Calling Dunbar's numbers

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ContentslistsavailableatScienceDirect

Social

Networks

j ou rn a l h o m e p 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

Calling

Dunbar’s

numbers

P. Mac

Carron

a,∗

_,

_K.

_Kaski

b

_,

_R.

_Dunbar

a,b

a_SENRG,_Department_of_Experimental_Psychology,_University_of_Oxford,_OX1_3UD_United_Kingdom b_Department_of_Computer_Science,_Aalto_University_School_of_Science,_P.O._Box_15500,_Espoo,_Finland

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Availableonline16July2016 Keywords: Social Brain Hypothesis Communication Ego Networks

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Thesocialbrainhypothesispredictsthathumanshaveanaverageofabout150relationshipsatany giventime.Withinthis150,therearelayersoffriendsofanego,wherethenumberoffriendsina layerincreasesastheemotionalclosenessdecreases.Hereweanalyseamobilephonedataset,firstly, toascertainwhetherlayersoffriendscanbeidentifiedbasedoncallfrequency.Wethenapplydifferent clusteringalgorithmstobreakthecallfrequencyofegosintoclustersandcomparethenumberofalters ineachclusterwiththelayersizepredictedbythesocialbrainhypothesis.Inthisdatasetwefindstrong evidencefortheexistenceofalayeredstructure.Theclusteringyieldsresultsthatmatchwellwith previousstudiesfortheinnermostandoutermostlayers,butforlayersinbetweenweobservelarge variability.

1. Introduction

In recent years the availability of communication data has allowedustoanalysethenatureofhumanrelationshipsand inter-actionsonamuchlargerscalethanpreviouslyavailable(see,for example,Onnelaetal.,2007).Althoughmodesofcommunication havechangedhowever,ourbrainsizeshavenot,anditissuggested thereisacognitiveconstraintonthenumberofface-to-facesocial interactionsonemayhave(Dunbar,1993;Robertsetal.,2009).This constraintﬁtsinabroadsensewiththe‘socialbrainhypothesis’ whicharguesthattheevolutionofprimatebrainswasdrivenby theneedtomaintainincreasinglylargesocialgroups(Humphrey, 1976;Dunbar,1992,1998;BartonandDunbar,1997).

Individuals do not give equal weight to each relationship andevidencefromthesocialbrainhypothesissuggeststhatego networksarestructuredintoasequenceoflayerswiththesizeof eachlayerincreasingasemotionalclosenessdecreases(Dunbar, 1998;HillandDunbar,2003).Themeannumberoffriendsineach hasbeenfoundtobearound5,15,50and150inthecumulative lay-ers(i.e.onaverage10peopleinthesecondlayertomakeatotalof 15)(Zhouetal.,2005;Hamiltonetal.,2007).Beyondthisthereare evenlargergroupingssuggestedat500and1500(Dunbar,1993; Zhouetal.,2005).

Recently these Dunbar layers have been observed in online social media, such as Facebook and Twitter (Dunbar et al., 2015)and anonlinecomputergame (Fuchset al.,2014).These

∗Correspondingauthor.

E-mailaddress:[email protected](P.MacCarron).

relationshipsaretemporal,however,andthe150inparticular rep-resentstheamountoffriendsatagiventime.Ifanewfriendismade, anoldoneismostlikelydropped,andthestrengthrelationships changesquickerintheouterlayersthantheinnerones(Sutcliffe etal.,2012;Saramäkietal.,2014).However,othermethodsfor estimatingpersonalnetworksizeshavefoundnumberslargerthan theouterDunbarlayer,thesestudiessuggestanaveragepersonal networksizeofaround290forAmericans(Killworthetal.,1984;

McCartyetal.,2001).

Hereweuseamobilephonecalldatasetinitiallytoascertain whetherlayersoffriendsaredetectableinanofflinecontext.Ifwe findevidenceoftheselayers,wethentestiftheymatchthelayer sizespreviouslyidentifiedusingdifferentclusteringalgorithms.

AEuropeanphone-calldatasetoverall12monthsof2007is used.Thishas34.9millionuserswithalmost6billioncalls.About6 millionoftheseusersarewiththecompany(whoprovidecoverage toapproximately20%ofthecountry’spopulation)forwhomwe havedataonallcallstheymake.

The call frequency between two individuals represents the strengthofarelationshipandhasbeenshowntocorrelatewith emotionalcloseness(RobertsandDunbar,2011;Arnaboldietal., 2013).Saramäkietal.(2014)havealsoshownthatsocial signa-turesincellphonedataremainrobustovertimeevenwithidentity changesinthealters.

2. Methods

Toeliminatecasualcallsandbusinesscalls,thedataareﬁltered sothatonlycallsarecountedifthereisatleastonereciprocalcall betweenthetwousers.

http://dx.doi.org/10.1016/j.socnet.2016.06.003

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Peoplevaryintheextenttowhichtheyusetheirphones,with someusingitasaregularmeansofcommunicationwithfamily andfriends,andothersusingitonlyforsocialemergenciesorto arrangemeetings.Whiletheformerarelikelytoprovideafull cov-erageoftheirsocialnetwork,thelatterwon’t.Toavoidthiskind ofunder-reporting,wecensoredthedatasetsoastoincludeonly thoseindividualswithaminimumnumberofalters.Sincethe aver-agenumberofaltersatagiventimeinpersonal,orego-centric, networksis150,withanaturalrangeofapproximately100–250 (Hill andDunbar,2003;Zhouetal.,2005; Robertset al.,2009), weseta value of100alters astheminimum cut-off. Bydoing so,weaimedtohaveamorecompletedistributionofactualego networks,whilenotbiasingagainstindividualswho have natu-rallysmallnetworks.Afterthiswelowerthecut-offto50altersto observetheresultsforlowerfrequencyusers.

Thedegreekofanegorepresentsthenumberofalterscalled andtheweighteddegreewrepresentsthetotalnumberofcalls anegomakes.Thedegreedistributionpkandweighted

distribu-tionpwarethefractionofverticesinanetworkwithdegreekand weighteddegreew,respectively.Notethatinempiricalnetworks, thedegreedistributionsareoftenfoundtohavepositiveorright skew(Newman,2003).

Inordertoestimatethefunctionalformsofdegreedistribution, themethodofMaximumLikelihoodEstimatorsis used(Clauset etal.,2009;Edwardsetal.,2007).Herewetestdifferent distri-butions;namelypowerlaw,exponential,stretchedexponential, Gaussian(ornormal) andlog-normaldistributions, andusethe AkaikeInformationCriteriatoselectthebestmodel(Akaike,1974; BurnhamandAnderson,2002).

Thedataforeachuserisconsideredasaonedimensionalarray whichwedenotebyWsuchthattheminimumpossibleweightis

wmin=1whenanegocallsanalteronce.Thereisnorealupperlimit (beyondﬁnancialortimeconstraints)tothemaximumnumberof callsausercanmaketotheirpreferredalter.Inordertocompare users,thedataforuseriisnormalisedby

W= Wi−Wimin Wimax−Wimin,

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whereWiisthenumberofcallsmadetoeachalterandWiminand

Wimaxareminimumandmaximumnumberofcallstheymaketo

anyoftheiralters.Thisensuresthat,foreachego,thestrongest interactionwithanalteris1andtheweakestis0.Aﬁrstestimate toidentifythelayersistoplottheprobabilitydensityofall differ-entweightsforalluserstoascertainifanypatternexists.Akernel densityestimateisappliedtothetrueprobabilitydensityandthe localminimaareusedtoidentifyclusters(Rosenblattetal.,1956; Parzen,1962).

Manymethodsexistfordataclustering,(see,forexample,Jain etal., 1999; Ganet al.,2007).Thevast majority of these algo-rithms,however,areforhigh-dimensionaldatasets (Jain, 2010). Here,althoughwearedealingwithbigdata,weseektobreakeach individual’scallsintoclustersorlayers.Thuswearedealingwith one-dimensionalclusteringforeachuser,andfromthisweanalyse theaveragelayersizes.

Acommonmethodforone-dimensionalclusteringistheJenks naturalbreaksalgorithm(Jenks,1967).TheJenksalgorithmis simi-lartok-meansclusteringinonedimension(Khan,2012).Itsearches fortheminimumdistancebetweendatapointsandthecentres oftheclusterstheybelongtoaswellasformaximumdifference betweenclustercentresthemselves.Thegoodnessoffitcanbe cal-culatedtooptimisethenumberofclustersfound(Coulson,1987). Agoodnessoffit of1.0canonlybeattainedwhenthereiszero within-classvariation(oftenwhenthenumberofclustersisthe samesizeasthedata).Tochoosetheoptimalnumberofclusters wetakeathresholdof0.85forthegoodnessoffitassuggestedin

Coulson(1987).

WealsouseaGaussianMixtureModelwhichassumesthatthe dataaregeneratedfromanumberofGaussiandistributions(Day, 1969).Naively, wemayassume thatthelayersare madeupof GaussiandistributionswiththeirmeansontheDunbarnumbers. Theexpectationmaximisationalgorithmisimplementedforthis (Dempsteretal.,1977)and,again,theAkaikeInformationCriterion isusedtoassessthenumberofclustersinthedata.

Another method for clustering the data, used here, is the head/tailbreaks(Jiang,2013).Thismethodwasdevelopedfordata withheavy-taileddistributions.Itsplitsthedataatthemeanand takingthehead(allvaluesabovethemean),itrecursivelysplits eachconsecutiveheadatitsmean.Ourdataisheavytailed(Onnela etal.,2007),withmostuserscallingmanypeopleasmallnumberof timesbutcallingtheircloserfriendsfrequently.Anadvantageofthe head/tailbreaksisthatthenumberofclustersisderivednaturally fromthedistributionofthedata.

3. Results

Althoughthemobilephonecalldatasetwestudyherecontains almostcomplete dataonover 6millionusers,onlyafractionof thesehaveadegreek≥100.Inordertotestthehypothesisofthe layersofdifferentlevelsofemotionalcloseness,weanalyseusers

Fig.1. Ontheleftpanel:Thedegreedistributionandalog-normalfit.Theinset showsuserswithdegreek≥100andasimilarfit.Ontherightpanel:Theweighted degreedistributionisshown,againwithafittedlog-normaldistribution.

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thatareinthelowerboundoftheouterlayer,i.e.havecalled100or morealters.Thisleaves26,680userswithameannumberofalters of129.9andstandarddeviationof37.7.Themeannumberofcalls anegomakesis3553.8withastandarddeviationof1894.1.Wealso analyseuserswith50<k<100(N=301,190).Thesehaveanaverage weighteddegreeof1964.2.Theremaininguserswithk<50have anmeanweighteddegreeof148.8indicatingtheyusetheirphone onaveragelessthanonceeverytwodays.

Thedegreedistributionandtheweighteddegreedistribution fortheentiredatasetareshowninFig.1withlog-normal distribu-tionsﬁtted.Theinsetintheleftpanelshowsthedegreedistribution foruserswithdegreek≥100,whichfollowsassimilar distribu-tion.Bothtruncatedpower-lawandlog-normalmodelsyieldhigh Akaikeweightsforthedegreedistributionbutwithslightlymore supportforthelog-normalbehaviour(atruncatedpowerlawis ﬁttedtothedatasetinOnnelaetal.(2007)).Henceweconsider that,ofthecandidatemodels,thelog-normalmodelhasthehighest supportfortheweighteddegreedistribution.

Log-normaldistributionsareassociatedwithmultiplicative pro-cesses,incontrastwithGaussiandistributionswhichareadditive. Animportantconsequenceofalog-normaldistributionisthatina growthprocess,thegrowthrateisindependentofthesize(Sutton, 1997).Intermsthedegreedistribution,thismeansthattherateof growthofanego’snumberoffriendsisindependentoftheir cur-rentnumberoffriends.Log-normaldistributionsarefoundinmany empiricaldatasets(Eeckhout,2004;Mitzenmacher,2004;Clauset etal.,2009).

Having establishedthatthedatais log-normallydistributed, beforenormalisingtheweightarrayinEq.(1),thelogofeachweight istaken.Fig.2showsthehistogramforthenormalisedlogofthe weights ˆwforallusers.Therearepeaksat0and1aseveryuserhas atleastonealtertheycallaminimalnumberoftimes(usually mul-tiple)andatleastonealtertheycallamaximumnumberoftimes. Althoughthedataisverynoisy,weobservepeaksindicatingthat therearesomegroupingswithinthedata.Thebluelineisa Gauss-iankerneldensityestimateofthedistribution.Thelocalminima onaveragesplitthenumberofaltersintogroupsof15untilthe normalisedstrengthofarelationship ˆw=0.28afterwhichthere areanaverageof68.8alters.Thiscouldroughlycorrespondtothe thirdDunbarlayerof50.Beyondthisregion,thedataistoonoisy tosplitupfurtheronagrouplevel.

Fig.2.Thehistogramofthenormalisedweightsofeachcallforallusers.Theblue lineisaGaussiankerneldensityestimatortothedata.(Forinterpretationof refer-encetocolorinthisﬁgurelegend,thereaderisreferredtothewebversionofthis article.)

Assumingthenthattherearesomekindsofgroupingswithin eachuser’scalldataW,wenextusemoretraditionalclustering algorithmstoattempttoidentifythelayers.Herewedonotneed tonormaliseWaswespliteachuser’sweightarrayindividually andanalysetheoveralldistributionsonebyone.

TheanalysiswiththeJenksalgorithmisfoundtosplitalmost halfoftheusers(13,209)into4–5clusters,butitalsoﬁndsofthe orderofahundredusersineveryclusterfromclustersof7andup.

Fig.3showsthenumberofusersineachcluster.

Weidentifythemostcommonnumberofclusterstobe4for 7226(27.1%)users.Theaveragenumberofusersandtheirstandard deviationineachclusterarereportedinTable1andtheaverage cumulativelayerturnsouttohold4.1,11.0,29.8and128.9users. Thesenumbersarealittlesmallerthantheconventionalnumbers forDunbarlayers,butwithintheirnaturalrangeofvariation.The nextmostcommonnumberofclusters,asshowninFig.3,is5 clus-tersfor5983(22.4%)userswithcumulativelayersholding2.9,7.4, 17.7,43.0,and134.3users.Thesenumbersarequitesimilartothe Dunbarnumbers,butwithananotherlayerbetweentheﬁrsttwo.

Table1

TheaveragenumberofusersineachclusterusingtheJenksNaturalBreaksalgorithm,GaussianMixtureModel(GMM)andtheHead/TailBreaks(H/T).Thenumberofusers inaclusterisgivenbyN,thetotalnumberofclustersthealgorithmﬁndsisdenotedbycandnigivestheaveragenumberofaltersinclusteri.Thecumulativenumberof

usersineachclusterisgiveninthesecondpartofthetable

N n1 n2 n3 n4 n5 Jenks c=4 7226 4.1(2.0) 6.8(4.0) 18.9(8.7) 99.1(32.7) c=5 5983 2.9(1.3) 4.6(3.2) 10.2(6.1) 25.4(11.4) 91.3(32.7) GMM c=2 9289 43.0(15.5) 79.9(21.3) c=3 3334 18.0(10.1) 41.8(15.4) 66.6(19.8) H/T c=3 12,951 4.3(1.9) 18.1(5.5) 98.5(22.9) c=4 11,616 2.2(0.9) 5.9(2.5) 21.6(7.7) 103.7(32.9) Cumulative N n1 n2 n3 n4 n5 Jenks c=4 7226 4.1 11.0 29.8 128.9 c=5 5983 2.9 7.4 17.7 43.0 134.3 GMM c=2 9289 43.0 122.9 c=3 3334 18.0 59.7 126.4 H/T c=3 12,951 4.3 22.4 120.9 c=4 11,616 2.2 8.2 29.8 133.5

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Fig.3. NumberofusersineachclusterusingtheJenksalgorithm.Themajorityof usershavefourlayersoffriends.Thealgorithmdoesnotconvergewellforlarge numbers(>7)ofclusters.

Theprobabilitydistributionsfor4and5clustersareshownin Fig.4.Eachclusterislog-normallydistributed.Thisisperhapsnot surprisinggiventhattheoriginaldistributionislog-normal;there isalreadyalargevariationwithsomeusersperhapshaving100 friendssplitinto4clustersandothershavingalmost500splitinto 4clusters.Thereforeeveniftheestimatesforthemeansizeofeach layerareaccurate,thereislargevariancewithnumbersofalters goingfartotherightofthesemeans.

NextweuseaGaussian MixtureModeltosplitthedata.For thisalgorithm,wehowevertakethelogofthedataasitis log-normallydistributedand notGaussian.In thiscase 9289ofthe usersaresplitinto2clustersand3334usersinto3clusters.(The modelﬁndsabout1000thousandusersineachclusterabovethis upto12clusters.)Theaverageandstandarddeviationsarereported in Table1. For 2 clusters, thecumulative layer-size meansare 43.0 and 122.9users, which again are quitecloseto theouter twoDunbarlayers.For3clustersthemeansareat18.0,59.7and 126.4userswhichareclosetotheouterthreelayersbutmissthe innerone.

Lastlyweapplythehead/tailbreaksalgorithm.Foruserswith morethan100alters,itﬁnds12,951usersin3clusterswith cumu-lativelayersizesof4.3,22.4and122.9usersand11,616usersin4 clusterswithcumulativelayersizesof2.2,8.2,29.8and133.5users. ThefulldetailsarereportedinTable1.Again,thelayersarefound tobelog-normallydistributedusingthismethod.

Eachofthethreealgorithmsﬁndssmallnumberoflayersforthe majorityofthedata.TheJenksandhead/tailsalgorithmsﬁndan innerlayerrangingfrom2.2to4.3altersandalmostallalgorithms yieldalargeouterlayerof80–100altersattheend.TheJenks algo-rithmandGaussianMixtureModelbothgivegoodevidenceforthe outertwolayers.

Ifweassumethatthemajorityof4clustersistheappropriate value,wecanusetheJenksalgorithmtoforceeveryoneinto4 clus-ters.Thevaluesofthecumulativelayersarethen3.5,10.6,31.1 and129.9users,whichareslightlysmallernumbersthanthose conventionallyconsideredfortheDunbarlayers,buthavevirtually thesamescalingratiobetweenthelayers,i.e.onaverage,3.3here, comparedtoanaverageof3.2foundbyZhouetal.(2005)and3.2 inFacebooktrafﬁcbyDunbaretal.(2015).

Theseresultsarefromcallfrequency.Wealsohavetheduration ofeachcall.Thedurationandcallfrequencyarehighlycorrelated witha Pearson correlationcoefﬁcient of r=0.71. For egoswith

k≥100thisisslightlyhigheratr=0.76.UsingtheJenksalgorithm thisagainyieldsamajorityofegoswith4clusters(27%ofegos)with cumulativelayersof3.9,10.1,27.2and129.3.Theseareslightly lowerthantheresultsforcallfrequency.

Fig.4. Theprobabilitydistributionsforthenumberofusersineachclusterfor4 clustersontheleftand5clustersontherightusingtheJenksalgorithmandshown onalog-logscale.Log-normaldistributionscanbeﬁttedtoeachasseenbythe dottedlines.

FinallyweapplytheJenksalgorithmtoegos’callfrequencies with50≤k<100(N=301,190).Inthiscasethemajorityofusers arefoundtobebestclusteredinto3(23.5%)withcumulativelayers of3.9,11.9and63.91.ThismatchestheﬁrstthreeDunbarlayers well.Thenextmostcommonis4clusters(14.5%)withcumulative layersof2.7,6.7,17.9and64.3.Thesealsocontaintheﬁrstthree Dunbarlayerswellbuthaveanadditionallowervalueclosetothe 1.5layerfoundinDunbaretal.(2015).Onceagaintheduration givesverysimilarresultstothecallfrequency.

4. Conclusions

Inthisstudy,byapplyingdifferentclusteringalgorithmstoa mobilephonedatasetweﬁndstrongevidenceforalayering struc-ture.Fig.2,forexample,makesnopriorassumptionsandshows thatthereissomestructurewithinthedatasetinspiteofallthe noise.However,ﬁndingdiscretelayersisstillaconsiderable chal-lenge.

Althoughtheclusteringmethodsyieldslightlydifferentresults, asshownatthebottomofTable1,theyhaveimportantsimilarities. Theyallﬁndasmallnumberofclustersandshowgoodsupportfor

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theoutertwolayers.Whilethedataisnoisy,allmethodssupport twodifferentgroupingswell.Thiscould,forexample,mean intro-vertsandextrovertshaveadifferentnumberoflayersoffriends. Furtherworkcouldinvestigatethispossibility.Anothersuggestion isthatoverayear,friendshipsaremoretransient.Alterscouldmove upordownfromonelayertothenextona regularbasis.This wouldreﬂectthetemporalnatureofemotionalcloseness, espe-ciallyamongone’snon-closestfriends.

TheJenksalgorithmandGaussianMixtureModelforfourlayers, theygiveresultsclosetotheDunbarlayers.Inaddition,theyhave thesamescalingpattern(∼3)ashaspreviouslybeenreportedfor thestructureofofﬂineegocentricsocialnetworksandthe organi-sationofnaturalcommunities(Zhouetal.,2005;Hamiltonetal., 2007),FacebookandTwittertrafﬁc(Dunbaretal.,2015)andonline gamingenvironments(Fuchsetal.,2014).Westilldonothaveany principledexplanationforwhythesestructurallayersshouldhave suchaconsistentpattern,buttheyarecloselytiedintothe psycho-logicalaspectsofrelationshipslikeemotionalcloseness(Sutcliffe etal.,2012).

Themeansintheclustersaregenerallysmallerthanthe pre-dictedmeansfromtheDunbarlayers,thoughtheymatchtheranges foundinHillandDunbar(2003).Areasontheycouldbesmaller hereisduetothefactthatamobilephonecalldatasetonly cap-turesaportionofanego’ssocialnetwork,evenwithtakingusers whocallmorethan100alters.Withmanyothermodesof com-municationavailable,itisunlikelythatauserwouldonlyresortto phoningtheirfriends.Weemphasisehowever,thattheyear2007is agoodtimetousecellphonedataforthiskindofanalysisasitisjust priortosmartphones(theﬁrstiPhonewasreleasedafewmonths beforethisdatasetends)whichfacilitatesmanyotheravenuesof onlinecommunication.Itisalsobeforeplatforms,suchasSkypeor Facebook,wereattheheightoftheirpopularity.Wealsoshowthat userswhocall50–100peoplethroughouttheyearmatchtheinner threeDunbarlayerswell.

Thisstudyhasstrongimplicationsforthesocialbrain hypothe-sisas,regardlessofthemodeofcommunication,similarstructure isobserved.Futureworkonthiswillinvolveanalysingtheturnover timeinthelayersusingtemporaldataindifferentcommunication datasets.Webelievethattheturnovertimewillrelatetothe emo-tionalcloseness,forexampletheinnerlayerislikelytobemore robustinphonedatathanonTwitter.

Aﬁnalpointofnotehereisonthestructureofthelayers.From eachalgorithmweﬁndthatthelayersarelog-normallydistributed forallnumberofclusters.This,tothebestofourknowledge,hasnot beenobservedbefore.Itisimportanttopointoutthatthis right-wardskewandlargestandarddeviationexist.Thelog-normality isduetotheentiredegreedistributionbeinglog-normalandthus alreadyhavingconsiderableskew.Thispotentiallyalsoshowsa dif-ferencebetweenextroverts,whotendtohaveanumberoffriends fargreaterthanthemean.Regardlessoftheirnumberoffriends however,theystillshowevidenceoflayers.

Acknowledgements

TheauthorswouldliketothankHang-HyunJoforhelpful discus-sions.P.MC.andR.D.aresupportedbyaEuropeanResearchCouncil (No.295663)AdvancedInvestigatorgrant toR.D. K.K. acknowl-edgesﬁnancialsupportfromtheAcademyof Finland’sCOSDYN project(no.276439).

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