SocialandTechnologicalNetworkDataAnalytics

Lecture2:SmallWorldandWeakTies

Prof.CeciliaMascolo

InThisLecture

• Wewillcomparerandomnetworkswithrealnetworks

• Wewillintroducetheconceptofsmallworldnetworks

• Wewillintroducetheconceptofweaktiesandillustratetheirimportance

ClusteringCoefficientofRealNetworks

• From[WattsandStrogatz,1998]• Characteristicpathlengthandclusteringcoefficientforsomerealnetworksandforrandomnetworkswithsamenumberofnodesandaveragenumberofedgespernode.

• Aimistocheckifrandomgraphscanmodelrealnetworks.

RealNetworksvs RandomNetworks

• FilmActors:actorsinmoviestogether• Powergrid:thenetworkoftheelectricitygenerators

• C.elegans:networkofneuronsofaworm• LiscomparablewhileCisverydifferent

RandomNet:samesizeandaveragedegree

SmallWorldModel

• Watts&Strogatz builtamodelwhichwasabletocapturethesecharacteristics.

• Startwithregularlattice– Increaseaprobabilitypof“rewiring”anodetoanothernode.

–Whenpveryhighthelatticewouldbecomearandomgraph.

SmallWorldModel(2)

HowareLandCinthismodel?• ThereisazonewhereCishighandLislow• Thesearesmallworldnetworks

RandomNets

Lattice

OtherRealNetworksExamples

AnalysisofMessengerNetwork

• [Leskovec andHorvitz2008]analyzedalargedatasetoftheMicrosoftMessenger.

• CommunicationNetworkcontained180millionusersand1.3billionconversationsin1month.

• BuddyNetworkcontained240millionusers.• 99.9%usersbelongedtoaconnectedcomponent.

AnalysisofaMessengerNetwork• Averageshortestpathis6.6(confirmingMilgram’s study).

• Althoughsomelongerpathsupto29.• Averageclusteringcoefficientisquitehigh:0.137.

AgainonClusteringCoefficient• Wehaveintroducedtheclusteringcoefficient.Thisindicates:– ThenumberoftrianglesincludingnodeA.– HowconnectedthefriendsofAare.

• Triadicclosure:ifCandBareconnectedtoAthereisanincreasedlikelihoodthattheywillbeconnectedamongthemselvesinfuture.

A B

C DE F

[Granovetter’74]

• Granovetter interviewedpeopleabouthowtheydiscoveredtheirjobs– Mostpeopledidsothroughpersonalcontacts– Oftenthepersonalcontactsdescribedasacquaintancesandnotclosefriends

• Basicintuitiononthisis:closefriendsarepartoftriadclosuresandwouldknowwhatyouknowandwouldknowotherswhowouldknowwhatyouknow

• Wewillexplainthismoreformally…

Bridges

• EdgebetweenAandBisabridge if,whendeleted,itwouldmakeAandBliein2differentcomponents

A

B

LocalBridges

• Anedgeisalocalbridgeifitsendpointshavenofriendsincommon– Ifdeletingtheedgewouldincreasethedistanceoftheendpointstoavaluemorethan2.

A

B

StrongTriadicClosureProperty(STPC)

• Linksbetweennodeshavedifferent“value”:strongandweakties– E.g:Friendshipvs acquaintances

• StrongTriadicClosureProperty(Granovetter):IfanodeAhastwostronglinks(toBandC)thenalink(strongorweak)mustexistbetweenBandC.

LocalBridgesandWeakTies• IfnodeAsatisfiestheSTCPandisinvolvedinatleasttwostrongtiesthenanylocalbridgeitisinvolvedinmustbeaweaktie.(Proofbycontradiction)

(assumingSTCP)Ifthereareenoughstrongtiesinthenetworkthenlocalbridgesmustbeweakties

B

CA

S

S

ForACandABtobeastronglinkSCTPsaysBCmustexistbutlocalbridgedefinitionsaysitmustnot

RealDataValidation

• Granovetter’s theoryabouttheimportanceofweaktiesremainednotvalidatedforyearsforlargesocialnetworksduetothelackofdata.

• [Onnela etal’07]testeditoveralargecell-phonenetwork(4millionsusers):– Edgebetweentwousersiftheycalledeachotherwithinthe18monthsperiod.

– Dataexhibitsagiantcomponent(84%).– Edgeweight:timespentinconversation.

Onnela etal.2007

• Extendingthedefinitionoflocalbridge• Given:• Neighbourhood overlap:

Numberofnodeswhoareneighbours ofbothA&BNumberofnodeswhoareneighbours ofatleastAorB

• Whenthenumeratoris0thequantityis0.– Numeratoris0whenABisalocalbridge

• Thedefinitionfinds“almostlocalbridges”(~0)

A B

Neighbourhoodoverlap

RelationshipofOverlapwithTieStrength

• Red:randomshuffledweightsoverlinks.

• Blue:realones.Correlationwithtiestrength.

Anomaly

Tiestrength:cumulativetiestrengthsmallerthanw

Overlap

Realtieweightsinaportionofthegraph(aroundarandomnode)

A=RealB=Randomlyshuffled

Effectofedgeremoval

Overlapbasedlinkremoval

Weaktiesmatter!

• Wehavejustseenthatweaktiesmatterandiftheyareremoved,theyleadtoabreakdowninthenetwork.

• Ifstrongtiesareremovedtheyleadtoasmoothdegradingofthenetwork

Differenceofimportanceofweaktiesinsocialandothernetworks

• Theimportanceofweaktiesisspecifictosocialnetworks

• Inbiologicalandspatialnetworks:– Deletinganimportantroad[strongtie]damagesthenetworkmore

– Acentralveininaleafismoreimportantthansmallerveins

Tiestrengthmatters:FacebookExample

• Facebook dataanalysisofonemonthofdata• Fournetworks:– Declaredfriendship– Reciprocalcommunication(messages)– Onewaycommunication–Maintainedrelationship:clickingoncontentonnewsfeedfromotherfriendorvisitingprofilemorethanonce.

Whatdoesitlooklike?(onerandomuser)

ActiveNetworkSize:numberoflinks

Declaredfriends

Newsfeedeffect

AnotherstudyonFBshowstheimpactoftiesoverinformationdissemination• 3monthsofFBdata• 253millionusers(profileandlocation)

• Measuringeffectoftiestrengthonsharing

Howdidtheymeasuretiestrength?

• Privateinteractions• Publicinteractions(comments)

• Coappearance inpictures

• Involvementinthesamepostwithcomments

Strongtiesaremoreinfluential

However…

Strongtiesaremoreinfluential buttheireffectisnotlargeenoughtocompensatetheabundanceofweakties…

TwitterAnalysis

• Huberman atal.haveanalyzedstrongandweaktiesinTwitter.

• The“followers”graphinTwitterisdirected– Someonecanfollowsomeoneelsewhodoesnotfollowhim

• Messagesof140charscanbeposted• Messagescanbeaddressedtospecificusers(althoughtheystayreadabletoall)

• Weakties:usersfollowed• Strongties:userstowhomtheusersentatleast2messagesintheobservationperiod

Twitter

Followees

Numbero

fuser’sstrongties

Numberofstrongtiesstaysbelow~50

Summary

• Smallworldnetworkmodelsareabletocaptureagoodquantityofrealnetworks– Theyhavecharacteristicpathlengthcomparabletorandomnetworks.

– Butmuchhigherclusteringcoefficient.• Wehaveintroducedweakandstrongtiesandshownexampleofapplicationonrealnetworks

References• Kleinberg’sbook:Chapter3and20.

• Collectivedynamicsof'small-world' networks. Watts,D.J.;Strogatz,S.H.(1998).Nature 393(6684):409–10.

• Structureandtiestrengthsinmobilecommunicationnetworks.J.P.Onnela,J.Saramaki,J.Hyvonen,G.Szabo,D.Lazer,K.Kaski,J.Kertesz,A.L.Barabasi.ProceedingsoftheNationalAcademyofSciences,Vol.104,No.18.(13Oct2006),pp.7332-7336.

• Maintainedrelationships onfacebook.CameronMarlow,LeeByron,TomLento,andItamar Rosenn.2009.On-lineathttp://overstated.net/2009/03/09/maintained- relationships-on-facebook.

• Theroleofsocialnetworksininformationdiffusion.Eytan Bakshy,ItamarRosenn,CameronMarlow,andLada Adamic.2012.InProceedingsofthe21stinternationalconferenceonWorldWideWeb (WWW'12).ACM,NewYork,NY,USA,519-528.

• Socialnetworksthatmatter:Twitterunderthemicroscope. BernardoA.Huberman,DanielM.Romero,andFangWu.FirstMonday,14(1),January2009.