©CopyrightJASSS

AndrewWhite(2013)

AnAbstractModelShowingThattheSpatialStructureofSocialNetworksAffectstheOutcomesofCulturalTransmissionProcesses

JournalofArtificialSocietiesandSocialSimulation 16(3)9<http://jasss.soc.surrey.ac.uk/16/3/9.html>

Received:09-Oct-2012Accepted:04-Feb-2013Published:30-Jun-2013

Abstract

Spaceplaysanimportantroleinthetransferofinformationinmostsocietiesthatarchaeologistsstudy.Socialnetworksthatmediatelearningandthetransmissionofculturalinformationaresituatedinspatialenvironments.Thispaperusesanabstractagent-basedmodeltorepresentthetransmissionofthevalueofasingle"stylistic"variableamonggroupslinkedtogetherwithinasocialnetwork,thespatialstructureofwhichisvariedusingafewsimpleparameters.Thepropertiesofthenetworksareshowntoclearlyaffectboththeoverallamountofvariabilitythatisproducedbytheculturaltransmissionprocessandthespatialorganizationofthatvariability.Therelationshipsbetweennetworkstructure,networkproperties,andassemblagevariabilityinthissimplemodelarepatternedandpredictable.Thissuggeststhatchangesinthespatialstructureofsocialnetworksmayhaveimportantimplicationsforinterpretingpatternsofartifactvariabilityinlarge-scalearchaeologicalassemblages.

Keywords:SocialNetworks,CulturalTransmission,SpatialAnalysis,NetworkStructure,NetworkProperties,Archaeology

Introduction

1.1 Allhumansocietiesarecomprisedofindividualsconnectedtooneanotherbyoverlappingarraysofsocialtiesthattogetherconstituteasocialnetwork.Socialnetworksareemergentphenomenathatbothinfluenceandareproducedbythebehaviorofindividuals(Barnes1972;BlauandScott1962;Radcliffe-Brown1940).Theychannelinformation,people,genes,andresourcesandcanbeusedtodefinetheextentofasocialsystem.Theimportanceofsocialnetworkstoallhumansocietiesmakesthemafundamentalandenduringtopicofanthropological,archaeological,andsociologicalinterest.

1.2 Socialnetworksarticulatewithmaterialculturebecauselearningissituatedinsocialcontexts.Closelyrelated,co-residentpersonsaretypicallyinstrumentalinthegenerational(i.e.,"vertical")transferofcrafttraditions:parents(andrelatedindividuals)teachchildren(BamforthandFinlay2008;BerryandGeorgas2009;Minar2001;ShennanandSteele1999).Thehighdensityoftieswithinlocalgroupsallowsfortherapid"horizontal"peer-to-peertransferofinformationrelatedtotechnologicalinnovationsandimprovements(seeBerryandGeorgas2009;BettingerandEerkens1999:237;alsoSchifferandSkibo1987:597).Inneithercaseisinformationtransferrandom:interactionsamongindividualsaremediatedbysocialties.Transferofbothkindsofinformationissubjecttohumanerror,whichisasourceofvariabilityinmaterialculture.Spaceplaysanimportantroleinsociallearninginmostsocietiesthatarchaeologistsstudybecauseoftheimportanceofface-to-faceinteractionforthetransferofinformation.Thuswecanviewarchaeologicalartifactsasthematerialresiduesofspatially-situated,network-mediatedsystemsofsociallearningandinformationexchange.

1.3 Archaeologically-visiblechangesinthepatternsofvariabilityinmaterialcultureareoftenattributedtochangesinthescaleorstructureofsocialnetworks(e.g.,seeBanksetal.2009;Conkey1980;Gamble1986;Isaac1972;KoldehoffandLoebel2009;White2006;Wobst1976,1977).Relationshipsbetweennetworkstructureandpatternsofvariabilityinmaterialculturearenotwell-understood,however,asitisimpossibletodirectlyobservehowlarge-scaletemporalandspatialpatternsemergefromthe"rules"ofculturaltransmissionandsocialnetworkformationthatguidebehaviorattheleveloftheindividualorgroup.Bothethnographicstudies(e.g.,WhiteandJohansen2005;YellenandHarpending1972)andstudiesofabstractmodelnetworkedsystemswithaspatialcomponent(e.g.,Axelrod1997;Klemmetal.2003)suggestthattheserelationshipsarelikelytobeneithersimplenorintuitivelyobvious.Makingsoundinferencesaboutthestructureofprehistoricsocialnetworksorthepresenceof

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socialboundariesbasedonmaterialremainswillrequirebuildingatheoreticalbasisforunderstandingtherelationshipsbetweenthesephenomena(seeWhite2012).

1.4 Currentapproachestostudyingculturaltransmissionanditsmaterialcorrelatesgenerallyfallateitherendofa"micro"to"macro"continuumoftemporalandspatialscalesofanalysis/interpretation(Starketal.2008).Culturaltransmissionamonglivingpopulationsis,necessarily,studiedatthesmallscalesoftimeandspacethatarepossibleusingethnographicmethodsofdirectobservation(e.g.,seeMinarandCrown2001;Starketal.2008).Archaeologicalstudies,conversely,aredominatedbytheuseofequation-basedmodelstointerpretlong-termchangesinartifactvariability(e.g.,BentleyandShennan2003;EerkensandLipo2005;HamiltonandBuchanan2009;Neiman1995;butseealsoPremoandScholnick2011).Thesemodelsoftenmakeassumptionsthatindividualsinteractrandomly(i.e.,"panmixia")orthatindividualscanmore-or-lessaccuratelycopythemeanofalargepopulation.

1.5 Thegapbetweenthesetwoscalesofanalysisisanimportantone:themacro-scale(archaeologicallyvisible)outcomesofmicro-scaleculturaltransmissionbehaviorsarepotentiallyaffectedbythestructured(i.e.,non-random),network-mediated,spatially-situatedinteractionsthataretypicalofhumansystems.Numerousstudieshaveshownthatdifferencesinnetworkstructureaffectthebehaviorofbothmodelandreal-worldsystemsmediatedbythesenetworks(e.g.,seeKlemmetal.2003;Newman2000;Watts2004;WattsandStrogatz1998).Relativelysmallalterationstonetworkstructurecanhavealargeeffectonhowinformationistransferredacrossthenetwork:itisoftenthestructureoftheconnectivitythatdeterminesbehaviorratherthanthedetailsoftheparticularsystem.Thissuggeststhatthestructureofinteractionmayhaveimportant,observableeffectsonthepatternsofartifactvariabilityproducedbyculturaltransmissionprocessesmediatedbynetworks.

1.6 Agent-basedmodelingoffersawaytobridgethegapbetweenthemicro-andmacro-scalesofanalysisandanalyzehowmicro-scalebehaviors"mapup"tomacro-scaleoutcomesinenvironmentswhereinteractionisstructuredratherthanrandom.Thispaperusesasimple,"abstract"agent-basedmodel(ABM)toconsiderhowthespatialstructureofinteractionaffectstheoutcomesofabasicculturaltransmissionprocess.Themodelallowsparametersaffectingthecreationofasocialnetworklinkingapopulationofgroupstobeadjustedsothatcause-and-effectrelationshipsbetweennetworkstructure,networkproperties,andpatternsofartifactvariabilitycanbesystematicallyinvestigated.Analysisofresultsfromthemodelshowsthatnetworkstructurehaspatternedeffectsonboththeamountandspatialorganizationofvariabilitythatisproducedbyaculturaltransmissionprocess.Themodelisalsousedtoexplorethe"costs"and"benefits"ofdifferentlystructurednetworks.Experimentationwithabstractmodelssuchasthisoneconstitutesafirststepinbuildingthetheoryrequiredforinterpretingpatternsofarchaeologicalartifactvariabilityintermsofthesocialnetworksthatwereinoperationwhenthosepatternswereproduced.

TechNet_04:Asimplemodelofculturaltransmissioninaspatially-situatednetwork2.1 TheTechNet_04modelisoneofaseriesofabstractmodelsthatwasdevelopedtobeginexaminingthebasicrelationships

betweenthestructureofprehistoricsocialnetworksandpatternsofvariabilityinmaterialculture.Itwasconstructedtobeasimple,flexibleplatformforrunninganarrayofexperimentstoinvestigaterelationshipsbetweennetworkstructure,networkproperties,artifactvariability,andthespatialorganizationofartifactvariability.

2.2 TheTechNet_04modelwasbuiltusingRepastJ.Thecodeforthemodelisavailableonlineathttp://www.openabm.org/model/3257/version/1/view.Table1suppliesadescriptionofthemainparametersandvariablesofthemodel.

Table1:MainparametersandvariablesoftheTechNet_04model.

Level Parameter/Variable DescriptionGroup id Uniqueidentifierforeachgroup

X,Y Group'sXYcoordinates;usedbythemodeltoidentifylocationswithinthegrid

A,B Group'sABcoordinates;usedtocalculategeographicdistancesbetweencells

networkList ListofallgroupstowhichaparticulargroupislinkedvariableA Group'scurrentvalueoftheculturaltraittobetransferred

Link fromGroup,toGroup GroupsconnectedbythelinklinkDistance Geographicdistancespannedbythelink

System meanNetworkSize Meansizeofallgroup-levelnetworksinthesystemnumLinks Totalnumberoflinksinthesystem-levelnetworklinkDistMean MeangeographicdistancespannedbylinksinthesystemlongestLink LongestgeographicdistancespannedbyalinkmeanPathLength Meanpathlengthofsystem-levelnetwork

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(MPL)clusteringCoefficient(CC)

Clusteringcoefficientofthesystem-levelnetwork

meanSD MeanstandarddeviationofvariableA,averagedover500timestepsModel p Parametersettingprobabilityofreplacementoflocallinkswithrandom

links(inrewireMode1)oradditionofnon-locallinks(inrewireMode2)rewireMode Parameterdeterminingprocessforrewiringthelocal-onlynetworkthatis

establishedatthestartofamodelrunrewireRadius Radius(intiersofhexagonalcells)withinwhichlinkstonon-localgroups

canbeaddedwhenrewireMode=2copyMode Parametercontrollingthepopulationagroupwillcopy:

1=groupcopiesmeanvariableAofgroupsonitsnetworkList;2=groupcopiesmeanvariableAofentirepopulation;3=groupcopiesitsownvariableA;4=groupcopiesmedianvariableAofentirepopulation;5=groupcopiesdeterministicmeanthatchangesaccordingtoformula

pConform Parametersettingprobabilitythatagroupwillcopysomeotherpopulationofgroups(determinedbycopyMode)

copyError Percentage+/-errorappliedwhencopying

Spatialenvironmentandpopulation

2.3 The"world"ofthemodelisatwo-dimensional,bounded,hexagonalgrid.Eachcellisthelocationofasingle"group".Ahexagonalgridwaschosenoverasquaregridbecauseitreflectshowspaceisusedinidealizedmodelsof"packed"hunter-gatherersystems(Wobst1974,1976)andequalizesthespatialdistancesoftheneighborsofanyparticularcellatagivennumberoftiers.Aboundedgridwasusedinsteadofatorusbecausethemodelisintendedtorepresentasocialworldthathasspatiallimits.

2.4 Apopulationofgroupsiscreatedandplacedintheworldduringtheset-upphaseofthemodel.Becauseeachcellcontainsasinglegroup,thenumberofcellsandthepopulationoftheworldareidentical:a40x40worldcontains1600groups.Eachofthesegroupshasan"address"consistingofXandYcoordinatesthatspecifyitslocationinthegrid.EachgroupalsohasasetofAandBhexagonalgridcoordinatesthatareusedtodeterminespatialdistancesbetweengroupsintermsofthenumberofcells(Figures1.Aand1.B).

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Figure1.Conceptualillustrationofcoordinatesystems,stagesofnetworkformation,andcalculationoflinkdistanceandpathlengthinthemodel.

Networkstructureandproperties

2.5 Thismodelisbuiltaroundtheconceptthateachgroupcaninteractwithaunique,finitearrayofothergroupstowhichitislinked.Thisarrayconstitutesagroup'sindividualnetwork.Whentwogroupsarelinked,informationcanbetransferredbetweenthem.Theoverlappingnetworksofindividualgroupscompriseasystem-levelnetworkthatconnectsallgroupswithinthesystem.

2.6 Networksareformedatthebeginningofamodelrunanddonotchangeduringamodelrun.Networksareformedthroughatwo-stageprocess.Intheinitialstage,eachgroupaddsthegroupsintheimmediatelyadjacenttierofhexagonalcellstoitsnetworkList(Figure1.C).Thisproducesa"localonly"networkwhereeachgroupislinkedtoonlythosegroupswhoareimmediatelyadjacenttoitinspace.

2.7 Asecondstageofnetworkformation"rewires"thislocal-onlynetworkinoneoftwoways,controlledbytheparameterrewireMode.IfthevalueofrewireModeis1,agroup'slocallinkstoitsneighborscanbereplacedbylinkstorandomgroups(Figure1.D).Theprobabilityofeachlocallinkbeingreplacedbyarandomconnectioniscontrolledbythevariablep,whichisaparameterthatcanbevariedcontinuouslybetween0and1.EachgroupgoesthroughitsnetworkListandputseachmemberofitsnetworkinjeopardyofbeingreplacedbygeneratingauniformlydistributedrandomnumberbetween0and1andcomparingittothevalueofpsetinthemodel.Ifthegeneratednumberislowerthanp,thelinkwiththatgroupisdissolvedandanewlinkisformedwitharandomlyselectedgroup.Thuswhenp=0(nochanceofreplacement)thenetworkremainslocal-only,whileapof1(certaintyofreplacement)producesanetworkwhereeachlinkisrandomlydetermined.Ifagrouptriestocreatealinktoagroupwithwhichitisalreadylinked,thatchancetoaddalinkislost(i.e.,a"duplicate"linkcannotbecreated).Becauseeachlinkbetweentwogroupsistwo-way,eachconnectioncanbeinjeopardyofbeingreplacedtwice.Thelinkbetween"GroupA"and"GroupB",forexample,isinjeopardyofbeingreplacedoncewhen"GroupA"goesthroughtherewiringmethodand(assumingthelinkwasnotreplaced)oncewhen"GroupB"goesthroughtherewiringmethod.Thisprocedureofnetwork"rewiring"to

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interpolatebetweenaregularandarandomnetworkissimilartothatemployedinstudiesexploringthesmall-worldproperty(e.g.,Klemmetal.2003;WattsandStrogatz1998).Themaindifferenceisthat,inthosepapers,eachlinkbetweenverticeswasonlysubjecttopotentialrewiringonce.

2.8 Theboundednatureofthegridandthespatialcomponentofnetworkformationmeanthatgroupslocatedalongtheedgesofthegridhavefewerthansixothergroupsintheirnetworks.Ina40x40grid,themeansizeofagroup'snetwork(meanNetworkSize)is5.8andthereare4641directtwo-waylinksbetweengroups.Therewiringofthenetworkcontrolledbypdoesnotaffectthetotalnumberofdirectlinksinthenetwork:foreachlinkthatisdissolvedanewoneiscreated.ThusmeanNetworkSizeremainsconstantinrewireMode1.

2.9 RewiringinrewireMode1doesaffectthedistributionoflinksamongthegroupsinthenetwork,however.Whenalocalgroupisreplacedwitharandomgroup,thelocalgrouplosesalinkwhiletherandomgroupgainsone.Becauseofthis,itispossibleforindividualgroupstobecomedisconnectedfromtherestofthepopulation.Thisoccurredwithsomefrequency.Inagroupof3000runs,forexample,oneormoregroupsbecamedisconnectedfromtheremainderofthepopulationabout15percentofthetime.Thedistributionofthenumberofdisconnectedgroupswasright-tailed,withthemajorityofcases(abouttwothirds)havingonlyasinglegroup(i.e.,lessthan0.06percentofthetotalpopulationof1600)disconnected.Themaximumnumberofdisconnectedgroupsinasinglerunwas6(i.e.,about0.4percentofthepopulation).Highernumbersofdisconnectedgroupstendedtobeproducedathighervaluesofp.Groupsthatbecomeunlinkedduringrewiringareremovedfromtheworldpriortothestartofamodelrun,resultinginsmallvariationsinpopulationsize.Whileitistheoreticallypossibleforlargerportionsofthenetworktobecomeunlinkedfromeachother,thisdidhappeninpracticeduringuseofthemodel.

2.10 IfthevalueofrewireModeis2,eachgrouphasthechancetocreatelinkswithnon-localgroupsthatarewithinaspecifieddistanceofthegroup'slocation.Inthismode,agrouppreserveslinkswithallofitslocalneighborsandaddslinkstoother,non-localgroups.Theprobabilitythatagroupwillcreatealinkwithanon-localgroupis,again,controlledbytheparameterp.Thenumberofchanceseachgrouphastoaddnon-locallinkswasheldconstantat6(themeansizeofalocal-onlygroup-levelnetwork,roundedupfrom5.8).Inotherwords,whenp=1,eachgroupwilladdlinksto6nonlocalgroups.ThegeographicdistancebetweenlinkedgroupsisconstrainedbytheparameterrewireRadius,whichspecifiesthemaximumseparationoftwolinkedgroupsintermsofthenumberofhexagonalcells.IfthevalueofrewireRadiusis8,forexample,agroupwillformalinkwitharandomlyselectedgroupthatisbetween2and8tiersdistant(itisalreadylinkedtoallthegroupswithin1tier).Becauselinksareaddedratherthanreplaced,themeannetworksizewillvarybutwillalwaysbegreaterthan5.8whenp>0.00.Ifagrouptriestocreatealinktoagroupwithwhichitisalreadylinked,thatchancetoaddalinkislost(asinrewireMode1).

2.11 Afterthenetworkisformed,themodelcomputesthemeanpathlength(meanPathLength,orMPL)ofthenetwork.MPLdescribestheaveragesocialdistancefromonegrouptoanothergroupintermsoftheshortestnumberofsocial"steps"betweenthegroups(Figure1.E).Meanpathlengthisastandardmeasureoftheoverall"closeness"orinter-connectednessoftheentirenetwork(LovejoyandLoch2003).Whenagroupisdirectlylinkedtoanothergroup,thelengthofthepathbetweenthemis1.Whenasingleintermediaryisrequiredtoreachonegroupfromanothergroup,thepathlengthbetweenthemis2.Themodelcomputesthesumofthepathlengthsbetweeneachuniquepairofgroupsandthendividesthetotalpathlengthbythenumberofpaths.Thereare1,279,200group-grouppathsinagridof1600groups.ThemeanPathLengthofa40x40gridwithalocal-onlynetworkis21.5.

2.12 ThevariablelinkDistancedescribesthegeographicaldistancetraversedbyalinkintermsofthenumberofspatialsteps(Figure1.E).ThevariablelongestLinkdescribesthegeographicdistancetraversedbythelongestgroup-grouplinkthatexistsinthegrid.ThevalueoflongestLinkinalocal-onlygridwillalwaysbe1.Whennon-localconnectionsarepossible,thedistanceofthelongestpossiblelinkwillbelimitedbythesizeoftheworldinrewireMode1andbythevalueofrewireRadiusinrewireMode2.ThevariablelinkDistMeanisthemeandistanceofallgroup-to-grouplinks.

2.13 Themodelalsocomputestheclusteringcoefficient(clusteringCoefficient,orCC),whichmeasuresthemeaninter-connectednessoflocalneighborhoods(seeWattsandStrogatz1998).Theclusteringcoefficientofeachlocalneighborhood(i.e.,theareaoccupiedbyagroupandtheimmediatelyadjacentgroups)isdefinedastheratioofthenumberoflinksthatexistamongagroup'slocalneighborstothenumberoflinksthatarepossibleamongthoseneighbors(Figure2).Whenagrouphassixneighbors,atotalof15linksarepossibleamongthem.If11ofthoselinksexist,theclusteringcoefficientinthatneighborhoodis0.73.TheclusteringCoefficientofthesystem-levelnetworkisthemeanoftheclusteringcoefficientofallneighborhoods.TheCCofa40x40gridwithalocal-onlynetworkis0.4122.

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Figure2.Calculationofclusteringcoefficient.

Informationtransfer

2.14 Thetransmissionofculturalinformationinthismodelisrepresentedbythetransferofthevalueofasinglerealnumber(variableA).VariableAismeanttorepresentsomecontinuouslyvariable,"stylistic"aspectofartifactsize,shape,etc.,thatissubjecttocopyingerrorandisfreetovarythroughprocessessuchasdrift(i.e.,changethroughaccumulationofrandomcopyingerror).Becausethepurposeofthismodelistounderstandonlytherelationshipsbetweenthestructureandpropertiesofsocialnetworksandtheamountandspatialorganizationofthevariabilitythatisproducedthroughsimplecopyingerror,noattemptismadetorepresentsourcesofvariabilityotherthancopyerror.

2.15 Everygroupintheworldbeginsamodelrunwiththesamevalue(5)ofvariableA.Thiswasanarbitraryvalueselectedsotheresultsofsomeexperimentswouldbedirectlycomparabletothoseofpreviouswork(e.g.,HamiltonandBuchanan2009).BecausevariabilitygeneratedduringarunisproportionaltothevalueofvariableA,beginningwithadifferentvaluewouldaffecttheresultsintermsofabsolutevaluesbutnottheoverallpatterns.

2.16 Thetransferofinformationoccursthroughcopyingevents.Ateachstep,eachgroupcopiesthevalueofvariableAeitherfromitselforfromsomepopulationofgroupsotherthanitself(seeTable1).Twoparameterscontrolthesechoices:pConformspecifiestheprobabilitythatagroupwillcopysome"pool"ofgroupsotherthanitself,whilecopyModecontrolswhichpoolthatis.Inotherwords,thevariablepConformcontrolstherelativestrengthof"horizontal"vs."vertical"transmission,whilecopyModedeterminesthepopulationthatiscopiedifagroupcopiesanoutsidepopulation.

2.17 ThevariablepConformcanbesetbetween0and1.Thisvariableisconceptuallythesameasthevariablelambda(or"strengthofbias")usedbyHamiltonandBuchanan(2009)and"strengthofconformance"usedbyEerkensandLipo(2005).WhenpConformequals0.30,anagentwillcopythevalueofvariableAfromapopulationofgroupsotherthanitselfwithaprobabilityof0.30.Ifthegroupdoesnotcopyfromanoutsidepopulation(i.e.,doesnotundertake"horizontaltransmission"),itwillcopyitselfbydefault("verticaltransmission").

2.18 IntheeventthatagroupcopiesthevalueofvariableAfromanoutsidepopulation,themodeofcopying(copyMode)determinesthecompositionofthatpopulation:themeanvariableAofthegroupsinitsnetworkorthemean(ormedian)variableAoftheentirepopulationintheworld.Thelattermodeofcopyingdoesnotincorporateanyaspectsofnetwork-structuredinteraction(asthegroupissimplycopyingsomeglobalmeasureofthecentraltendencyofthepopulation),butallowsthemodeltobeusedtoreproducetheresultsofequation-basedculturaltransmissionmodelsthatdonotrepresentinteractionasastructuredphenomenon(e.g.,EerkensandLipo2005;HamiltonandBuchanan2009).

2.19 VariabilityinvariableAisgeneratedthroughcopyingerror(controlledbytheparametercopyError)thatisappliedeachtimeagroupcopies(whethercopyingfromanoutsidepopulationorcopyingitself).Theapplicationoferrorduringeachcopyingeventisbasedontheideathatthereareinherentconstraintsinhumanperceptionthatpreventthedetectionofslightdifferencesbetweenanytwoobjects,shapes,colors,etc.Theamountoferrorisrelativeandistypicallysetatamaximumof+/-3to5percentbasedonempiricalstudiesofhumanperception(e.g.,Eerkens2000;EerkensandLipo2005;HamiltonandBuchanan2009).Thismodelallowscopyingerrortobeeitheruniformlyornormallydistributed(Figure3).Whencopyingerrorof+/-5percentisnormallydistributed,themeanerroris0andthe5percenterroristwostandarddeviationsfromthemean(errorsgreaterthan5percentcanoccurwhentheerrorisnormallydistributed).Auniformdistributionoferrorplacesahardlimitontheamountoferrorthatcanoccur.Theexperimentsperformedhereusedanormaldistributionoferrorsimplytofollowthestandardpracticeinequation-basedstudiesofculturaltransmission.

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Figure3.Histogramsshowingnormalanduniformdistributionsof+/-5percenterror.

Modeloperation

2.20 Followingcreationoftheworldandthegroups,two-stageformationofgroup-levelnetworks,andsettingofparameterscontrollinginformationtransfer,themodelissetintomotion.Ateachtimestep,eachgroupgoesthroughasequenceofactionsduringitsturn.ItfirstdetermineswhetheritwillcopyvariableAfromitselforfromsomeoutsidepopulation(basedonpConform).Ifitcopiesfromanoutsidepopulation,itdeterminesthepopulationtocopy(basedoncopyMode),calculatesthevalueofvariableAtocopy,appliescopyingerror,andcopies.Ifitcopiesitself,itappliescopyingerrortoitsownvalueofvariableA.Theorderingofgroupsisrandomlyshuffledeachtimestep.Mostoftherunsdiscussedherelastedfor1000timesteps.

2.21 Thedataoutputofthemodelcanbeadjustedtoincludedifferentmeasuresandmoreorlessdetaileddatadependingonwhatisrequiredforanalysis.Outputscanrangefromsummarydataproducedattheendofaruntodataabouteachgroupateachstep.

Experimentsandresults

3.1 Experimentsperformedwiththismodelwereprimarilyintendedtoinvestigatetherelationshipsbetweenthe"rules"ofgroup-levelnetworkformation,theresultingstructureandpropertiesofsystem-levelnetworks,andtheamountandspatialpatterningofvariabilityproducedbyasimpleculturaltransmissionprocess.Theywerealsointendedtoinvestigatewhetherspatially-structuredinteractionproducesresultsthataredifferentfromthoseofequation-basedmodelsofculturaltransmissionthatincorporateneitherspacenorstructuredinteraction.Thesectionsbelowsummarizetheexperimentsandanalysesthatwereusedtoinvestigatetheseaspectsofthemodel.Manyoftheexperimentalresultspresentedherewereproducedtoshowtherelationshipsamongmultiplevariables.Becausetheserelationshipswereoftennonlinear,somegroupsofexperimentalrunswereperformedtoproduceadditionaldatapertainingtospecificconditions(i.e.,runstoproduceadditionaldatawherepislow).

Implementationofequation-basedACEandBACEmodels

3.2 Equation-basedmodelsarethedominantmodelsofculturaltransmissionusedtointerpretarchaeologicaldata.Thesemodelsaretypicallycastas"accumulatedcopyingerror"(ACE)or"biasedaccumulatedcopyingerror"(BACE)models(e.g.,EerkensandLipo2005;HamiltonandBuchanan2009).IntheACEmodel,transmissionislimitedtoverticaleventsthatrepresentagenerationallearningphenomenon,suchasbetweenaparentandachildoramasterandanapprentice.Individual"lines"oftransmissionareindependent.Theoverallresultofthisindependenceisthattheamountofvariabilityamongagivensetoflinesincreasesthroughtimewithoutconstraint.IntheBACEmodel,horizontalorobliquecopyingeventsalsooccurwhereinformationistransferredbetweenlines,constrainingtheamountofvariabilitythatispossible.

3.3 TheparametersofACEandBACEmodelscanbeimplementedwithintheABMemployedhere.Thegoaloftheseexperimentswasnottoreproduceallthemathematicalnuanceoftheseequation-basedmodels,buttoimplementthemconceptuallyusingacomputationalplatformsothatresultscouldbecomparedtothoseofthespatially-structuredinteractionmodelsdiscussedbelow.Replicatingtheresultsoftheseequation-basedmodelsservesvalidatetheagent-basedmodelutilizedhereandprovideabaselineforfurthermodelingefforts.

3.4 InACEmodel,eachindividual(or"line")isrepresentedbyanequationwhichisrunthroughaseriesofdiscretetimesteps.TheunbiasedACEmodelcanberecreatedsimplybyhavingeachgroupinthemodelbeginwiththesameinitialvalue(5)forvariableAandcopyitself(with+/-5percenterror,normallydistributed)eachtimestep.Inthiscase,eachgroupcreatesitsownACE"path"whichcanbeplottedvs.time(Figure4.A).Whenalargenumberofruns(n=1000)isconsidered,thedistributionofoutcomesafter100timestepsislognormal:themediandecreasesbutthemeanstaysapproximatelyconstant(Figures4.B,4.C,and5).Varianceincreaseslinearlywithtime(Figure4.D).Thesearetheexpectedoutcomesofadriftprocessbasedonaccumulated,proportionalerroractingindependentlywithineachline(e.g.,EerkensandLipo2005:322).

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Figure4.ResultsfromimplementationoftheACEmodel(+/-5%error,normallydistributed):(A)resultsfrom10individualruns;(B)meanandmedianof1000runs;(C)distributionsofvariableAatseveralpointsintime;(D)varianceof1000runs.Compare

toHamiltonandBuchanan(2009:Figure1).

Figure5.HistogramofvariableAatstep100intheACEmodel(1000runs).

3.5 TheBACEmodelinvolvesthepossibilityofinformationtransferbetweenlines.AsintheACEmodel,anindividualintheBACEmodelisrepresentedbyanequationwhichisrunthroughaseriesofdiscretetimesteps.TheconceptoftheBACEmodelcanbe

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implementedintheagent-basedmodelbyhavingeachgroupbeginwiththesameinitialvalue(5)forvariableAandthencopyitselfeachtimestepunlessitchoosestocopythemeanofthepopulation(n=100)withprobabilityof0.30(0.30waschosentomimicthedegreeofbiasusedbyHamiltonandBuchanan2009:Figure2).AsampleoftenindividualpathswithinoneexperimentusingcopyMode2isshowninFigure6.A.NotethatthemeanofthisbatchofindividualsdoesnotappeartodecreasethroughtimeasintheresultsofHamiltonandBuchanan(2009:Figure2A).Figures6.B,6.C,and6.Dshowthemean,median,andvariancethroughtimeforaseriesofruns(eachpointistheaverageof20runseachcontaining100agents).Themeanandmedianstayapproximatelyconstantwhengroupscopytheactualpopulationmean(copyMode=2).Themeanandmedianbothdecreasegraduallywhenagentscopythepopulationmedian(copyMode=4).Itisonlywhencopyingadeterministicequation(decreasingbyonehalfthevarianceofthecopyerroreverytimestep,copyMode=5)thatthemeansteadilydecreasesataratecomparabletothatshownbyHamiltonandBuchanan(2009:Figure2).Thepatternofchangeinthevariance(Figure6.D)issimilartothatdescribedbyHamiltonandBuchanan(2009:Figure2).

Figure6.ResultsfromimplementationoftheBACEmodel(pConform=0.30,+/-5%errornormallydistributed):(A)superimposedresultsof10individualgroupswithinonesamplerun;(B)meanofvariableAusingthreedifferentmodesof

copying;(C)medianofvariableAusingthreedifferentmodesofcopying;(D)varianceofvariableAusingthreedifferentmodesofcopying(copyMode2=copypopulationmean;copyMode4=copypopulationmedian;copyMode5=copyformula).

3.6 ThestandarddeviationofvariableA(n=1600agents)wascalculatedateachtimestepandplottedagainsttimeforsixrunsvaryingpConform(Figure7).TherelationshipsbetweenpConformandassemblagevariabilityintheserunsisverysimilartothosedepictedbyEerkensandLipo(2005:Figure2)andHamiltonandBuchanan(2009:Figure3).IntheBACEmodel,the"strengthofbias"constrainsvarianceandtheeffectisanonlinearone.

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Figure7.EffectofvaryingpConformintheBACEmodel(onerunforeachsetting,1600groupsperrun).TheACEmodelconditionsarereproducedwhenpConform=0.

3.7 Insummary,acomputationalimplementationoftheACEandBACEmodelsproducedsimilarpatternsofchangeinvariabilitytoequation-basedimplementations(e.g.,EerkensandLipo2005;HamiltonandBuchanan2009).Morecopyingbias(morehorizontalinformationflow)constrainsvariabilityandtheeffectisnonlinear.Anegativedriftwaspresentinthemedian(butnotthemean)oftheACEmodelwhenthefirst100timestepsof1000runswereconsidered.AnegativedriftinthemeanwasproducedintheBACEmodelonlywhentheactualpopulationmeanwasreplacedwithadeterministicequation(cf.Kempeetal.2012).Thetake-awaypointoftheseequation-basedmodelsisthatanincreaseincopyingbias(horizontaltransmission)resultsinaloweroverallvariability.

Networkstructureandproperties

3.8 Aseriesofexperimentswasusedtoinvestigatehowthecreationofnon-locallinksaffectsthestructureandpropertiesofthenetworksinthemodel.Resultsfromthetwodifferentmodesof"rewiring"(seeabove)willbediscussedseparately.

3.9 WhenthemodelisinrewireMode1,thevalueoftheparameterpistheprobabilitythateachlinktoalocalgroupwillbereplacedwithalinktoarandomgroup.Whenp=0,alllinksinthenetworkarebetweenspatially-adjacentgroups.Whenp=1,alllinksarerandomlydetermined.Varyingpbetween0and1interpolatesbetweenaregular,locallatticeandarandomnetwork.

3.10 Thechangesinnetworkpropertiesthatresultfromchangesinpwereevaluatedbycomparingptotheresultingmeanpathlength(meanPathLength,orMPL)ofthenetworkinaseriesofruns(n=3000)withina40x40grid(Figure8.A).TherelationshipbetweenpandMPLisclearlynonlinear:replacementofrelativelyfewlocalconnectionswithrandomconnectionsresultsinareductioninMPLthatisdisproportionatetothenumberoflinksreplaced.Randomlinksare"shortcuts"thatcanhavealargeeffectoninformationflowwhentheyspanlongdistances.TherelationshipsbetweenMPL,p,andthegeographicdistancespannedbythelongestlink(longestLink)areshowninFigures8.Band8.C.Thelikelihoodofaddingatleastonerelativelylongconnectiontothegridgrowsrapidlyaspincreasesfrom0.TherelativelylooserelationshipbetweenMPLandlongestLinkwhenpislowreflectstheinfluenceoftheorientationandarticulationofeachrandomlinkwhenthenumberofrandomlinksissmall.TherelationshipbetweenpandlinkDistMeanisshowninFigure8.D.

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Figure8.Relationshipsbetweenvariablesrelatedtonetworkformationrules,networkstructure,andnetworkproperties(rewireMode=1).

3.11 Figure9showsMPLandclusteringCoefficient(CC)plottedagainstponalogarithmicaxis.ThevaluesofMPLandCChavebeennormalizedbydividingeachdatapointbythevaluesofMPLandCCwhenp=0(21.5and0.4112respectively).Therelationshipbetweenp,MPL,andCCcompareswelltothedatapresentedbyWattsandStrogatz(1998:Figure2)intheirdescriptionofthe"small-world"phenomenon.TherapiddropinMPLassociatedwithlowvaluesofpproducesalmostnochangeinCC:thereplacementofafewlocallinkswithrandomlinkshasasignificanteffectontheglobalpropertiesofthenetworkbutanegligibleeffectonhowthenetworkappearsfromthelocallevel.

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Figure9.PlotofpversusmeanPathLength(MPL)andclusteringCoefficient(CC)withthehorizontalaxisdisplayedonalogarithmicscale,rewireMode=1.Datawerenormalizedbydividingbythevalueswhenp=0(21.5forMPL,0.4122forCC)in

ordertoplotbothvariablesonthesameaxis.

3.12 InrewireMode1,locallinksarereplacedwithrandomlinkswithprobabilityp.Theselinkreplacementsproducebotha"benefit"(areductioninMPL)anda"cost"(areductioninCC)intermsofnetworkproperties.Asimpleratioof"benefit:cost"canbecalculatedforeachrunbymultiplyingtheproportionalreductioninMPLbytheproportionalreductioninCC(Figure10.A).Ifweassumethatlongerlinksaremore"costly"toform/maintain,wecanalterthiscalculationtoincludelinkDistMean(Figure10.B).Ineithercase,itisclearthatnetworkscreatedatvaluesofpbetween0.001and0.10offerthebesttrade-offbetweenbenefitandcost.

Figure10.Relationshipsbetweenpandmeasuresof"benefit:cost"inrewireMode1.Valuesin(A)werecalculatedas:((21.5-MPL)/17.08)*(CC/0.4122):21.5isthemaximumvalueofMPL,17.08isthemaximumreductioninMPL,and0.4122isthemaximumvalueofCC.Valuesin(B)werecalculatedbydividingthevaluescalculatedin(A)by(linkDistMean-18.24):18.24

ismaximumvalueoflinkDistMean.

3.13 WhenthemodelisinrewireMode2,pistheprobabilitythatagroupwilladdanewlinktoanonlocalgroup(seeabove).ThegeographicdistanceofgroupstowhichnewlinkscanbeaddedisconstrainedbythevalueofrewireRadius,whichspecifiesthenumberofhexagonaltiersfromthegroupthatareincludedinthesearchforanewgroupwithwhichtocreatealink.Becauselinksareaddedratherthanreplacedinthisrewiringmode,thevalueofphasadirect,positiveeffectonboththetotalnumberoflinksandthemeansizeofgroup-levelnetworks.

3.14 AseriesofexperimentalrunswasusedtogeneratedataforseveralvaluesofpandvaluesofrewireRadiusvaryingfrom2to57

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(i.e.,uptothemaximumgeographicseparationofanytwogroups,thehypotenuseofa40x40).Figure11showstherelationshipbetweenMPLandrewireRadiusforfivevaluesofp(n=500runseach).ThecombinationofthenumberofnewlinksaddedandthegeographicdistancethatthoselinksarepermittedtospanreducesMPLinapredictablefashion.Whenp=0.001,between2and18links(meanof9.6)wereaddedtotheexistinglocal-onlynetworkcomprisedof4641two-waylinks.TheadditionoftheselinkshasanegligibleeffectonMPL,especiallywhenthevalueofrewireRadiusislow.HighervaluesofpcauseamorerapiddropinMPLevenwhentherewireRadiusisconstrainedtojust2,3,or4tiers.IncreasingthevalueoftherewireRadiusover10haslittleeffectonMPLwhenp>0.10.TherelationshipsbetweenpandMPLareshownforseveralvaluesofrewireRadiusinFigure12.

Figure11.RelationshipsbetweenrewireRadiusandmeanPathLengthforfivevaluesofpinrewireMode2.

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Figure12.RelationshipsbetweenpandMPLforseveralvaluesofrewireRadius,rewireMode2.3.15 InrewireMode2,reductionsinMPLaretheresultofacombinationofthenumberoflinksaddedandthegeographicdistancethat

thoselinksspan.TheadditionofnewlinkshasthepotentialtoincreaseCC,asgroupswithinlocalneighborhoodsmaybecomemoreinter-connectedaslinksareadded(CCcanonlyincreaseorstaythesamebecauseexistinglocallinksarenotremoved).ReductionsinMPLandincreasesinCCcanbothbeviewedas"benefits"arisingfromthealterationofalocal-onlynetworkstructurebytheadditionofnewlinks."Costs"associatedwithaddingnewlinksincludethoseassociatedwithforming/maintaininglargergroup-levelnetworksandthoseassociatedwithforming/maintaininglinksspanninggreatergeographicdistances.

3.16 AratioofbenefittocostcanbecalculatedbyincorporatingreductioninMPLandincreaseinCCas"benefits"andincreasesinmeanNetSizeandlinkDistMeanas"costs"(Figure13).Whilehighvaluesofp(i.e.,p=0.5andp=1.0)arerelativelycheapstrategieswhenrewireRadiusis2,theyreturnprogressivelylessbenefitperunitofcostas rewireRadiusincreasespast2.Verylowvaluesofp(i.e.,p=0.001andp=0.01)deliverlessbenefitperunitofcostatlowvaluesofrewireRadiusbutgetprogressivelybetterasrewireRadiusincreases.Avalueofp=0.10iscomparabletohighvaluesofpwhenrewireRadiusis2,butcontinuestoimproveuntilthevalueofrewireRadiussurpasses5.Thelowbenefit:costratiosofhighvaluesofpstemfromthelargeincreasesinthesizesofthegroup-levelnetworksthatarecreated.WhentherewireRadiusisgreaterthan2,theselargegroup-levelnetworksaddcostwithoutaddingmuchbenefitintermsofeitherareductioninMPLoranincreaseinCC.Thesparsewebofnon-locallinkscreatedatlowvaluesofp,conversely,isrelativelycheapbutproducesrelativelylittlebenefituntiltherewireRadiusislargeenoughtopermittheestablishmentofrelativelylonglinksthathaveasignificantimpactonMPL.Ofthevaluesofpthatareinvestigated,p=0.10producesthebestratioofbenefit:costwhentherewireRadiusislessthan8.Thedensityofnon-locallinksishighenoughtohaveasignificantimpactonMPLbutnotsohighthatadditionallinksdonotproduceabenefit.

Figure13.Relationshipbetweenmeasureof"benefit:cost"andrewireRadiusforfivevaluesofpinrewireMode2.Valueswerecalculatedas(((21.5-MPL)/18.64)*(CC/0.8103)))/((linkDistMean/15.08)*(meanNetSize/17.78)):21.5isthemaximumMPL,18.64isthemaximumreductioninMPL,0.8103isthemaximumCC,15.08isthemaximumlinkDistMean,and17.78isthe

maximummeanNetSize.

3.17 WhenthevalueofrewireRadiusissettothemaximum(i.e.,57),theadditionoflinksisarandomprocesscontrolledbyp.ThusmanyoftherelationshipsbetweenvariablesseeninrewireMode1canbereproducedinrewireMode2whennewlinksarecreatedwithnospatialconstraint.

Amountofvariabilityproduced

3.18 Asshownabove,thestructureofanetwork(howgroupsareinter-connected)affectsitsproperties(the"ease"ofinformationflow).AlowerMPLindicatesthat,onaverage,fewerstepsarerequiredtogetfromonegrouptoanothergroup.InformationtravelingthroughanetworkwithalowerMPLtakesfewerstepstotraveltoreachallthegroupsinthenetwork.AsMPLdecreases,informationflowshouldconstrainthevariabilitythatispossible(assumingthatthepropensityforthehorizontaltransferofinformationremainsconstant),resultinginapositiveassociationbetweenMPLandthestandarddeviationofvariableA.Becausestandarddeviationfluctuatesateachtimestepinanygivenrun,themodelwasfirstrunforsteps0-499andthenthestandarddeviationofvariableAwascalculatedduringeachofsteps500-999andaveragedoveraperiodof500steps.Thiswasdonetocharacterizethemeanbehaviorofeachrun.

3.19 Figure14showsthestandarddeviationofvariableAplottedagainstp,MPL,longestLink,andlinkDistMeanfor3000runsinrewireMode1.ApositiveassociationbetweenMPLandthestandarddeviationofvariableAisshowninFigure14.A.Inallofthesecomparisons,networkswheretheflowofinformationiseasier(i.e.,networkswithlongerlinksandlowermeanpathlengths)producelowerstandarddeviationsofvariableA.TheoutcomeswiththemostvariabilityinvariableAareproducedbynetworkswithhighmeanpathlengthsandfewlonglinks.

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Figure14.StandarddeviationofvariableAplottedagainstmeanPathLength,p,linkDistMean,andlongestLinkfor3000runsinrewireMode1.Standarddeviationisaveragedoverthelast500timestepsofeachrun.

3.20 Figure15showsthestandarddeviationofvariableAplottedagainstMPLandlinkDistMeanfor4000runsinrewireMode2.TherelationshipsareverysimilartothoseproducedinrewireMode1:culturaltransmissionacrossnetworkswithlongerlinksandashortermeanpathlengthresultsinlessvariability.Figure16plotsthestandarddeviationofvariableAvs.pforfourvaluesofrewireRadius(2,4,8,and16).AsrewireRadiusincreases,thenonlinearrelationshipbetweenpandthestandarddeviationofvariableAbecomesmoreapparent:smallincreasesinpfrom0havealargereffectastheradiusavailableforformingnon-locallinksincreases.

Figure15.StandarddeviationofvariableAplottedagainstmeanPathLengthandlinkDistMeanfor4000runsinrewireMode2.Standarddeviationisaveragedoverthelast500timestepsofeachrun.ThevalueofrewireRadiusvariedrandomlybetween2

and57.

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Figure16.StandarddeviationofvariableAplottedagainstpforfourvaluesofrewireRadius(2,4,8,and16),rewireMode2.

3.21 Theseinter-relationshipsbetweennetworkstructure,networkproperties,andthestandarddeviationofvariableAdemonstratethatstructuredinteractionaffectstheamountofvariabilitygeneratedbyasimpleculturaltransmissionprocessinapredictableway:networkswhereinformationflowiseasier(i.e.,wheremeanpathlengthislow)producelessvariability.Thisistrueinbothmodesofnetworkformation(compareFigure14.AandFigure15.A).Thissuggeststhatthepropertiesofnetworks,ratherthandetailsofthestructurethatproducesthoseproperties,aretheprimedeterminantsofhowmuchvariabilityisproduced.

Spatialpatterningofvariability

3.22 Severalgroupsofexperimentswereruntoevaluatehowassemblagevariabilityisstructuredwithregardtospaceunderdifferentconditionsofpinbothmodesofnetworkformation.TheMoran'sIstatisticwaschosentodescribethedegreeofspatialautocorrelationamongthevaluesofvariableAatthelasttimestepinaseriesofexperimentalruns.Spatialautocorrelationisthedegreetowhichthevalueofavariableatapointinspaceisrelatedtothevaluesofthesamevariableinadjacentpointsinspace(seeCliffandOrd1973;Moran1950).Moran'sIisanindex:valuesrangefrom+1(perfectcorrelation)to-1(perfectdispersion).Avalueof0indicatesarandomarrangementinspace(i.e.,spatialproximityhasnoeffectonthevalueofavariable).Valuesbetween0and1indicatesomedegreeofspatialclustering,wheresimilarvaluesarelocatedclosertoeachotherinspacethanwouldbeexpectedrandomly.

3.23 Moran'sIwascalculatedonsetsofmodeldatausingtheSpatialStatisticstoolsinArcMap9.2.ModeloutputswereadjustedtoproducealistcontainingthecoordinatesandvalueofvariableAforeachgroupatthelasttimestep.ThislistwasimportedintoGISandconvertedtoanXYshapefile.Moran'sIwascalculatedonthese40x40gridsofdatausingthedefaultsettings.TheXYcoordinatesystemusedforthehexagonalgridemployedinthemodelintroducesaslightdistortiontothespatialrelationshipsbetweenagents(theYcoordinateofeveryothercolumnofagentsisoffsetby0.5units)whenthedataarereadinaCartesiancoordinatesysteminGIS.BecausethisintroducesonlyanegligibledifferenceinMoran'sIforeachrun,however,ithasnosignificanteffectontheanalyticalresultsandthereforenoeffortwasmadetocorrectthisdistortion.

3.24 ThefirstoftheseexperimentswasdesignedtoproducedatatocompareMoran'sItothevalueofp,theresultingMPL,andtheresultingamountofvariabilityinvariableAattheendofrunsinrewireMode1(Figure17).Forthese40x40grids,arelativelyslightdeviationfrom0isastatisticallysignificantdifferencefromthenullhypothesisofrandomness.TherelationshipbetweenMoran'sIandpisverysimilartothatbetweenpandothervariables:thelargemajorityofthevariabilityinMoran'sIwithintheseresultsoccurswherep<0.10(figure17.a).thewidestrangeofvaluesofmoran'siwasproducedwhentheMPLofanetworkwasrelativelyhigh(Figure17.B).TherelationshipbetweenMoran'sIandthestandarddeviationofvariableAisrelativelylinear,suggestingthatthedegreeofassemblagevariabilityisrelatedproportionallytotheorganizationofvariabilityacrossspace(Figure17.C).Inotherwords,the"extra"variabilitythatisproducedthroughinteractionswhenpislowisnotrandomlydistributedacrossspace,butispatternedinastatisticallysignificantway.

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Figure17.RelationshipsbetweenMoran'sIandp,meanPathLength,andthestandarddeviationofvariableAatstep1001for162runsinrewireMode1.Individualrunresultsmarked"Low","Medium",and"High"correspondtoresultsshowninFigures18

and19.

3.25 Figure18showsgraphicrepresentationsofexamplerunsproducinglow,medium,andhighvaluesofMoran'sI(designatedas"Low","Medium",and"High"inFigure17).DatafromtheserunsarecodedusingagrayscalecolorramptorepresentthevalueofvariableAatthelocationofeachgroup.Figure19depictsthesesamedataintheformofinterpolatedrasterscreatedinGIS.Intheleftcolumn,thecolorrampsareindividuallycalibratedtothehighandlowvalueswithineachgrid.Therastersintherightcolumnaredisplayedusingasinglecolorrampthatincorporatesthefullrangeofvaluespresentinallthreegrids.

Figure18.ResultsfromthreeexamplerunsshownwithacircleshadedtorepresentthevalueofvariableAforeachagent.

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Figure19.RasterscreatedinArcGISforthesamethreeexamplerunsshowninFigure18.Rastersontheleftusehighandlowvaluesspecifictoeachgrid.Rastersontherightusehighandlowvaluesencompassingvariabilityinallthegrids,showingboth

thegreateramountofvariabilitywhenpislow(MediumandHigh)andthespatialsegregationofthatvariability.

3.26 Differencesinthespatialorganizationofvariabilityareapparentinthesevisualdepictionsofthedata.BoththeunmodifiedandtheinterpolateddatasetsillustratehowhighervaluesofMoran'sIareassociatedwithincreasingspatialsegregationofhigherandlowervaluesofvariableA.ThepresenceofnonrandomspatialstructureislinkedtothemuchgreatervariabilitythatispresentingridswithhighvaluesofMoran'sI.

3.27 AsimilarsetofexperimentswasruntoproducedataontherelationshipsbetweenMoran'sI,p,MPL,andthestandarddeviationofvariableAinrewireMode2.Moran'sIwascalculatedfordatasetsproducedwithrewireRadiussetat2(n=105runs)and8(n=105runs).Whentheestablishmentofnonlocallinksislimitedtoasingletieroutsidethelocalneighborhood(i.e.,rewireRadius=2),thereisnodiscerniblerelationshipbetweenpandMoran'sI(Figure20.A).Meanpathlengthslessthan11arenotpossible(Figure20.B).WhenrewireRadius=8,anonlinearrelationshipbetweenpandMoran'sIispresent(Figure20.D),andMPLispositivelyrelatedtoMoran'sIinafashionsimilartothatseeninthedatafromrewireMode1(compareFigure20.E.andFigure17.B).Inbothcases,therelationshipbetweenMoran'sIandthestandarddeviationofvariableAislinear,alsosimilartotherewireMode1data(Figures20.C.and20.F).

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Figure20.RelationshipsbetweenMoran'sIandp,meanPathLength,andthestandarddeviationofvariableAatstep1001for210runsinrewireMode2.ToprowrewireRadius=2,bottomrowrewireRadius=8.

3.28 Becausetheequation-basedBACEmodeldiscussedearlierdoesnotincorporateaspatialcomponenttointeraction,thedataitproducesisnotstatisticallydifferentfromrandomintermsofitsspatialorganization(i.e.,Moran'sI).TheACEmodelhasnointeractionstructure(andthereforealsonospatialcomponent).Inotherwords,neitherofthesemodelscanbeusedtounderstandhowthespatialorganizationofvariabilityisrelatedtoculturaltransmissionwhentheinteractionsinvolvedinthattransmissionhaveaspatialstructure.

Discussion

4.1 Themodelhasbeenusedtoproducebaselineinformationoninter-relationshipsbetweennetworkstructure,networkproperties,andtheoutcomesofasimpleculturaltransmissionprocess.Itproducesrelationshipsamongnetworkpropertiesthatareconsistentwiththe"smallworld"phenomenonwhenconfiguredtomimicthemodelofWattsandStrogatz(1998).Resultsclearlyshowthatthepropertiesofthenetworksinthemodelhavesignificanteffectsonboththeoverallamountofvariabilitythatisproducedandthespatialorganizationofthatvariability.Thebehaviorofthemodelsuggestsasetofgeneral,directionalrelationshipsbetweenthe"rules"ofnetworkformation,thepropertiesofthenetworksthatareproducedbythoserules,andtheamountandspatialorganizationofthevariabilitythatisproduced(Figure21).Theformstheserelationshipstakearelargelyaresultofthenonlineareffectsthatthecreationofrelativelylongnonlocallinkshasonnetworkproperties.

Figure21.Conceptualsummaryofrelationshipssuggestedbymodelresults.

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4.2 Thecombinationsoflocalandnon-localtiesthatarepresentinrealhumannetworkssuggestthatthestructuresofthesenetworksfallsomewherebetweenthe"local-only"and"random"extremesconsideredhere.Modelresultssuggestthatthecreationofarelativelysparsewebofnon-localconnectionsisthemostefficientwaytoengineerasignificantimprovementintheeaseofinformationflowacrossaspatially-situatednetworklikethoseconsideredhere.Thisistruebothwhenthemodelisconfiguredtointerpolatebetweenlocal-onlynetworksandrandomnetworkswithnospatialconstraintsandwhenthecreationofnon-localconnectionsisspatially-constrainedtomorecloselymimicwhatisplausibleinrealhumansystems.

4.3 Followingtherelativelylargereductioninmeanpathlengththatiscausedbythecreationofasparsewebofnon-locallinks,thefurtheradditionofnon-locallinkshasrelativelylittlebenefitintermsofmeanpathlength.The"cost"ofestablishingand/ormaintainingnon-locallinks(e.g.,throughtravelrequiredforface-to-faceinteraction,etc.)wouldtendtolimitthecreationofnewlinksthatreturnnoaddedbenefit.Ifthemainpurposesofmaintainingnon-localcontactsistofacilitate"overthehorizon"informationflowandsecureaccesstoassistanceorresourcesindistantareasduringtimesofstress,thecostofmaintainingconnectionswouldbeanincentivetohaveasfewasnecessarytoservethepurposeofmaintainingsufficientinformationflow.Whileitwouldnotincreasetheefficiencyofinformationflow,however,havingmorethantheminimumnumberofconnectionswouldaddanelementofredundancythatmaybedesirableundersomecircumstances(e.g.,wherethecostofnetwork"failure"isveryhigh).Resultsfromthemodelareconsistentwiththeideathatsparsewebsofnonlocalconnectionsproducethebesttrade-offbetween"benefit"and"cost."

4.4 Linksinthemodelallowinformationtoflowdirectlybetweengroups,influencingthemathematicaloutcomesofindividualcopyingevents.Inactualsystems,sociallinksbetweengroupswouldfunctionasavenuesfacilitatingthebetween-groupmovementsofpeopleandfamiliesthroughmarriage,individualmobility,etc.Thesetransfersofpeoplewouldchangethecompositionofthelocal"pool"ofinformationthatisavailable.Non-locallinkswouldallowdirecttransfersofinformation(viathemovementsofpeople)betweengroupsthatarenotspatiallyadjacent.

4.5 Ifweacceptthatmanyhumaninteractionsinvolvedinculturaltransmissionarelocal(i.e.,thatpeoplelearnfromthosearoundthem)andthathumansocialnetworksarelikelytobestructuredinsomewaysimilartothelowpnetworksrepresentedbythemodel(i.e.,largelylocalnetworkswithafewnon-localconnectionsthatallowmovementsofinformationbetweenspatially-nonadjacentgroups),themodeldatasuggestthatchangesinthestructureofinteractionshouldbeconsideredamongthepossiblecausesofchangesin"stylistic"variabilityinarchaeologicalassemblages.Whenthelargemajorityofinteractionsarelocal(i.e.,whenpis0orlow),relativelysmallchangesinthenumberofnon-localconnectionscanhavearelativelylargeeffectonhowinformationflowsamonggroupsandthevariabilityoftheassemblagesthatareproduced.Thissuggestsanappropriate"nullmodel"forunderstandingtheoutcomesofculturaltransmissionprocessesmightbeonewhereinteractionsarestructuredthroughlocal-onlynetworks.

4.6 HistogramsofthestandarddeviationofvariableAshowthatthatincreasingthespatialradiusforaddingnonlocalconnectionsresultsinagreaterdifferenceintheoutcomesassociatedwiththeextremevaluesofp(Figure22.Aand22.B).Whentheradiusisrelativelylarge,thestatisticalpatternsofvariabilityarenotsubstantiallydifferentthanthoseproducedwhenlocallinksaresimplyreplacedbyrandomlinks(Figure22.C).

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Figure22.Histogramsofoutcomes(intermsofstandarddeviation)forp=0andp=1inthestructuredinteractionmodel(A,B,andC)andforthreesettingsofpConformintheBACEmodel(D).

4.7 Changesinthenon-localcomponentofinteractionare,ofcourse,onlyonepotentialcauseofchangesinassemblagevariabilitythroughtime.Changesinthestrengthofcopyingbias,changesinselection,anddeterministicmathematicalprocesseshaveallbeensuggestedasexplanationsforchangeinarchaeologicalassemblages(e.g.,EerkensandLipo2005;HamiltonandBuchanan2009;Neiman1995).HistogramsofoutcomesproducedbytheBACEmodelatvarioussettingsofpConformareshowninFigure22.D.Itisnotablethat,atcomparablevaluesofpConform,theBACEmodelproducesoutcomeslessvariablethanthoseofacompletelyrandomnetwork(comparetheresultsforpConform=0.25inFigure22.Dwiththoseforp=1inFigure22.C).Alocal-onlyinteractionstructure(i.e.,p=0inFigure22.A-C)producesassemblageswithmorevariabilitythanassemblagesproducedbyequation-basedmodelsthatmustassumerandominteractionsorpopulation-widecopyingofmeanvalues(neitherofwhichactuallyoccurinhumansystems).

4.8 Discerningbetweenalternativesourcesofchangeshouldbeagoaloffuturework.Independentlinesofevidenceareavailableforevaluatingnetwork-basedexplanationsofchangeinvariability.Itisoftenpossibletomonitorchangesinnon-localinteractioninarchaeologicalsettingsthroughindependentmeanssuchaschangesinrawmaterialuse,exchange,settlementorganization,etc.Iftherearearchaeologicalindicationsofchangingpatternsofnonlocalinteraction,wecanexpectthatthesechangesmayhaveadiscernibleeffectonpatternsofvariationinrealmsofmaterialculturethatarticulatewiththeseinteractions.Basedonthemodelresults,thisismostlikelytobetrueincaseswheresocialnetworksarepositionedwithinthe"small-world"(i.e.,lowp)rangewhereinformationflowisparticularlysensitivetotheadditionorsubtractionofsmallnumbersofnonlocallinks.

4.9 Themodelemployedhereisnotintendedtomimicinteractioninanyparticularculturalsystemorproducedatathataredirectlycomparabletodataderivedfromarchaeologicalassemblages.Itshouldbeclearthattherangeofnetworkstructuresthatcanbeproducedbythemodelisintendedtoneitherincludenorexhaustallpossiblepermutationsofhumansocialnetworks.Networkformationandmaintenance"rules"otherthanthoseconsideredheremayproducenetworksthathaveadifferentrangeofpropertiesthatarepotentiallyimportanttounderstandingartifactvariability.Thismodeldoesnotattempttorepresentmany

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aspectsofrealculturalsystemsthatmayalsohaveasignificanteffectonculturaltransmissionanditsoutcomes:theinteractionnetworksarestatic,allsocialconnectionsareofequalstrengthandtransmitthesameinformation,groupsaremonolithicentitiesthatdonotmove,styleisrepresentedasasimple"passive"quality,thereisnorepresentationoffunctionalvariability,andthegeographyoftheworldisuncomplicated.

4.10 Itisexactlytheabstractsimplicityofmodelssuchasthisone,however,thatmakethemusefulheuristictoolsforgeneratingabaselineunderstandingofhowvariablesarerelatedtooneanotherinuncomplicatedcontexts.Theexclusionofrepresentationsofphenomenasuchasagentmobility,variabilityinthestrengthandcharacterofsocialconnections,multi-dimensionalrepresentationsofstyle(i.e.,representingboth"active"and"passive"formsofstylisticvariability),networkdynamics,otherkindsofnetworkstructure,andcomplexgeographyensuresthatthesevariablesarenotresponsibleforanyofthebehaviorthatthemodelexhibits.Aspectsofnetworkstructureotherthanthoseinvestigatedhere(suchasdegree,forexample)mayalsoaffectinformationflowandberelatedtopatternsofartifactvariability.Anyorallofthesephenomenacanberepresentedandinvestigatedinfuturemodels.Someofthemodelbehaviordiscussedheremaychangewhenadditionalvariablesareincorporated,andsomemaynot.Agoaloffuturemodelingworkshouldbetoevaluatewhichrelationshipsarepresentunderavarietyofconditionsandwhicharemoresensitivetoparticularcircumstances.

4.11 Theusefulnessofabstractmodelssuchasthisonecanbejudgedbasedonthemodel'sabilitytoproducepatternsthatareexpectedandinterpretableanditscapabilitytocontributetothebuildingoftheorybygeneratingspecificideasforfurtherwork(Gilbert2008:p.41).Thismodelhasdonethesethings.TherelationshipsdepictedinFigure21,whileperhapsapplicabletomanydifferentkindsofarchaeologicalsystems,cannotbeassumedaprioritocharacterizeanysystemotherthanthatofthemodel.Theserelationshipscanbeusedasastartingpointforfurtherinvestigations,however.Modelsmorespecificandmorecomplicatedthantheonedescribedherewillberequiredtogenerateexpectationsthatcanbecompareddirectlytoarchaeologicaldata.Thevoluminousethnographicdataonkinship,marriage,exchange,andmobilityinhunter-gatherersystemswillmakeitpossibletocreatecomputationalmodelsusefulforunderstandingthelong-termandlarge-scaleimplicationsofthesedifferentbehaviorsintermsofnetworkstructure,networkproperties,andpatternsofartifactvariability(cf.WhiteandJohansen2005).

Conclusion

5.1 Thispaperaddsadditionalelementstothebasicviewthatdifferencesinhowinformationistransferredinaculturalsystemmighthaveoutcomesthatwecanobserveinpatternsofvariabilityinmaterialculture.Culturaltransmissionprovidesapointofarticulationbetweensocialnetworksandmaterialculture.Agent-basedmodelingallowsustoinvestigatehowmacro-levelpatternsofvariabilityareaffectedbytherulesgoverningindividualinteractionsincomplex,network-mediated,spatially-situatedsystems.Themodeldescribedhereshowsthatthespatialstructureofinteractionhaspatternedeffectsonartifactvariabilitythatarepotentiallysignificantandcomparableinmagnitudetothoseof"copyingbias."Theassumptionofunstructuredinformationtransferthatisfundamentaltoequation-basedmodelsisneitherareasonablesimplificationnorananalyticalnecessity.Thestructureofinteractionmakesadifferencetotheoutcomesofculturaltransmissionprocessesandshouldbeconsideredapotentiallyimportantcausalfactor.

5.2 Understandinghownetworkstructureaffectsthetransferofinformationandtheresultingtime-spacepatternsofvariabilitymayhelpusunderstandthecomplex,multi-dimensionalpatterningthatischaracteristicofsomanyaspectsofmodernhumansystems.Ifthe"natural"outcomeofopen,local-onlynetworksistoproduceregionalizedorclinalpatternsofvariability,thismaygosomedistancetowardsexplainingthemacro-fragmentationdescribedbyYellenandHarpending(1972)andthenon-isomorphicdistributionsoflanguage,biologicalvariation,anddifferentaspectsofmaterialculturethatarenotablefeatureofmanyhumanculturalsystems(e.g.,Boas1911;Hymes1967).Informationtransferineachofthesedomainsmaytakeplacelargelyattheleveloflocalizedinteractionsbetweenindividuals,mediatedthroughsocialnetworks.Thenon-linearbehaviorsofnetwork-mediatedsystemscouldalsoshedlightonthevariableratesanddimensionsofchangeacrosstimeandspacewhichareacharacteristicaspectoftheprehistoricarchaeologicalrecord(cf.Ramsey2003;SchifferandSkibo1987;Shott1996,2008).

Acknowledgements

Thispaperisastepinanongoingefforttousecomputationalmodelingtounderstandthecomplexrelationshipsbetweensocialnetworkstructureandmaterialcultureinhunter-gatherersystems.IamgratefulforthecontinuingencouragementandsupportofRickRiolo,JohnSpeth,BobWhallon,andHenryWrightattheUniversityofMichigan,andtootherfacultyandstudentswhohavetakenaninterestinthisworkandbravedtheconsequencesofinitiatingadiscussionaboutit.ThemodelingworkdescribedinthispaperwasconductedattheCenterfortheStudyofComplexSystemsattheUniversityofMichigan.IthankJohnSpethandRickRioloforprovidingvaluablefeedbackonearlydraftsofthispaper.Allerrorsandweaknessesremainthesolepropertyoftheauthor.

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