheretherehasbeensignicantcollapseandre-aggregationofpoliticalborder inthetwentiethcentury{thecollapseofYugoslavia,thedivisionofMoldova fromRomania,theSerbiancivilwarandthebirthofKosovoareallexamples. Consequently,thisphenomenonisnotsurprising. 55

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Figure4.7: SimulatedVoronoistates N k attime t =2forcontinental Europe. Attime t =2Figure4.7,afterallyingitselfwithLiechtenstein,Austria collapsesintoitsRUIs,implyingtheculturalcohesionscorewastoolowafter onlyonestep.Furthermore,Slovenia,Croatia,andSlovakiahavejoinedtogether,andthereismoreaggregationinEasternEurope.Atthispointthere are30countriesinEurope. 56

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Figure4.8: SimulatedVoronoistates N k attime t =3forcontinental Europe. Attime t =3Figure4.8,theAustrianstateshavebeguntore-coalesce, leavingthenumberofcountriesat28. Figure4.9: SimulatedVoronoistates N k attime t =4forcontinental Europe. 57

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ShowninFigure4.9arethecountriesattime t =4.TheCzechRepublic haspiecedopartofAustria,butmorenotably,amajorshakeupinWestern EuropehasoccurredasFrancehascollapsed.Thisissurprisinggiventherelative homogeneityoflanguageandreligioninFrance,butthefactthatFrancehas aprimatecityParisandthusweakeralliedcities,combinedwithitslocation betweenSpain,Germany,Switzerland,andItaly,meansitisstronglyinuenced byoutsiderforces.Thecollapseleave35countriesinEurope. Figure4.10: SimulatedVoronoistates N k attime t =5forcontinental Europe. 58

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Figure4.11: SimulatedVoronoistates N k attime t =6forcontinental Europe. ShowninFigures4.10and4.11arethecountriesattime t =5and t =6. Francelargelyregroupsby t =6,butnotbeforelosingterritorytoBelgium, Switzerland,and,surprisingly,Monaco. 59

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Figure4.12: SimulatedVoronoistates N k attime t =9forcontinental Europe. Noticebytime t =9Figure4.12thatBelgiumandtheNetherlandshave allied,whichisnotunfoundedgiventherelativesimilaritiesoftheircultures, andbytime t =12Figure4.13,GibraltarhasbeenannexedbySpain.This movementisalsonotsurprising,asGibraltarhasbeenasubjectofcontention betweentheUKandSpainforsometime. 60

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Figure4.13: SimulatedVoronoistates N k attime t =12forcontinental Europe. Figure4.14: SimulatedVoronoistates N k attime t =14forcontinental Europe. Attime t =14,showninFigure4.14,thecountrythathadoriginatedin thesouth{composedofMonacoandportionsofFrance{collapses,asdoesthe 61

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EasternEuropeancountrythathadbeencomposedofSlovakia,Slovenia,and Croatia.Attime t =15,showninFigure4.15,there-aggregationofpartof Slovakiaisshown,aswellasFranceannexingpartofitsformerterritory.Italy gainsMonaco,andtwoRUIsjustsouthofFranceally. Figure4.15: SimulatedVoronoistates N k attime t =15forcontinental Europe. Bytime t =16,weseeinthemapshowninFigure4.16there-aggregation ofFrance,whereFranceregainsthelostterritoryintheSouth. 62

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Figure4.16: SimulatedVoronoistates N k attime t =16forcontinental Europe. Figure4.17: SimulatedVoronoistates N k attime t =19forcontinental Europe. ShowninFigures4.17and4.18aregeneratedmapsofEuropeattime t =19 and t =30.Noticethatattime t =19,Switzerlandgainsterritoryfrom 63

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LuxembourgandFrance,whileat t =30,LatviaandLithuaniaaggregate. Figure4.18: SimulatedVoronoistates N k attime t =30forcontinental Europe. Figure4.19: SimulatedVoronoistates N k attime t =36forcontinental Europe. ThemapshowninFigure4.19showsthecollapseofItaly.This,similarly 64

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tothecollapseofFrance,isnotunwarranted;Italydidnotuniteuntilthelate nineteenthcentury,anditdidnotbecomearepublic,asitcurrentlystands,until 1946.Beforethistime,thecountryincludedseveralsmallerregions,housedthe Romanempire,theHolyRomanempire,thePapalStates,andmanyother politicalstates.ThiskindofmotionoccurringinItalyshouldbeexpected. Figure4.20: SimulatedVoronoistates N k attime t =37forcontinental Europe. Attime t =37Figure4.20,Italybeginstoreunite,thoughonlyafter losingterritorytoSwitzerland,whichnowextendsfromBelgiumatitsnorthern enddowntotheMediterraneanatVenicePadova,Italy. 65

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Figure4.21: SimulatedVoronoistates N k attime t =38forcontinental Europe. Thenaltwochangesinalliancesthatoccurredinthesimulationareshown inFigures4.21and4.22,ortimes t =38and t =39.At t =38Switzerlandgains evenmoreterritory,whileItalyiscontinuingtoreunite.By t =39,Italyhas,for themostpart,reunited,althoughthelossofVenicetoSwitzerlandisnotable; SwitzerlandisalargecountryinWesternEuropebythispoint.EasternEurope remainedfairlystableafteralotofchangeearlyon,andtheCzechRepublicnow isalargercountry.Polandhaslostsignicantterritory,andRomaniashowsan enormouspushintotheUkraine,wheretheUkrainealsoabsorbedBelarus. 66

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Figure4.22: Simulatednations N 0 k attime t =39forcontinentalEurope withsimulatedbordersegments. Someremarkablephenomenaoccurarerevealedinthesimulations.AlthoughSwitzerlandisaverylinguisticallyandreligiouslydiversecountry,atno pointdoesitcollapseinthesimulation.Infact,itcontinuestogrow,becoming averylargegeographicalpresenceinWesternEuropeby t =39,containingnot onlytheSwissRUIsrepresentedbyBasel,Bern,Geneva,Lausanne,andZurich, buttheadditionalRUIsofInnsbruck,Austria;Luxemburg,Luxembourg;Metz, France;PadovaVenice,Italy;Strasbourg,France;Trieste,Italy;andVaduz Liechtenstein.Switzerlandgainedthemostterritoriallyduringthesimulations. Itispossiblethatitsabilitytoadaptandembraceadiversearrayoflinguistic andreligiousbackgroundsatleastbyEuropeanstandardsisanassettoits nation'sgrowth.Thistypeofmotionisrecognizedinotherspotsaswell;Spain, whichwasexpectedtocollapseduetoitsarrayofdierentlanguagesGalician [Vigo],Basque[Bilbao],Catalan[Barcelona],andCastilianSpanish[Madridand 67

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allinitialSpanishRUIs],showednomovementatallduringthesimulation.Andorra,amicrostatelocatedinthePyreneesmountainrangeseparatingFrance andSpain,alsodidnotgetabsorbed.Theprimarylanguagesspokenthereare Catalan,Spanish,andFrench,anditsrelativeinitialdiversitymadeitableto withstandallreligiousandlinguisticexchange.Finally,notethatthenalstate endedwithgenerallylargercountries;thereareonlyafewsmallcountriesleft, butforthemostpart,smallerstatesjoinedtogethertocreatelargerstates.Each stateshowninFigure4.22tendtoreectlinguisticandreligioussimilarities,as expectedgiventhedynamicalmodel.Thisfurtherreinforcestheneedtoincorporatemorecultural,economic,governmental,andmilitarydataintothemodel toexplainthesustainedexistenceofsmallercountries,whicharestillrelatively similartoadjacentcountries,overtime. 68

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5.Conclusions Themodelpresentedhereincanbeusedtoexaminehowdynamicalchanges betweencountriescanimpactandchangenationstates.Withmoreaccurate dynamicalmodelsthantheonepresentedinSection3.3,moreaccuratemodels ofterritorialchangecanbepresented.Furthermore,thestructureofthemodel allowsterritorialchangesandnationalexpansetoeasilybedescribedusingany numberofapplicabledynamicalmodels,includingthoseineconomics,gametheoryandmodelsofwar,populationgrowthanddynamics,historicaldynamics, etc.Forexample,theworkofTurchinsee[37][38]and[39]forsomeexamples wouldbewellappliedheretomodeltheexpanseofhistoricalempiresandhow thoseempireschangeddynamically.Applyinghistoricalknowledgemayallow historianstoexaminehowtheseempiresandstateschangedinexpanseover time,whytheydid{ordidn't{collapse,whatmeansofinfrastructurearemore eective,andsoonandsoforth.Furthermore,agranddynamicalmodel,incorporatingeconomicconditions,theUnitedNationsHumanDevelopmentIndex [42],governmenttype,militaryspending,religion,language,education,other nationalindicators,and,importantly,geographicfeatures,couldprovevaluable inpredictingtheoutcomesofconictanddrawingsubsequentboundariessothat equilibriumbetweennationsorgroups{andthuspeace{couldbeachieved. 5.1PotentialImprovements Insummary,themodelpresentedhereinuseslatitude/longitudedataforthe locationofcitiesinadditiontocontinentalboundariestodeterministicallysimulatetheterritorialexpanseofnationstates.Whilethemodelisabletodescribe 69

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dynamicalevolutionofpoliticalbordersovertime,thesimulationrevealedsome keypointsofimprovement.Forexample,Andorra,Liechtenstein,andLuxembourgallhadvastlylargerareasofinuencethanthecountrieshavehadover time;thisislargeleyduetotheequivalenceofnodeweights.Also,Austria collapsedimmediatelyinthesimulation;thisisduetothenon-incorporationof geographiccomponentsinbordersegmentsandtheexclusionofotherproperties ofnationstatesthatbindthemtogethere.g.,history,ideology,politicalvalues, etc..However,thereareseveralimprovementstothemodelthatcanbemade toxtheseshortcomings,describedbelow. UseofWeightedVoronoiTessellations Thesimulationrevealedsomepoints ofpotentialimprovementofthemodel.First,andmostnotably,the weightingofVoronoitessellationswouldbevaluable.Forexample,the weightoftheregionalinuenceofAndorraLaVella,Andorra,should notbethesameastheregionalinuenceofParis,France,ortheRhine metropolitanareainWestGermany.Therearesomemethodsavailableto performthiskindoftessellation;see[5]and[12]fordescriptionsandapplicationsoftheseweightedtessellations.Furthermore,theworkofRicca, Scozzari,andSimeone[30]inusingweightedVoronoitessellationstodevelopcongressionaldistrictswouldhavevaluableinsightintohowtoapply thesestructurestothemodelofnationalexpanse.Furthermore,usingthe populationofacity'smetropolitanarea{asopposedtothepopulationof onlythecity{inconjunctionwithweightedVoronoitessellationswould allowformoreaccuraterepresentationsofregionsofurbaninuence. IncorporationofGeographicData Inadditiontoimprovingtheuseoftes70

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sellations,theincorporationofgeographicstructureswouldimprovethe model;thesestructures{suchasmountainranges,landmarks,rivers,and bays{arecrucialtounderstandingtheconstructionofnations.Forexample,thestructureoftheprimarybordersegmentseparatingCroatiaand BosniaandHerzegovinaisanoutlierwhenconsideringnetdisplacement andpusharea,butisexplainedbytheSavaRiver;similarly,theAlpsalmostentirelydenetheborders,anddefendtheexistences,ofSwitzerland andAustria.IfthismodelweretoutilizegeographicstructuresinconjunctionwithweightedVoronoitessellations,itwouldbebettersuitedto describetheterritoriesofmicrostateslikeAndorraandLiechtenstein.Note thatwhiletheincorporationofgeographicdatawasinitiallyattempted, alackofavailabilityand/oraccessibilityofGISdataformajormountain rangeswasasignicantinhibitor. ParametricSimulationofPoliticalBorders Thesimulationofpoliticalborderscanbeimprovedsignicantlybyincorporatinggeographicaldata.For example,ifthelinesegmentbetweentwoendpointsofaborderbetween adjacentRUIswasanywhereclosetopeaksinamountainrange,randompeakcoordinatescouldbeselected{aspartofageographicborder segment{toreplacethelinesegment.Furthermore,aparametricstochasticcomponent,suchastrueBrownianmotion,wouldbevaluable.Some segmentsexhibittopologicalpropertiesthatcannotbedescribedbyanonparametricfunction.Allowingcertainborderstodynamicallyprocessover timewouldbedesirableaswell.Forexample,anexclavemightbeformed afteraanothercountrystartspushinginbetweenterritories,pushinga 71

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portionofthecountrybeingpushedintotoformasortofdroplet"structure,whereeventuallythiswoulddrop",andtheremainingterritory, separatedbytheinvadingterritory,wouldbetheexclave. AccurateDynamicalSystems Improvementstothedynamicalsystemcould benumerous.Themodelpresentedinthispaperisnotarigoroustranslationoftheoreticalwork,butamodelbasedonreasonableassumptions designedtoshowdynamicaggregationandcollapseofnationstates.Incorporatingexistingeconomicdynamics,includingmodelsoftrade,tourism, interdependence,unemployment,andothersignicanteconomicfactors wouldhelpdescribetherelativepowerofanation.Incorporatingmilitary modelsandevolutionarygametheorywouldalsoleadtointerestingresults.ThehistoricaldynamicsdescribedbyTurchin[37][39]couldbe appliedaswelltomoreaccuratelydescribethechangeofnations.Finally, arigoroustreatmentoflinguisticandreligiousexchangewouldbevaluable;modelsoflanguageshiftandcontact,aswellasmodelsofreligious conversion,inconjunctionwithmodelsofgeospatialpopulationmovement anddynamicshavethepotentialtoaddrealisticdynamics. UsingPopulationDynamics Populationdynamicscanbeincorporatedto demonstratethevaryingrelativeinuenceofcitiesovertimetoadjacent andalliedRUIs.Furthermore,ifathresholdsizewereusedforcitiesand metropolitanareas,insteadofjustalistofprimateandglobalcities,then nodescouldbeusedtogeneratevariableRUIsovertime;forexample,by keepingtrackofallcitieswithpopulationsover50,000,andusingpopula72

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tiondynamicsbetweenthesecities,anycitiesthatgrewsignicantlycould evolveintogeneratingnodesforRUIs.Thiswouldallowforincreasing dynamicaltreatmentofnationstates. ImprovetheCohesionScore Thecohesionscorecanincorporateadditional data,suchasethnicdata,historicalcohesionandcooperationbetween groups,politicalandideologicalsimilarities,andsoonandsoforth.Improvingthecohesionscorewouldhelptoexplaintheexistenceofadjacent statesthatareverysimilartoeachotherlinguisticallyandreligiously,but arestillseparatestatese.g,KosovoandAlbania,RomaniaandMoldova, etc.. 5.2FutureWork Futureworkincludesimplementingthepotentialimprovements,beginning withgeographicdata.FindingGISdatawithmountainpeaksordevelopinga set,aswellasdevelopingaroutinefortherandomincorporationofpeaksinto aborder,wouldhelpsignicantlywiththetreatmentofborders.Incorporating thesebordersintoRUIs,muchasthecontinentalbordersare,wouldmakebordersegmentsimulationsimple. Finally,encouragingdynamicalmodelinginthesocialsciencesisoneofthe goalsofthismodel.Sharingtheframeworkisoneoftheimmediategoals,inthe hopethatitwillleadtofurtherunderstandingthenatureofnations,whichcan leadtoabetterunderstandingofthedrawingofpoliticalboundaries,onboth thenationalandsub-nationallevel. 73

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REFERENCES [1]XiaohuiAi,WenboV.Li,andGuoqingLiu.Karhunen-loeveexpansions forthedetrendedbrownianmotion. StatisticsandProbabilityLetters 82:1235{1241,July2012. [2]AlbertoAlesinaandEnricoSpolaore.War,peace,andthesizeofcountries. JournalofPublicEconomics ,89:1333{1354,2005. [3]PeterBoguckiandPamJ.Crabtree,editors. AncientEurope,8000B.C. toA.D.1000:EncyclopediaoftheBarbarianWorld ,volume2:Bronze AgetoEarlyMiddleAgesc.3000B.C.-A.D.1000.NewYork:Charles Scribner'sSons,2004. [4]PatrickBolton,GerardRoland,andEnricoSpolaore.Economictheories ofthebreak-upandintegrationofnations. EuropeanEconomicReview 40:697{705,1996. [5]B.N.Boots.Weightingthiessenpolygons. EconomicGeography 56:248{259,1980. [6]StevenBrakman,HarryGarretsen,andMarcSchramm.Thestrategic bombingofgermancitiesduringworldwariianditsimpactoncitygrowth. JournalofEconomicGeography ,4:201{218,2004. [7]NeilBrenner.Urbanlocationalpoliciesandpost-keynesianstatehoodin westerneurope.InDianeE.DavisandNoraLibertundeDuren,editors, CitiesandSovereignty ,chapter6,pages152{175.IndianaUniversityPress, 2011. [8]CentralIntelligenceAgency. TheWorldFactbook .C.I.A., https://www.cia.gov/library/publications/the-world-factbook/,2013. [9]NicholasCharron.Dejavualloveragain:Apost-coldwarempiricalanalysisofhuntington's'clashofthecivilizations'theory. Cooperationand Conict ,45:107{127,2010. 74

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[22]H.D.F.Kitto.Thepolis.InRichardT.LeGatesandFredericStout,editors, TheCityReader ,pages40{45.MPGBooksGroupLtd.,5edition,2011. [23]JamesKurth.Thesoldier,thestate,andtheclashofcivilizations:The legacyofsamuelhuntington. Orbis ,54:320{334,2010. [24]M.PaulLewis,GaryF.Simmons,andCharlesD.Fennig,editors. Ethnologue:LanguageoftheWorld .SILInternational,Dallas,Texas. http://www.ethnologue.com,seventeenthedition,2013. [25]ArnoldS.Linsky.Somegeneralizationsconcerningprimatecities. Annalsof theAssociationofAmericanGeographers ,55:506{513,September1965. [26]GeoreyJ.Martin.thelawofprimatecities"re-examined. Journalof Geography ,60:165{172,April1961. [27]IstanbulMetropolitanMunicipality. ConquestandIstanbul .Istanbul2010: EuropeanCapitalofCulture.http://www.ibb.gov.tr/sites/ks/en-US/0Exploring-The-City/History/Pages/ConquestandIstanbul.aspx,2008. [28]GillesPageesandJacquesPrintems.Functionalquantizationfornumerica withanapplicationtooptionpricing. MonteCarloMethodsandApplications ,11:407{446,2005. [29]FernandoTavaresPereira,JoeeRuiFigueira,VincentMousseau,and BernardRoy.Comparingtwoterritorypartitionsindistrictingproblems: Indicesandpracticeissues. Socio-EconomicPlanningSciences ,43:72{88, 2009. [30]FredericaRicca,AndreaScozzari,andBrunoSimeone.Weightedvoronoi regionalgorithmsforpoliticaldistricting. MathematicalandComputer Modeling ,48:1468{1477,2008. [31]GillianRobinson.Thegreekpolisandthedemocraticimaginary. Thesis Eleven ,40:25{43,1995. [32]FritzSchachermeyr.Thegenesisofthegreekpolis. Diogenes ,1:17{30, 1953. [33]GarySheeld. TheFallofFrance .BBCHistory. http://www.bbc.co.uk/history/worldwars/wwtwo/fall france 01.shtml, 2011. 76

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[34]DavidG.SirmonandPeterJ.Lane.Amodelofculturaldierencesand internationalallianceperformance. JournalofInternationalBusinessStudies ,35:206{319,2004. [35]LeslieSklair.Iconicarchitectureandurban,national,andglobalidentities.InDianeE.DavisandNoraLibertundeDuren,editors, Citiesand Sovereignty ,chapter7,pages179{195.IndianaUniversityPress,2011. [36]P.J.TaylorandG.Catalano. WorldCityNetwork:TheBasicData .GlobalizationandWorldCitesResearchNetwork, http://www.lboro.ac.uk/gawc/datasets/da11.html,2000. [37]PeterTurchin. HistoricalDynamics:WhyStatesRiseandFall .Princeton UniversityPress,2003. [38]PeterTurchin.Dynamicalfeedbacksbetweenpopulationgrowthandsociopoliticalinstability. StructureandDynamics ,1,2005. [39]PeterTurchin.Atheoryforformationoflargeempires. JournalofGlobal History ,4:191{217,2009. [40]EdwardUllman.Atheoryoflocationforcities. AmericanJournalof Sociology ,46:853{864,1941. [41]UnitedNations. WorldUrbanizationProspects,the2011Revision .DepartmentofEconomicandSocialAairs,PopulationDivision,PopulationEstimatesandProjectionsSection.http://esa.un.org/unpd/wup/CDROM/Urban-Rural-Population.htm.,2011. [42]UnitedNations.Humandevelopmentreport2013:Theriseofthesouth -humanprogressinadiverseworldstatistcalannex.Technicalreport, UnitedNations,2013. [43]LawrenceJ.Vale.Thetemptationsofnationalisminmoderncapital cities.InDianeE.DavisandNoraLibertundeDuren,editors, Citiesand Sovereignty ,chapter8,pages196{208.IndianaUniversityPress,2011. [44]EricW.Weisstein. VoronoiDiagram .FromMathWorld{AWolframWeb Resource.http://mathworld.wolfram.com/VoronoiDiagram.html. [45]EricW.Weisstein. VoronoiPolygon .FromMathWorld{AWolframWeb Resource.http://mathworld.wolfram.com/VoronoiPolygon.html. 77

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[46]SorenWichmannorenWichmann,AndreMuller,VivekaVelupillai, AnnkathrinWett,CecilH.Brown,ZarinaMolochieva,JuliaBishoberger, EricW.Holman,SebastianSauppe,PamelaBrown,DikBakker,JohannMattisList,DmitryEgorov,OlegBelyaev,MatthiasUrban,HaraldHammarstrom,AgustinaCarrizo,RobertMailhammer,HelenGeyer,David Beck,EvgeniaKorovina,PattieEpps,PilarValenzuela,andAnthony Grant.TheASJPDatabase,2012. [47]WolframResearch. CityDataSourceInformation .Wolfram, http://reference.wolfram.com/mathematica/note/ CityDataSourceInformation.html,2013. [48]WolframResearch. CountryDataSourceInformation .Wolfram, http://reference.wolfram.com/mathematica/note/ CountryDataSourceInformation.html,2013. [49]TangYijie.Ontheclashandcoexistenceofhumancivilizations. Procedia SocialandBehavioralSciences ,2:7381{7391,2010. 78

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AppendixA.CitiesIncludedinSimulation Thefollowingsymbolsindicatethesourceofthecitiesincludedinthesimulationofthemodel: y -GaWCWorld/Globalcity[36]; -Primatecity; x -Additionalcityaddedasproxy. Amsterdam y Netherlands AndorraLaVella x Andorra Antwerp y Belgium Arhus y Denmark Athens y Greece Barcelona y Spain Basel y Switzerland Belgrade y Serbia Berlin y Germany Bern y Switzerland Bilbao y Spain Bologna y Italy Bonn y Germany Bordeaux y France 79

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Bratislava y Slovakia Brno x CzechRepublic Brussels y Belgium Bucharest y Romania Budapest y Hungary Chisinau Moldova Cologne y Germany Constanta x Romania Dortmund y Germany Dresden y Germany Dusseldorf y Germany Essen y Germany Frankfurt y Germany Gdansk x Poland Geneva y Switzerland Genoa y Italy Gibraltar x Gibraltar Grenoble y France Hamburg y Germany Hanover x Germany Innsbruck x Austria Kaliningrad x Russia Katowice x Poland Kaunas x Lithuania 80

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Kiev y Ukraine Klaipeda x Lithuania Kosice x Slovakia Krakow y Poland Lausanne y Switzerland Leipzig y Germany Liege y Belgium Lille y France Linz y Austria Lisbon y Portugal Ljubljana y Slovenia Luxemburg Luxembourg Lviv x Ukraine Lyon y France Madrid y Spain Mainz y Germany Malaga x Spain Mannheim y Germany Marseille y France Metz x France Milan y Italy Minsk y Belarus MonteCarlo Monaco Montpellier x France 81

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Munich y Germany Naples y Italy Nuremberg y Germany Odesa x Ukraine Padova x Italy Paris y France Podgorica Montenegro Porto x Portugal Prague y CzechRepublic Pristina x Kosovo Riga y Latvia Rome y Italy Rotterdam y Netherlands Sarajevo y BosniaandHerzegovina Seville y Spain Skopje Macedonia Soa y Bulgaria Strasbourg y France Stuttgart y Germany Tallinn Estonia TheHague x Netherlands Thessaloniki x Greece Tirana y Albania Toulouse x France 82

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Trieste y Italy Turin y Italy Utrecht y Netherlands Vaduz x Liechtenstein Vienna y Austria Vigo x Spain Vilnius y Lithuania Warsaw y Poland Zagreb y Croatia Zilina x Slovakia Zurich y Switzerland 83