Exponential random graph models of preschool affiliative networks

SocialNetworksxxx (2012) xxx–xxx

ContentslistsavailableatSciVerseScienceDirect

Social

Networks

j o u r n al hom ep a g e :w w w . e l s e v i e r . c o m / l o c a t e / s o c n e t

Exponential

random

graph

models

of

preschool

affiliative

networks

João

R.

Daniel

_{, António}

_J.

_Santos

_{, Inês}

_Peceguina

_{, Brian}

_E.

_Vaughn

a_{UIPCDE,ISPA-InstitutoUniversitáriodeCiênciasPsicológicas,SociaisedaVida,RuaJardimdoTabaco34,1149-041,Lisboa,Portugal} b_{HumanDevelopmentandFamilyStudies,AuburnUniversity,203SpidleHall,Auburn,AL36849,USA}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received23May2011

Receivedinrevisedform24October2012 Accepted16November2012 Keywords: Affiliativestructures p*Models Preschoolchildren Structuraleffects

a

b

s

t

r

a

c

t

Exponentialrandomgraphmodelswereusedtoassesstherelevanceofreciprocity,popularity, transitiv-itycontrollingforeffectsbaseduponegoandaltersex,andsexhomophily,ontheformationofaffiliative tiesin19Portuguesepreschoolpeergroups.Thenumberoftimestwochildrenwererecordedasnearest neighborsinfocalsampleswasusedasanindicatoroftherelationship’sstrength.Independent parame-terestimatesofthedifferentmodels(oneforeachgroup)weresummarized,separatelyforthethreeage groups(“3-year-olds”,“4-year-olds”and“5-year-olds”)usingamulti-levelapproachtometa-analysis. Resultsshowedthataffiliativetiesbetweenchildrenweresexsegregated,highlyreciprocal,morelikely tobedirectedtoarestrictednumberofchildrenandwithatendencytocreatetransitivetriads.The structuralprocessesunderlyingtheformationofaffiliativetieswerequitestablebetweenclassrooms.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Casual observations in preschool settings easily reveal that interactionsbetweenchildren,both affiliativeand agonistic,are rarelydirectedatrandomtothelargenumberofavailable part-ners.Theirdyadicbehaviorreflectsindividualsocialdiscrimination, operationalizedintermsofthedifferentialallocationofbehavior towardotherclassroompeers(Strayer,1980).Evidencesof selec-tivityhavebeendocumentedformorethan70years(Challman,

1932;Hagman,1933),andasearlyasagetwo,childrendevelop

preferencesforspecificpeersthatcanlastforyears(Howes,1988). There are several studies describing affiliative structures of preschool peer groups from observational data (Santos and

Winegar, 1999; Santos et al., 2000, 2008a, 2008b; Strayer and

Santos,1996),butalthoughprocedureslikethehierarchical

clus-tering techniquesusedby Santos,Strayer andco-workers have proventobeavaluabletoolfordescribingpreschoolers’cohesive structures,welackstudiesthatlinksuchstructureswiththesocial processesthatmighthaveproducethem.Withoutstatistical mod-elsitisdifficulttoaddressthisquestion.

Exponentialrandomgraphmodels(ERGMs;orp*models)offer apromisingframeworkwithinwhichsuchmodelscanbe

devel-oped(RobinsandPattison,2005;Robinsetal.,2007a;Snijdersetal.,

2006;WassermanandRobins,2005).ERGMsareprobabilistic

mod-elsthatregardobservednetworksasonerealizationfromasetof possiblenetworkswithsimilarimportantcharacteristics(i.e.,an

∗ Correspondingauthor.Tel.:+351218811700;fax:+351218860954. E-mailaddresses:jdaniel@ispa.pt(J.R.Daniel),asantos@ispa.pt(A.J.Santos), inespeceguina@gmail.com(I.Peceguina),vaughbe@auburn.edu(B.E.Vaughn).

observednetworkisseenasoneparticularpatternoftiesoutofa largesetofpossibilities).Thestructureofagraphcanbeinterpreted asbeinggeneratedbytheoverlapoflocalconfigurationsthatcan beseenasoutcomesofstructuraleffectsinthenetwork.Strength anddirectionofparameterestimatesallowinferencesaboutwhich configurationsareimportant(e.g.,positiveandlargeparameters indicatethatthecorrespondingconfigurationismorefrequentthan expectedbychance,givenotherconfigurationsofthemodel) assist-ingthejudgmentaboutwhichstructuralprocessescouldexplain howthenetworkemerged.

Enteringpreschool,generallyatthreeyearsofage,represents for manychildren thefirstopportunitytointeractwitha great number of peers, and for researchers, an opportunity to study theprinciplesbehindnetworkformationonasampleof partici-pantswithlittlepriorexperiencewithpeers(Schaeferetal.,2010;

Snyderetal.,1996).Onarecentlongitudinalstudyconductedover

aschoolyearon11 North-Americanclassrooms,fromtheHead Startprogram,Schaeferetal.(2010)showedthattheimportance ofthestructuraleffectsofreciprocity,popularityandtransitivity onrelationshipformation(andchange)cascadesovertime,from thesimpletothemostcomplex.Reciprocityeffectsremained con-stant along theyear, popularitypeaked in importance midway throughtheschoolyear,whiletriadicclosureincreasedin impor-tanceovertime.Theirfindingsagreewithearlierethologicalstudies ofpreschoolaffiliativebehaviorshowingthattheincreased coor-dinationofchildren’sassociationprofilesgeneratedprogressively morecomplexaffiliativestructures(StrayerandSantos,1996).

Reciprocity,popularityandtransitivityarewell-known mech-anismsthatcanhelpexplaintheformationofnetworks(Snijders, 2011).Reciprocityisabasicfeatureofsociallifethathasbeenfound consistentlyinpreschoolpeergroups(Snyderetal.,1996;Strayer

0378-8733/$–seefrontmatter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.socnet.2012.11.002

2 J.R.Danieletal./SocialNetworksxxx (2012) xxx–xxx

andSantos,1996).Popularityeffectsinsocialnetworkanalysis

lit-eraturereferstotheexistenceofdegreedifferentials,differingfrom thesociometricdefinitionofpopularity,whichrelatestotheverbal assessmentsofpeerlikeabilityorstatus,andfromperceived popu-larityfromstudentreports.Unequalin-degreedistributioncanbe anemergentphenomenonofunderlyingvariationsinindividual characteristicsthatmakessomechildrenmoreattractiveas inter-actionpatterns,oraresultofchildrenchoosingtobefriendswith otherswhommanyotherhavealsochosen(BarabásiandAlbert,

1999;Gould, 2002).Transitivitymeasuresthetendency toward

triadicclosure innetworks–“friendsofmyfriendsarealsomy friends”.Transitivityinpeergroupsmayariseduetotheincreased propinquityof individualswho sharemutual friends,orfrom a psychologicalneedforbalance–convergenceofthirdparties’ eval-uation(Schaeferetal.,2010).

Thepurposeofthisstudywas,thus,todescribetherelevance ofdifferentlocalsocialprocesses– reciprocity,popularity, tran-sitivitycontrollingforeffects basedupon egoandaltersex,and sexhomophily-onthecreationofaffiliativenetworksof19 Por-tuguesepreschoolgroups,usingERGMs.Theseprocesseshavebeen showntosufficientlycapturethenetworkstructureofaffiliative

ties(Huitsingetal.,inpress;Schaeferetal.,2010).

Althoughboth, this and Schaefer et al. (2010) study, tackle thenetwork processes underlying children’s peerrelationships there are some different elements, described below, that will helpstrengthenthecasefortheimportanceofthefundamental processespresentedonrelationshipformationinpreschoolpeer groups.

Tocircumvent thedifficulty of obtainingreliable parameter estimatesduetorelativelysmallnetworks,Schaeferetal.(2010)

analyzedallnetworkssimultaneouslybyarrangingtheirdataas onelargematrixwithstructuralzerosbetweenchildrenin differ-entclassrooms(Snijdersetal.,2008).Thisimpliesthatparameter values(andconsequentstructuraleffects)wereconsideredequal acrossallclassrooms.Wefollowedadifferentapproach.AnERGM wasfittoeachoftheobservednetworksandthena multi-level approachtometa-analysis(Lubbers,2003;LubbersandSnijders, 2007)wasusedtosummarizetheindependentestimatesofthe differentmodels.This meta-analyticalprocedureallows assess-mentoftheextenttowhichtheprocessesstudiedvarybetween classrooms.

Preschoolclassroomscompositionofbothstudiesalsodiffersto somedegree.WhiletheHeadStartclassesobservedbySchaefer

etal. (2010)ranged in sizefrom 15to 21 and includedmixed

agechildren(range:37–60months),Portugueseclassroomswere slightlylarger(20–27)andincludedonly“same”agechildrenwith awiderage-span(“3-year-olds”,“4-year-olds”and“5-year-olds”; descriptionsprovidedinSection2.1).

Also,noneofthemodelspresentedbySchaeferandco-workers analyzedtheeffectsofpopularityandtransitivitysimultaneously. Giventhatbothprocessesleadtosimilarnetworkstructures(e.g., coresin networkscanarise eitherbyclosure-typeprocesses or frompopularity,orevenacombinationofthetwo),itisimportant toincludebotheffectssimultaneouslyinthemodeltodistinguish theindividualcontributionsofeachprocess(Robinsetal.,2007a), controllingfortheeffectoftheother.

2. Method

2.1. Data

Observationaldatawerecollectedof allchildren innine dif-ferentpreschoolclassroomsintwocentersservingmiddleclass familiesintheLisbonarea,Portugal,aspartofalargerstudyon preschoolchildrensocialdevelopment.Classroomswereobserved

once(twoclassrooms),twice(fourclassrooms)orinthree consecu-tiveyears(threeclassrooms),foratotalof19preschoolpeergroups observed:six“three-year-old”groups(i.e.,children<48monthsof ageatthestartoftheacademicyear),six“four-year-old”groups (i.e.,children between48and 60 monthsof ageatthestart of theacademicyear)andseven“five-year-old”groups(i.e.,children between60and72monthsofageatthestartoftheacademicyear). Intheclassroomsfollowedinconsecutiveyears,83%ofthechildren, onaverage,transitedtogetherfromoneyeartothenext.The over-allsampleconsistedof242differentchildren(122girlsand120 boys).Classroomsrangedinsizefrom20to27children,withthe proportionofboysrangingbetween.40and.68.

2.2. Socialproximityobservations

Using a focal individual sampling design, children were observedinarandomlydeterminedorderfora15-sinterval.At theendofthesamplinginterval,thechild’snearestpeer neigh-borwasidentified.Apeerwhowaswithinreach(ifbothchildren weretoreachout,roughly3–4feet)andengagedinthesameora similaractivityasthetargetchildwasconsideredthenearest neigh-borofthetarget.Whentwoormorechildrenwereexactlyequally closetothefocalchildthepeertothechild’simmediaterightwas consideredasnearestneighbor.Forinstancesinwhichachildwas interactingverballyorphysicallywithapeerattheendofthe 15-sinterval,theinteractingpartnerwasconsideredasthenearest neighbor,eventhoughanotherchildmightbephysicallycloser. Ifnopeerwaspresentintheseconditionsthechildwas consid-eredtobealone.Twoobserversmade200observationroundsper classroom(100roundseach;oneroundcorrespondstoobserving every child present once). Social proximity rounds were inter-spersedwithroundsofotherobservationaldata.Childrenabsent fromtheclassroomfor50%ormoreoftheobservationalrounds wereexcludedfromtheanalysis.Observationalassessmentsfora givenclassroomtookaroundthreeweekstocomplete.

Researchassistantsreceivedtraining(>80%agreementforthe identityofthenearestneighbor)intheobservationscheduleprior toinitiatingclassroomobservations.

2.3. Socialnetworkmeasurement

Childrenwereassignedrowsinadyadicmatrixandobserved frequenciesofproximitywitheachpeerasnearestneighborwere tabulated into columns.This produced an asymmetricaldyadic matrix. At the next step, the matrix wasrotated on its major diagonal and added to itself, resulting in a symmetric dyadic co-occurrence matrix. The number of times two children were recorded as nearest neighborswas used asan indicatorof the relationshipstrength.Dyadicco-occurrencedatawerefilteredas follows:

Yij=1if_ooij i+>

2 1

N−1 (1)

Atiefromchilditochildj(Yij)wastreatedashavingweight1

ifthenumberoftimesiandjwerenearestneighborsoijdividedby

thenumberoftimesiwasobservedwithanotherchildoi+exceeded

twicetheproportionexpectedbychance(Nequalsthenumberof childrenintheclassroom).

Filteringedgesalwaysinvolvessomedegreeofarbitrarinessand atthemomentthereisnotagenerallyacceptedapproachfordoing so.Thisprocedureallowscapturingrelationshipsthatare impor-tantforallchildren,regardlessoftheirattendancetoschool.Edges filteredthisawayallowasymmetricties,retainingonlythestronger connectionsforeachchild.

J.R.Danieletal./SocialNetworksxxx (2012) xxx–xxx 3 2.4. Modelspecification

Toassess the relevanceof the differentstructural processes we fitted an ERGM, to each observed classroom network, that includedthefollowingstructuralparameters:reciprocity, alternat-ingk-instar,alternatingk-outstar,alternatingk-triangletransitive andalternatingk-two-path,controllingforeffectsbaseduponego andaltersex,andsexhomophily.

Thealternatingk-starparametersareintendedtoassist model-ingthedegreedistribution(k-instarandk-outstarparametersfor in-degree/popularityandout-degree/activitydistributions respec-tively).Apositivealternatingk-starparameterindicatesthatthe networkhasaskeweddegreedistribution,containingsomehigher degreenodes,whereasanegativeparametersuggeststhatnodes withhighdegreeareimprobable(smallerdegreevariance).The alternatingk-triangletransitiveparametermeasuresthetendency toformtransitiverelationsandtheextenttowhichtriangles them-selvesgrouptogetherinthenetwork.Thepreconditionfortriadic closure istheoccurrence oftwo-paths(the sidesofa triangle). Alternatingk-two-pathparameterhelpsmodelingdegree distri-bution.Egoandaltersex,andsexhomophilyeffects,representthe effectsofsexonchildren’spropensitytoformties,receiveties,and choosesimilarothers,respectively.

Allmodelswerefittedusingthepnetsoftware (Wangetal., 2006), conditioned on the number of ties (i.e., the number of tieswerefixedduringMonteCarloestimationprocedures).Fixing densityisdesignedtodiminishtheriskofdegeneracyproblems (non-convergenceof theparameters; seebelow),havingminor effectsonotherparameterestimates(Robinsetal.,2007b;Snijders etal.,2006).Theweightingparameterofthethreealternating knetwork statisticswassetto2.Thisvaluehasbeenshownto workwellinotherinvestigations(Goodreau,2007;Hunter,2007;

HunterandHandcock,2006;LubbersandSnijders,2007;Robins

etal.,2007b,2009;Snijdersetal.,2006).Thevalueofimposes

a constraint onthe influence of higher order k configurations. Forexample,apositivealternatingk-triangletransitive parame-terindicatesanincreasedtendencyfortransitivetriangleclosure tohappenalongsidewithhighernumberofsharedpartners.This increase,however,isnotlinearand,beyondacertainnumberof additionalsharedpartnersthechancesofclosureareonlyslightly increased(dependingonthevalueof).

2.5. Modelselectionandgoodnessoffit

ParameterswereestimatedusingMarkovchainMonteCarlo methods(Snijders,2002;Snijdersetal.,2006).Parametersaresaid tohaveconvergedwhenthemeannumberofconfigurationsinthe sampleofsimulatedgraphsissimilartonumberofconfigurations intheobservedgraph:

t−ratio=(observation_standard−_deviationmeansample) (2) Goodconvergenceisindicatedbyat-ratioforallparameter esti-matesbeinglessthan(orcloseto).1inabsolutevalue.

Afterparametershavebeenestimatedfromanobserved net-work,weinvestigatedhowwellthemodelparameterssucceeded inreplicatingotherfeaturesoftheobservedgraphthatwerenot explicitlymodeled.Thereareagreatnumberofthesefeatureson which toexamine anymodel and nogeneralrulefor selecting suchfeatures.Thenetworkstatisticscomparedincluded:standard deviations andskewnessof both in-degreeand out-degree dis-tributions; correlation between the in-degree and out-degree distribution;andasetofeightglobalclusteringcoefficients(see

Robinsetal.,2009fordetails).At-ratioforthesestatisticslessthan

2inabsolutevalueisnotregardedasbadfit,althoughpreferably t-ratiosshouldliebetween−1and1(Robinsetal.,2007b,2009).

Table1

Descriptivestatistics.

3-year-olds 4-year-olds 5-year-olds

Classrooms 6 6 7

Avg.classroomsize 23.67(1.86) 23.83(1.94) 24.29(2.06)

%boys .42–.60 .40–.68 .40–.56 Avg.in/outdegree 1.72–3.67 2.48–3.75 2.72–4.07 OutdegreeSD .55–1.24 .89–1.24 .75–1.27 IndegreeSD 1.14–2.16 1.43–1.74 1.12–2.27 Reciprocityindex .68–.84 .63–.84 .75–.88 Transitivityindex .11– .46 .16– .38 .17– .47 %samesexties .58– .79 .69–.86 .76–.96 Note.Statisticsrepresentbetweenclassroomsrange.Thetransitivityindex meas-urestheproportionofalternatingk-two-pathsthathaveabasepresenttocomplete analternatingk-triangletransitive.

2.6. Meta-analysis

TheERGMsyieldedasetofparameterestimatesandstandard errors for each of the19 models. In order tosummarize these independent estimates, we followed the multi-level approach to meta-analysis described in Lubbers (2003)and Lubbers and

Snijders(2007),usingMLwiNsoftware(Rasbashetal.,2009).

Sinceasubstantialproportionofchildrenwereobservedin con-secutive years,the19 setof estimatesare not independent.As such,averageeffectsizesarepresentedseparatelyforthethree agegroups.

At-ratiooftheaverageparameterestimatetestedwhetherthe averageeffectsizewaszero:

t= ˆ WLS  SE(ˆWLS  ) (3) BothˆWLS

 andSE(ˆWLS )areproducedbyMLwiN.Thisstatistichas

approximatelya standard normaldistribution.Ratios exceeding 1.96inabsolutemagnitudeindicateanon-zeroaverageeffectsize. Totestwhetherbetweenclassroomsvariance2

waszero,weused

theQstatisticdescribedinSnijdersandBaerveldt(2003).

3. Results

3.1. Descriptivestatistics

Afterfilteringco-occurrencematricesindividualout-degreeand in-degree rangedfrom0to7 and0 to10 respectively.Table1

presentsasetofdescriptivestatisticsforthepreschoolclassrooms studied.

3.2. Summaryofexponentialrandomgraphsmodel

ERGMswereestimatedwiththesamenetwork effectsforall the19preschoolnetworks.Modelsconvergedsuccessfully with allt-ratiosbetween−.1and.1(M±SD=.04±.03,absolutevalues).

Table2resumestheaverageeffectsizesforeachoftheparameters.

Theestimatedaverageeffectsizesindicatestatistically signif-icant effects for all parameters, excepting for some sex effects (altersexfor“3-”and“4-year-olds”).Byfar,thestrongesteffect observedwasreciprocity.Ahighvalueforthereciprocity parame-ters(range:4.54–4.61)wastobeexpectedbecausealmost80%of thetiesobservedweremutualties.

Significantpositivevaluesforalternatingk-instarparameters (range: .81–1.10) indicate skewed in-degree distributions with somehighlypopularnodes,i.e.,somechildrenwereclearly pre-ferred than others as association partners. On the other end, significant negativevalues for alternating k-outstar parameters (range:−1.12to−1.75)indicatelessvarianceinoutdegreethan wouldbeexpectedbychance.

4 J.R.Danieletal./SocialNetworksxxx (2012) xxx–xxx Table2

Meta-analysisofparameterestimates.

“3-year-olds” “4-year-olds” “5-year-olds”

ˆ WLS SE 2 ˆWLS SE 2 ˆWLS SE 2 Reciprocity 4.61** _{.28 .00 4.59}** _{.33 .26 4.54}** _{.24 .00} Alternatingk-instar 1.10** _{.22 .02 1.12}** _{.20 .00 .81}** _{.20 .00} Alternatingk-outstar −1.75** _{.09 2.82}** _−1.12** _{.48 .35 −1.70}** _{.60 .78}* Alternatingk-trianglet .30** _{.08 .02}* _.21** _{.04 .00 .26}** _{.06 .01} Alternatingk-two-path −.51** _{.10 .03 −.60}** _{.07 .01 −.34}** _{.06 .01} Egosex(♂) −.63** _{.21 .00 −.54}** _{.21 .00 −.84}** _{.27 .04} Altersex(♂) −.27 .16 .00 −.13 .15 .01 −.59** _{.19 .00} Sexsimilarity .87** _{.15 .00 1.21}** _{.17 .00 1.57}** _{.18 .00}

Note:ˆWLS–estimatedaverageeffectsize;SE–standarderrorassociatedtoestimatedaverageeffectsize;2–estimatedvarianceoftheeffectsizebetweenclasses. * _p<.05.

** _p<.01.

Positivealternatingk-triangletransitive(range: .21–.30) and alternatingk-two-pathparameters(range:−.34–.60)canbe inter-pretedtogether, suggesting a tendency for triadic closure. The positivek-triangleparametertellsusthatchildrentendtospend time withthepreferredchoices of theirpartners. Thenegative alternatingk-two-path indicatesa tendency against non-closed paths(asexpectedbythesignificantpositivek-triangleparameter). Positivevaluesforalternatingk-instarandalternatingk-triangle transitive parameters indicate the existence of a core in the preschoolnetworks,createdbybothpopularityandtriadicclosure effects.

Significantsexsimilarityeffects,reflectingatendencyforsex segregated ties, were found in all age groups (range: .89–1.57 respectively).Additionally,boysofallagegroupsformedfewerties thangirls(egosexrange:−.54and−.84respectively),and “5-year-old”boys,furthertendedtoreceivefewertiesthangirls(altersex: −.59).

Thevarianceofnetworkeffectsobservedwassmall,excepting forthealternatingk-outstarparameters,indicatingsome consis-tencyofparametervaluesacrossclasseswithinagegroups.Dueto thesmallnumberofclassroomsanalyzedforeachagegroupitis notpossibletomakeamoresubstantialanalysis,testingfor possi-bleeffectsofgroupcompositionvariables(e.g.,classroomsize,sex proportion)onthenetworkstructureofpreschoolers.

Aqualitativecomparisonoftheaverageeffectsizeofthe dif-ferentparametersbetweenthethreeagegroupsseemstoindicate anincreasingtendencyofsexsimilarityeffects.Noneoftheother parametersrevealedaconsistenttrend.

Models had generally a good fit, reproducing well a set of featuresoftheobservednetworks:standarddeviationsand skew-nessofbothin-degreeand out-degreedistributions;correlation between the in-degree and out-degree distribution; and eight globalclusteringcoefficients.Absolutevaluesoft-ratiosforthese configurationswereallbelow1in10classrooms.Oneclassroom hadonet-ratio higherthan2(skewin-degreedistribution),and theremaininghadeitherone(n=5)ortwo(n=4)t-ratiosranging between1and2inmagnitude.

4. Discussion

Thepurposeofthisstudywastoexaminetheinfluenceof differ-entsocialprocessesonthecreationofaffiliativesocialstructures withinPortuguesepreschoolclassrooms.

Withaffiliativerelationshipsdefinedusingfrequencyof associa-tionsinnaturalsettings,resultsshowedthataffiliativetiesbetween preschoolchildrenweresexsegregated,highlyreciprocal,more likelytobedirectedatpopularchildrenandwithatendencyto cre-atetransitivetriads.Structuraleffectswereverysimilarbetween classeswithin agegroups, exceptingfor activity(alternating k-outstarparameter).

Anapparentincreaseofsexsegregationwithageagreeswith theubiquityofsex-segregatedpatternsofsocialinteractionand itscentralroleinthesocialorganizationofpreschoolpeergroups, especiallyaschildrengrowolder(Martinetal.,2005).Appearing aroundthreeyears ofagesexsegregationisprobably themost noticeablefeatureofearlypeerrelationships(Bohn-Gettleretal.,

2010; Fabes et al., 2003; LaFreniere et al., 1984; Maccoby and

Jacklin,1987;MartinandFabes,2001;Martinetal.,2005;Pellegrini

etal.,2007).Althoughthetendencytoseekoutsame-sexpartners canbothreinforceand shapedevelopmentalsextype behavior, preschoolchildrendonotchooserandomplaymatesfromthe avail-ablesame sexpartners. Behavioral homophilyhas beenshown toalsoinfluenceselection–preschoolerstendtobeattractedto peerswhosebehavioraltendenciesaresimilartotheirown(Hanish

etal.,2005;Martinetal.,2005;Pellegrinietal.,2007).However

behavioralhomophilesareoftensextyped,makingitdifficultto distinguishbothprocesses(Fabesetal.,2004).

OurresultsareconsistentwithSchaeferetal.’s(2010)findings thattheselectivepeerpreferencesarenotonlydeterminedby indi-vidualcharacteristics,suchassex,butarealsocontingentonthe patternofexistingrelationships(i.e.,structuraleffects).Reciprocity isadefiningfeatureofsociallife(Molm,2010)andverycommon phenomenainpreschoolnetworks(Schaeferetal.,2010;Snyder etal.,1996).Theemergenceofreciprocalrelationsinvolves min-imalinformation-processionrequirementsgiventhatindividuals onlyneedtoreturnhis/herpartners’gestures,notneedingtobe awareofotherrelationships(Schaeferetal.,2010).Filteringedges fromsymmetricco-occurrencematrices,aswasdoneinthisstudy, mightbeboostingtheeffectofreciprocity.Thelogicbehindtheuse ofasymmetricmatrixforthistypeofdata(i.e.,nearestneighbor) hastodowiththefactthatnoinformationiscollectedofwhoisthe initiatoroftheinteractionthatledtoproximity,andsoweconsider iandjtobeassociatedifeitheriisj’snearestneighbor,orjisi’s nearestneighbor(althoughinsomecasesicanbethenearest neigh-borofjbutjisnotnecessarilythenearestneighborofi).Thatis, although,twoindividualsmightbeseentogetherinaregularbasis, itmaybethatonlyonememberofthedyadisresponsibleforallthe proximityseeking.Thoughanalysesforweightednetworkshave beendevelopedinthelastyears(Barthélemyetal.,2005;Newman, 2004)thegreatmajorityofsocialnetworkanalysesrequiredirected orundirectedbinarynetworks.Thisisadrawback,especiallyfor thosewhoinfersocialrelationshipsfromobservationaldata. Fil-teringedgesisalwayssomehowarbitraryandinformationislost alongtheway.

Asstatedontheinitialsectionsofthispaper,existenceofhigher (in-)degreenodescanbeanemergentphenomenonofvariation in individual characteristics that influence the probability of a childbeingselectedasapartner,oritcanreflectamorecomplex processofpreferentialattachment.Despitethedifferences,both processesmakesimilarpredictionsforsinglenetworkobservations

J.R.Danieletal./SocialNetworksxxx (2012) xxx–xxx 5 (existence of higher in-degree nodes) that cannot be

distin-guishedusingtheERGMsframeworkusedhere.Inalongitudinal perspective,preferentialattachmentwouldcontinuouslybias in-degreedistributions,assomechildren wouldbecomemoreand more popular as time goes by. If what shapes relationships is the assessment of individual characteristics that make a child more or less preferred, then it is expected that thepopularity effect maxes out after children had time to assess the “qual-ity”of theirpeers.Schaeferet al.’s(2010) findingssupportthe later process, with popularityeffects peaking in importance in the middle of the school year and remaining constant there-after.In reality theprocess might even bemore dynamic than thissincethe“quality”ofachildaspartnermaynot bea fixed property, co-evolving alongside the creation of new relational ties.

Thepositivepopularityandtriadicclosureeffectsfoundinour study indicate the presence of core-periphery structure in the networks, created by the conjunction of both effects. Previous applicationsofERGMspointedoutthattheformationofacorewas oftentheresultof,either,popularityoratriadicclosureeffect,

sel-domboth(Robinsetal.,2009).Thecombinationofbotheffectsmay

turnouttobenotasunusualaspreviouslythought(Robinsetal.,

2009).

Comparing the average effect sizes of popularity and tran-sitivity acrossage groups didnot reveal a consistent upwards trend, indicative of possible developmental processes or long termcascadingeffects.WesuspectthatunlessPortuguese “three-year-olds” classrooms (where most children are unacquainted witheachother)aresampledveryearlyintheschool-year,time relatedvariationsinstructuraleffectswillnotbedetected.Despite the cascading effects found by Schaefer et al. (2010), a closer look at their models reveals that the change in the popular-ityand transitivity parameters value,acrossthree timeperiods in a school year, only slightly changes the odds of a tiebeing created(parameterestimates:popularity_×period=.011, transitiv-ity×period=.010).

Insum,this isthefirstapplicationofERGMsin thestudyof preschoolchildren’s social networks andits goal wastoreveal theimportanceofdifferentsocialprocessesthatshapethe emer-gence of preschoolaffiliative peerrelations. We hopewe have contributedtoshowthebenefitsofastochasticmodelingapproach in order to complement the traditional descriptive analysis of preschoolchildrensocialnetworks.Futurestudiesshouldlookfor waysinwhichtheseprocessesinfluencepreschoolchildrensocial development.

Acknowledgements

Special thanks are also due to colleagues for primary data collection. We gratefully acknowledge the contribution of the preschool children and teachers for their active partic-ipation. Financial support for the conduct of the research and preparation of the manuscript was provided by F.C.T. grants PTDC/PSI/66172/2006, SFRH/BD/27489/2006 and SFRH/BPD/82522/2011.

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