Empirical Studies on the Network of Social Groups

19 Pages·2015·7.99 MB·English

RESEARCHARTICLE
Empirical Studies on the Network of Social
Groups: The Case of Tencent QQ
Zhi-QiangYou1,Xiao-PuHan1*,LinyuanLü1,ChiHoYeung2
1AlibabaResearchCenterforComplexitySciences,HangzhouNormalUniversity,Hangzhou311121,
China,2DepartmentofScienceandEnvironmentalStudies,TheHongKongInstituteofEducation,Hong
Kong
* [email protected]
Abstract
a11111
Background
Participationinsocialgroupsareimportantbutthecollectivebehaviorsofhumanasa
grouparedifficulttoanalyzeduetothedifficultiestoquantifyordinarysocialrelation,group
membership,andtocollectacomprehensivedataset.Suchdifficultiescanbecircumvented
OPENACCESS
byanalyzingonlinesocialnetworks.
Citation:YouZ-Q,HanX-P,LüL,YeungCH(2015)
EmpiricalStudiesontheNetworkofSocialGroups:
Methodology/PrincipalFindings
TheCaseofTencentQQ.PLoSONE10(7):
e0130538.doi:10.1371/journal.pone.0130538 Inthispaper,weanalyzeacomprehensivedatasetreleasedfromTencentQQ,aninstant
Editor:FredericAmblard,UniversityToulouse1 messengerwiththehighestmarketshareinChina.Specifically,weanalyzethreederiva-
Capitole,FRANCE tivenetworksinvolvinggroupsandtheirmembers—thehypergraphofgroups,thenetwork
Received:November28,2014 ofgroupsandtheusernetwork—torevealsocialinteractionsatmicroscopicandmeso-
scopiclevel.
Accepted:May21,2015
Published:July15,2015
Conclusions/Significance
Copyright:©2015Youetal.Thisisanopenaccess
articledistributedunderthetermsoftheCreative Ourresultsuncoverinterestingbehaviorsonthegrowthofusergroups,theinteractions
CommonsAttributionLicense,whichpermits betweengroups,andtheirrelationshipwithmemberageandgender.Thesefindingsleadto
unrestricteduse,distribution,andreproductioninany
insightswhicharedifficulttoobtaininsocialnetworksbasedonpersonalcontacts.
medium,providedtheoriginalauthorandsourceare
credited.
DataAvailabilityStatement:Datasetofthispaperis
availableonlinefromtheurl:http://datadryad.org/
review?doi=doi:10.5061/dryad.pc537.
Introduction
Funding:ThisworkissupportedbytheNational
NaturalScienceFoundationofChina(11205040, Socialinteractionsareessentialtoourdailylife,yetourunderstandingontheorganizationof
11205042,11105024,11305043),andtheresearch socialcontactsislimited.Majorreasonsincludethedifficultiestoquantifyindividualsocial
startupfundofHangzhouNormalUniversity,andthe relationshipandtocollectacomprehensivedataset.Nevertheless,therapiddevelopmentofthe
EUFP7Grant611272(projectGROWTHCOM),and
Internethasrevolutionizedtheformofsocialinteractionsfrompostalmails,telephonevoice
CCF-TencentOpenResearchFund.ZQY
calls,physicalmeetingandgathering,toemails,instantmessaging,onlineforumandonline
acknowledgestheXinmiaoTalentProgramof
socialnetworks.Throughtheinternet,interactionsarequantifiedintodatawhichgreatlyfacili-
ZhejiangProvince(GrantNo.ZX13005002062).CHY
acknowledgestheInternalResearchGrantofHKIEd. tatesthestudiesofsocialnetworks.Manyexcitingfindingsarerevealed.Asanexample,the
PLOSONE|DOI:10.1371/journal.pone.0130538 July15,2015 1/19
EmpiricalStudiesontheNetworkofSocialGroups
CompetingInterests:Theauthorshavedeclared hypothesisofsixdegreesofseparationwasinitiallyconsideredin1930s[1],whichstatesthat
thatnocompetinginterestsexist. anytwopersonscanbeconnectedbyasmallnumberofacquaintances,wasonlyrecentlytested
onFacebooknetworkwhichgivesanaveragedegreeofseparationofroughly4[2].Otherfea-
turesrevealedononlinesocialnetworksincludepower-lawdegreedistribution[3–7],commu-
nitystructure[8,9]andspecialcommunicationpatterns(e.g.thenon-Poissonpropertieson
contactactivities)[10–16].
Sofarthestudiesononlinesocialnetworksfocusmainlyonindividualsocialrelationship,
leavinganotherimportantaspect—participationinsocialgroups—lessunderstood.Itis
becausethecollectivebehaviorofhumanasagroupisdifficulttostudyintraditionalsocial
networksduetotheambiguityinquantitativelyaffiliatingindividualstospecificgroups.This
problemdoesnotoccurontheInternetsincegroup-basedapplicationshaveadefinitemem-
bershipidentityforindividuals.Forinstance,prototypeonlineapplicationssuchaschatrooms
andbulletinboardsystems(BBS)involveindividualusersjoiningandpostingmessageswhere
membershipidentityiswelldefined[17,18].IncludingWindowsLiveandGooglemessengers,
Whatsapp,Skype,FetionandTencentQQ,theseapplicationssetthebasisforexistingsocial
applicationsandinstantmessengers.Intheseapplications,userscreatesocialgroupson
demand.Itdrivessocialnetworkstoastatewhicharemoreextensiveandcomplicatedthan
theirphysicalcounterparts.
TwodifferenttypesofonlinesocialgroupscanbeformedontheInternet.Thefirstoneis
similartoordinarysocialnetworks,whicharejoinedbyfriendswithrealpersonalrelationship.
CirclesinGoogleplus,SkypeandWhatsappgroupsbelongtothistype[19].Thesecondoneis
moreuniquetoonlinesocialnetworks,consistingofgroupsofindividualswithcommoninter-
estsbutwithoutpriorpersonalrelationship,forinstance,membershipinforumsandstudent
bulletins.Theyconnectindividualsbeyondordinarysocialnetworksandextendthesocial
scopeofindividuals.Despitethedifferenceintheirnature,thetwotypesofnetworksareinter-
dependentoneachother[20–22].Forinstance,twousersinthesameforummaybecomeinti-
matefriendsandparticipateineachothers’personalsocialnetworks.Bymakingnewfriends,
anindividualmayfindnewinterestsandjoinnewforums.Thisnewformofsocialorganiza-
tionisuniquetoonlinesocialnetworksandhasgreatlysupplementedorevenreplacedits
physicalcounterpart.
Inthispaper,weanalyzeacomprehensivedatasetobtainedfromTencentQQ,aninstant
messengerwiththehighestmarketshareinChina.Bothtypesofsocialnetworksareestab-
lishedonQQandtheinteractionsbetweenthetwonetworksareexpected.Specifically,weana-
lyzethreedifferentnetworksinvolvinggroupsandtheirmembers—thehypergraphofgroups,
thenetworkofgroups,andtheusernetwork—torevealsocialinteractionsatmicroscopicand
mesoscopiclevels.Ourresultsuncoverinterestingbehaviorsonthegrowthofusergroups,the
interactionsbetweengroups,andtheirrelationshipwithmemberageandgender.Thesefind-
ingsrevealuniquephenomenoninonlinesocialnetworks,aswellasinsightswhichareother-
wiseinaccessibleinordinarysocialnetworks.Here“ordinarysocialnetworks”refertothe
socialnetworksdirectlybasedonpersonalusers.
Method
Datadescription
TencentQQ(commonlyabbreviatedasQQ,thewebsiteofTencentQQ:http://www.qq.com)
isaninstantcommunicationtooldevelopedbyTencentHoldingsLimitedin1999.Todate,it
hasover700millionactiveusersandhasbecomethelargestonlineapplicationinChina.QQ
userscansendmessages,sharephotosandfiles,postmicroblogs,andvoiceorvideochatwith
friendsusingcomputersorsmartphones.
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EmpiricalStudiesontheNetworkofSocialGroups
SocialgroupisoneofthemainfeaturesofQQwhichallowsmultipleuserstocommunicate
instantly.Amessagepostedbyamemberisimmediatelyreceivedbyalltheothergroupmem-
bers.Whennecessary,anytwomemberscancommunicateviaindividualchannel.Depending
ontheactivenessofauser,eachofthemcancreatenomorethansixgroups.Groupscanbe
searchedbytheirID’sornamesandotheruserscanjointhegroupupontheapprovalbythe
administrator,i.e.thegroupcreator.QQlimitsthegroupsizeby100,200,500,and1000,also
dependingontheactivenessofthegroupcreator.Forexample,accordingtothelatestruleof
Tencent,auserwithlevel0(theactivenessislessthan5)cancreateonlyonegroup(thelimit
ofgroupsizeis200),andauserwithlevel48(theactivenessishigherthan2496)cancreate5
groups(thelimitsare200foroneofthegroupsand500fortheother4groups).Otherthan
personalrelationships,somegroupsareformedbymemberswithcommoninterests,e.g.mov-
ies,orbelongtothesameorganizations,e.g.universitiesorcompanies.Thelattersareusually
exclusivesocialcirclesbasedonphysicalorganizations.
TheQQdataset(itwasreleasedfromtheonlineopendatabase[23]andcanbeavailable
usingwebcrawler)weexaminecoversmorethan58,523,079groupsand274,335,183users,of
which48,676,355groupshastheinformationwithallID,memberlist,anddate.Duetothe
limitof2000groupswhichareallowedforanordinaryusertojoin,34userswhojoinedmore
than2000groupsmusthavesuperiorpermissiongivenbyTencent,andthustheyareconsid-
eredastherobotsorthecustomerservicessetbyTencentandareexcludedfromouranalyzes.
Sincesomeusersdonotindicatehis/hergenderorage,orprovidesomeseeminglyfalseinfor-
mation,e.g.0yearold,weexcludeuserswithoutgenderinformationoryoungerthan10or
olderthan70.Overall,thereare273,204,518userswithgenderinformation,ofwhich42.5%
(116,135,972)arefemales,and244,521,321userswithagebetween10and70.Formostofthe
QQgroups,itsID,itsmemberlistwithgenderandage,andthedateofwhichitwasestablished
areknown.Theoldestandtheyoungestgroupsinourdatasetareformedon22ndSeptember,
2005and25thMarch,2011,respectively.Wethusonlyusedataupto25thMarch,2011.
Networksconstruction
Weexaminethefollowingtypesofnetworksembeddedinthedatasets:
1. User-grouphypergraphandbipartitenetwork—Ahypergraph[24,25]isagraphofnodes
andhyperedgeseachofwhichconnectstwoormorenodes.AsshowninFig1(a),thehyper-
graphinourdatasetdescribestheuser-grouprelationshipwithnodesrepresentingindivid-
ualusersandhyperedgesrepresentinggroups.Forinstance,userBisamemberofgroupG
1
andG ,andarethusconnectedtoAviahyperedgeG aswellastoCandDviahyperedge
2 1
G .Inthispaper,welabeltheresultsobtainedontheuser-grouphypergraphbysuperscript
2
H.Wealsoshowthecorrespondingbipartitenetwork[26,27]inFig1(b),whichisanequiv-
alentrepresentationofthehypergraphH.Thenodesinuppersideandbottomsiderespec-
tivelyareusersandgroups.TheresultsonthebipartitenetworkislabeledbysuperscriptB.
2. Groupnetwork—AsshowninFig1(c),groupnetworksinourcontextrefertoweightednet-
workswherenodesrepresentindividualgroups,andtwogroupsareconnectediftheyhave
atleastonecommonmember.Theweightontheedgeisdefinedasthenumberofcommon
usersbetweenthetwogroups.Forinstance,groupG andG inFig1(a)have3common
3 4
users,theedgeconnectingG andG inFig1(c)hasaweightof3.Inthispaper,welabelthe
3 4
resultsobtainedonthegroupnetworkbysuperscriptG.
3. Usernetwork—Tofocusonthebehaviorsofsocialgroups,theusernetworkinourcontext
isnottheordinaryfriendshipnetworkinQQ,butinsteadisaweightednetworkwhichonly
connectstwousersiftheyaremembersofatleastonecommongroup.Hence,allmembers
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EmpiricalStudiesontheNetworkofSocialGroups
Fig1.Schematicdiagramshowing(a)theuser-grouphypergraphH,(b)thebipartitenetworkB,(c)thegroupnetworkG,and(d)theusernetwork
U.Thedataiscomposedoffivegroupsdenotedbythecoloredellipsesin(a)andelevenusers.Thethicknessofedgesin(c)and(d)isproportionaltothe
weightontheedges.
doi:10.1371/journal.pone.0130538.g001
inagrouparefullyconnectedtoeachother.Theweightofanedgeconnectingapairof
usersisequaltothenumberofgroupstheybothjoin.AsshowninFig1(a),bothuserCand
DaremembersofG ,G andG ,andhencetheweightontheedgeconnectinguserCandD
2 3 4
inFig1(d)is3.Inthispaper,welabeltheresultsobtainedontheusernetworkbysuper-
scriptU.
ThenotationsusedthroughoutthepaperaresummarizedinTable1.
Results
TheStructuralPropertiesoftheUser-GroupHypergraphH
Thedistributionofsocialgroupsizeisoneofthemostinterestingfeaturesinasocialnetwork.
AsshowninFig1(a),thegroupsizesHisthetotalofnodenumberscoveringbyahyperedge.
AswecanseeinFig2(a),thedistributionP(sH)showsaslowandsmoothdecayintherange0
(cid:1)sH(cid:1)50.ThedecaybecomesfasterforsH>50andthecurvebecomesdiscontinuousatsH=
100,200,500and1000,duetothelimitationofgroupsizebyQQ.Wefindthatthebroken
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Table1.Notationsinthepaper.
Notations Description
sH ThesizeofgroupH,namelythetotalnumberofusersinagroup
hsHi Theaveragevalueofsizeofgroups
kH Thenumberofgroupsthatauserjoined,thenode’shyperdegree
hkHi Theaveragevalueofnode’shyperdegree
kH Themaximumvalueofjoinedgroupsofusersinagroup
max
kG Thenodedegreeingroupnetwork,namelythenumberofconnectedgroupsofagroup
KG Theweightednodedegreeingroupnetwork,namelythenumberofcommonusersbetweena
groupandothers
wG Theedge’sweightingroupnetwork,whichisdefinedasthenumberofcommonusers
betweentwogroups
wG Theeffectiveedge’sweightingroupnetwork,whichiscalculatedbytheresource-allocation
e
process.
dG Thedistancebetweengroupsingroupnetwork
hdGi Theaveragedistancebetweengroupsingroupnetwork
CG Thelocalclusteringcoefficientingroupnetwork
hCGi Theaveragevalueoflocalclusteringcoefficientingroupnetwork
kU Theuser’sdegreeinusernetwork,foranarbitraryuseri,kU(i)isthenumberofuserswhoare
inthesamegroupswithi
hkUi Theaverageuser’sdegreeinusernetwork
wU Theedge’sweightinusernetwork,whichisdefinedasthenumberofcommongroups
betweentwousers
wU Theeffectiveedge’sweightinusernetwork,whichiscalculatedbytheresource-allocation
e
process.
dU Thedistancebetweenusersinusernetwork
hdUi Theaveragedistancebetweenusersinusernetwork
a Theageofuseri
i
hai Theaveragevalueofuser’sage
P(a) Thedistributionofusers’ages
c Thecoefficientofvariationofage,c =σ /haiwhereσ isthestandarddeviationofmember
va va a a
age
c Thecoefficientofvariationofthenumberofneighboringusers,c =σ /hkUiwhereσ isthe
vu vu u u
standarddeviationofmemberdegreeinusernetwork
hc i Theaveragecoefficientofvariationofneighboringusers’ages
va f
hc i Theaveragecoefficientofvariationofneighboringusers’degree
vu f
doi:10.1371/journal.pone.0130538.t001
partsofthecurvecanbeenclosedbytwopowerlawswithexponent−3.5and−5.0,i.e.thetwo
dashedlinesinFig2(a).Theseexponentsaremorenegativethansimilarexponentsobserved
inothersocialnetworks,suggestingthatitismoredifficultforagrouptomaintainalarge
membercommunitythanforanindividualtomaintainalargenumberoffriends.Theresults
indicateamorehomogeneousnatureinthedistributionofgroupsize,probablybecausemain-
tainingsuchcloserelationshipinalargegroup,e.g.clubsororganizations,isnoteasy,which
limitsthegrowthofgroup.Ontheotherhand,weshowinFig2(b)adatacollapseofthediffer-
entbrokenpartsafterre-scaling,implyingthatformationmechanismsofgroupsaresimilar
regardlessoftheirsize.Andalso,therelationbetweenthenumberofgroupsandthenumberof
usersobeyspowerfunctionwithexponent1(TheinsetofFig2(c)).
Intuitively,weexpectoldergroupstohavealargersizesincetheyhavealongertimeto
accumulatemembers.Torevealthecorrelationbetweenthesizeofagroupandthedateof
whichitisformed,wecomputetheaveragesizeofgroupsestablishedonthesamedate.Aswe
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Fig2.StatisticsforthehypergraphH.(a)P(sH),thedistributionofgroupsizesH,withthedistributioninsemi-logscaleshownintheinset.Thetwodashed
linesinshowtherangeofthetailexponentofP(sH),namely−3.5(orange)and−5.0(magenta).(b)ThedatacollapseofthedifferentbrokenpartsonP(sH)
afterre-scaling,inwhichsH¼ð1þsH(cid:3)sH Þ=ð1þsH (cid:3)sH Þ,heresH andsH respectivelyaretheminimumandmaximumvalueofsHineachsection,and
c min max min min max
P (sH)isthecorrespondingre-scaledprobability.(c)Theaverage(pink)andthestandarddeviation(grey)ofgroupsizegivenspecificdateofestablishment,
c
andtheinsetshowsthescalingrelationshipbetweentotalofgroupsandtotalofusersateachdate.(d)ThedistributionP(kH)ofthenumberofjoinedgroupby
individualusers.P(kH)formaleandfemaleusersareshownintheinset.Thepinklinescorrespondtopower-lawfitswithexponent−3.82.
doi:10.1371/journal.pone.0130538.g002
canseeinFig2(c),theaveragesizehsHiisalmostindependentofthedateofestablishment,
whichiscontrarytoourbelief.Thisresultmayimplythatmostgroupsdonotgrowsignifi-
cantlyafterestablishment,andthegroupsizeismainlydeterminedbythenumberofusers
whojoinedthegroupshortlyafterthegroupwascreated.Itisbecausewhenagroupiscreated,
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itsinformationusuallyspreadsrapidlyinthecreator’ssocialcircle.Asaresult,mostinterested
usersjointhegrouponcetheyheardaboutit.Occasionally,asmallnumberofusersmayjoin
existinggroupsbutontheotherhand,someexistingmembersmayleavethegroupleadingto
anequilibriumgroupsize.Thiscertaintyongroupmemberscreatessomedifficultiesintothe
studiesonrecommendationalgorithmsforQQgroups.
Theabovepicturesarefurthersupportedbythestandarddeviationσ ofgroupsize,which
s
againdoesnotincreasewiththeageofagroup.Moreover,afterexcludingthegroupswithsize
closetothesizelimits(i.e.excludinggroupswithsizeintherange90–100,180–200,450–500),
theaveragesizehsHioftheremaininggroupsalsoshowsthesamephenomenon(theviolet
r
curveinFig2(c)).Thisbehaviorofconstantsizeisdifferentfromthoseobservationsinmany
otherslow-growingsocialnetworks.
Otherthanthegroupsize,thenumberofgroupsjoinedbyanindividualuserisalsoan
importantcharacteristicofasocialnetwork.Inthecontextofhypergraph,onecanrepresent
thenumberofgroupjoinedbyauserbythehyperdegreekHoftheuser.Fig2(d)showsthedis-
tributionP(kH)withatailwellfittedbyapowerlawwithexponent−3.82.Althoughtheexpo-
nentismorenegativethanmostoftheothersocialnetworks,apower-lawdecaydoesimply
thatuserswhichjoinedalargenumberofgroupsarepresent.Unlikepreviousstudieswhich
revealeddifferencesbasedongender,weobservedsimilarP(kH)forbothmaleandfemale
users.Inaddition,wefindobviouspositivecorrelationontherelationshipbetweentheaverage
valueofkHamonggroupmembersandthecorrespondinggroupsizesH.Furtheranalysis(see
Section1ofMaterialsandMethods)indicatesthatthispositivecorrelationreflectstheprefer-
enceofactiveusersinjoininglargegroups.
TheStructuralPropertiesofGroupNetworkG
Afterexaminingthemacroscopiccharacteristicsofgroups,wemoveontorevealtheirmicro-
scopicinteractions.Inthisrespect,theweightedgroupnetworkcharacterizesanindirectinter-
actionbetweengroupswhentheysharesomecommonmembers.Asareminder,twogroups
areconsideredconnectediftheysharesomecommonneighborsandtheweightoftheedgeis
thenumberofuserswhojoinedbothgroups.
AsshowninFig3(a),thedistributionP(kG)ofthegroupdegreekGshowsapowerlawwith
exponent−0.8whenkG<120andanotherpowerlawwithexponent−2.23whenkG>120.
Similarly,asshowninFig3(b),theweighteddegreedistributionP(KG)showsapowerlawwith
exponent−0.81whenKG<160andanotherpowerlawwithexponent−2.33whenKG>160.
Theresultsimplythatagrouponlysharememberswithasmallnumberofgroups,usuallyat
mostoftheorderO(102)amongthe58milliongroupsintheQQnetwork.Ontheotherhand,
thenumberofcommonmembersbetweenapairofgroups,i.e.theweightofedge,alsoobeysa
two-regionpower-lawasshowninFig3(c),withanexponent−5.94atthetail.Thisimpliesthat
thenumberofuserswhohaveinterestsinacommonpairofgroupsarelimitedtotheorderof
O(102).Furthermore,usingbipartitenetworkprojection,wecalculatetheeffectiveedge’sweight
wGthatreflectstheinfluenceofagrouponanotherone[28],andfindthatthefittingpower-
e
lawexponentofthedistributionofwGissmallerthantheoneofwG,indicatingthattheinflu-
e
encebetweengroupsismoreheterogeneous(seeSection2ofMaterialsandMethods).
Thedegreeofagroupisdependentontwofactors,namely(i)thenumberofusersinthe
group,and(ii)thetotalnumberofothergroupsjoinedbyitsmembers.Fig4(a)showstherela-
tionbetweenthegroupdegreekGandthegroupsizesH,suchthattherelationbetweensHand
thecorrespondinghkGiisgivenbythepinkcurve.Theresultsshowthatgroupdegreeincreases
withgroupsize,whichisexpectedsincethenumberofdifferentgroupsjoinedbythemembers
ofalargergroupshouldbeproportionatelyhigher.InFig4(b),asimilarstatisticsshowsthe
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Fig3.PropertiesofthegroupnetworkG.Thefiguresshow(a)thedistributionP(kG)ofgroupdegree,(b)thedistributionP(K )ofweightedgroupdegree
G
K ofG,and(c)thedistributionP(wG)ofedge’sweight.Theinsetsshowthesamecurvesinsemi-logscale.
G
doi:10.1371/journal.pone.0130538.g003
relationbetweenthegroupdegreekGandkH ,thelargestnumberofjoinedgroupsbyanindi-
max
vidualmemberinagroup.ThereasonissimilartothatinFig4(a),sincealargergrouphaspro-
portionatelymoreactivemembers,thelargestnumberofgroupjoinedbyanindividual
memberishigher.TheaverageofkGhasanobvioustransitionfromafastergrowthtoaslower
growth(seeFig4(b)),indicatingthatk ismorestronglydependentonkH whenkGissmaller
G max
Fig4.Theheatmapswhichshowthecorrelation(a)betweenkGandsH(a),and(b)betweenkGandkH .Thecolorscalecorrespondstothelog-
max
frequencyofoccurrence.ThePearsoncorrelationcoefficientsforlog(kG)vs.log(sH)andforlog(kG)vs.logðkH Þare0.92and0.91,respectively.Thepink
max
linesshowthecurvesontheirmeansalongverticalvalues,andthebluedashedlinein(a)showsthefittingpowerfunctionwithslope1.14.
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than100.ThisresultsimplythatwhenthegroupdegreekGissmall,theactiveusershaveasig-
nificantroleinimprovingkG.
Finally,weshowthattheQQgroupnetworkissparsebutshows“small-world”phenome-
non,similartothefriendshipnetworkofFacebook[2,29].ComparingtoFacebook,theaver-
agedegreeandaverageweighteddegreeareslightlysmallerinQQgroupnetwork,withvalues
108.8and133.6respectively.Thesedegreesaresmallgiventhelargesizeofthenetwork,indi-
catingthenetworkissparse.Toshowthe“small-world”phenomenon,werandomlysample
2×104pairsofgroupsandremarkablyfindthattheiraveragedistanceisonly3.70±0.004,indi-
catingtheupperlimitoftheaveragedistancebetweeneachtwousersisonly4.70,similartothe
fourdegreesofseparationobservedinFacebook(theaveragedistancebetweenusersis4.74)
[2].WealsocomputethelocalclustercoefficientCG
2n
CG ¼ T ð1Þ
kGðkG(cid:3)1Þ
for104randomchosengroups,suchthatn isthenumberofconnectionamongtheneighbors
T
ofthegroup.Weshowthefrequencyofthevalues(kG,CG)forindividualgroupinFig5.Aswe
cansee,CGisnegativelyrelatedtokGinaroughpower-lawrelationwithexponent−0.62,
whichissimilartotheFacebookcase[2].TheaveragevalueofCGis0.35,whichishighcom-
paredtotheothersocialnetworks.
Fig5.TheheatmapwhichshowsthecorrelationbetweenlocalclusteringcoefficientCGandthedegreekGingroupnetworkG.Thecolorscale
correspondstothelog-frequencyofoccurrenceover104randomlysampledgroups.Thepinkdashedlineshowsthefittingcurvewithslope−0.62onthe
meansalongverticalvalues.
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TheStructuralPropertiesofUserNetworkU
Asimilaranalysisisconductedfortheweightedusernetwork.WeshowinFig6thedegreedis-
tributionP(kU),whichhasapower-lawtailwithexponent−3.22,andanaveragevalue135.3.
Thedegreedistributionsformaleandfemaleusersdonotshowobviousdifferenceandare
showninthebottominsetofFig6.Byaveraging1600randompairsofusers,weobtainedthe
averagedistancebetweenapairofuserstobe4.36±0.015,whichissmallerthantheaverage
distance4.74observedinFacebookfriendshipnetwork[2].Theseresultsshowthattheuser
networkissparseandexhibitsasmall-worldphenomenon.Bycomparingthedegreedistribu-
tionP(kU)andtheweighteddegreedistributionP(wU)asshowninthetopinsetofFig6,we
observethatthelattercanbefittedwellbyadecayfunctionP(wU)(cid:4)10−5.45[log(wU)]−7.96,
whichisslowerthanpower-law.Itimpliesthatuserswithlargedegreearemorelikelytoshare
groupswithotherusers,resultinginalargeedgeweight,andthusashiftofthetailparttothe
right.Nevertheless,theeffectiveedge’sweightwUcalculatedbytheprojectionofbipartitenet-
e
workBobeysarapid-decayingtypeofdistribution,indicatingthatthedifferenceonthe
Fig6.DegreedistributionP(kU)intheusernetworkU.Thebottominsetshowsthesamedistributionovermaleandfemaleusersrespectively.Thetop
insetshowsthedistributionP(wU)ofedge’sweightwU.
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