CSE316:SOCIALNETWORKANALYSIS

Fall2017MarionNeumann

MIDTERMREVIEWQUESTIONS

Contents inthese slidesmaybesubject tocopyright. Somematerialsareadopted from: http://www.cs.cornell.edu/home/kleinber/networks-book, http://web.stanford.edu/class/cs224w/, http://www.mmds.org.

SIGNEDNETWORKS&BALANCE• Canyouexplainsignednetworksandthestructuralbalance?

Likemembersacrossagrouphave+edges.(Harary)Explainmoreaboutthat.Youkindofrushedthistowardstheendofclasswhenweweretalkingaboutthis.

• Canyoure-explainwhystablesignedgraphsareclusterable?Whycantheyalwaysbesplitintogroupsofedgeswithinternalpositiveedgesandexternalnegativeedges?

ànextslide

• Ifagraphisbalanced,doesthathaveanydirectimplicationsonnodecentrality?Ifso,why?à No

• Whendoexceptionstoabalancednetworkoccur,andhowcanwerecognizewhenimbalanceisrealisticallyexpected?à Never (in(large)realworldnetworks),expecttheentirenetworktobealways imbalanced

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SIGNEDNETWORKS&BALANCE

à Balancednetworksareclusterable [Harary,1953]

Simplerversion:balancedcomplete graphsareclusterableinto2groupswhereeveryonewithinthe2gropus (𝑋&𝑌)isfriendsandeveryoneacrossthe2groupsareenemies.

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à wecandividethenodesintogroupssuchthat• withineachgroupwehaveonly+edges• acrossgroupsyouhaveonly- edges

SIGNEDNETWORKS&BALANCESimplerversion- ProofSketch(cf.NCM Ch5p124):

• LetGbeacompletegraphwithatleastonenegativeedge• let’sconstructthetwogroups(constructiveproof)• pickanynodeinthenetwork,say𝐴

• put𝐴 andallitsfriendsinto𝑋• putitsenemiesinto𝑌CHECK(1) allnodesareassignedtoeither𝑋 or𝑌

à YES,because(ii) holds✔

(2) allconditions (i),(ii),(iii) holdà satisfiedbecause(i) holds✔ 4

Conditions tobesatisfied(i) Everytwonodes inXarefriends(ii) Everytwonodes inYarefriends(iii) EverynodeinXisanenemyofeverynodeinY

(i) (ii)

(iii)

Given(i) networkisbalanced(ii) networkiscomplete

Everytrianglehasaneven numberof– signs.

HOMOPHILY• Howdohomophilyandcharacteristics contributetothe

graphs,ifatall?• I'mabitconfusedaboutsocialinfluence,howit'srepresented

inagraph,andinwhatwaysthegraphmaybealteredduetosocialinfluence(whatwouldthatchangeinthegraphlooklike?).

à theyareresponsibleforedge(relationship)formations

• Whatistherelationshipbetweenhomophilyandstrongtriadicclosure.ifanetworkshowssignsofeitherdoesthatimplytheother?à yessomewhatà e.g.wecouldpostulatethathomophily causestriadicclosureà triadicclosureisamathconceptà homohily isaconceptfromsocialstudies

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AFFILIATIONNETWORKS• Canwegooveraffiliationnetworksagain

à affiliationandsocial-affiliationnetworksincorporatecontextualfactorsintothenetwork

à thisrepresentationhelpstoexplainrelationshipsbetweenactors(focalclosurevs.triadicclosure)

à NCM 4.3&4.4

• Howdotherolesofactorsandfocidifferinasocialaffiliationnetwork?(aresomemorelikelytobehubs?)à goodquestion(thisdependsmostlikelyonthenetworkyouarelookingat.Youcananswerthisquestionbydoinganempiricalstudy.)

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STRONGTRIADICCLOSURE• CanyouwalkthroughtheproofbycontradictionofHomework

2Problem1b?(i.e."Ifstrongtriadicclosureissatisfied,thenlocalbridgesbetweennodeswithatleastoneotherexistingstrongtieareweakties.")à Proofbycontradiction• Assume𝑨 satisfiesStrongTriadicClosure

andhasonestrongedgeto𝑪• Let𝑨 −𝑩 belocalbridgeandastrongtie• Now:Triadhas2strongties• Then𝑩 −𝑪mustexistbecauseofStrongTriadicClosure• Butthen𝑨−𝑩 cannot bebridge!à contradiction

• What'sthedifferencebetweenstrongandweakedge.Andwhat'stheirfrequency'simpactonthegraph?à edgeformationsaremorelikelyiftriadswithstrongedgesare

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EIGENVECTORCENTRALITY• I'mconfusedabouteigenvectorcentrality.Pleaseexplain.• Couldyoupointustoanexampleforacalculationordooneforus?• I'mstillconfusedonwhatEigenvectorcentralityrepresentsaswellas

howtocalculateit.Ialsodon'tunderstandhowpoweriterationworks.Thanks!

• CanyoureviewEigenvectorcentrality:Howthecalculationworks(PowerIterationAlgorithm?)andwhatdoesitmeaninthecontextofthesocialnetworks?

à Example3.2&3.3inSSMCH3.1.2• Canyouexplaineigenvectorcentralityintermsofitsrelationshipto

degreecentrality?Whenwelookatgraphsofcentralitymeasuresitlooksliketheareasofhighesteigenvectorcentralityareinthespotswheretherearethemostnodesthatscorehighindegreecentrality.Arethesetwoactuallyrelatedoristhisjustcoincidence?

à YES!EigenvectorcentralitygeneralizesdegreecentralityDEMO:MeasuresOfNetworkCentrality

• Canyoureviewdifferentcentralitymeasuresandhoweachlookslikeinactualpicturesofgraphs?àQuiz (linkedfromCourseCalendar)

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EIGENVECTORCENTRALITY

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Seealso:https://www.youtube.com/watch?v=DGVvm-j-NG4

POWERITERATION

Information Propagation in Graphs• Random walks (RWs) used for learning on the node level

• iteratively update node-label distributions Pt

• e.g. label di�usion or label propagation Pt+1 TPt

Initialization:

?

?

?

November 9, 2015 Background 910

POWERITERATION

Information Propagation in Graphs• Random walks (RWs) used for learning on the node level

• iteratively update node-label distributions Pt

• e.g. label di�usion or label propagation Pt+1 TPt

1. Iteration:

November 9, 2015 Background 911

POWERITERATION

Information Propagation in Graphs• Random walks (RWs) used for learning on the node level

• iteratively update node-label distributions Pt

• e.g. label di�usion or label propagation Pt+1 TPt

2. Iteration:

November 9, 2015 Background 912

POWERITERATION

Information Propagation in Graphs• Random walks (RWs) used for learning on the node level

• iteratively update node-label distributions Pt

• e.g. label di�usion or label propagation Pt+1 TPt

3. Iteration:

November 9, 2015 Background 913

POWERITERATION

Information Propagation in Graphs• Random walks (RWs) used for learning on the node level

• iteratively update node-label distributions Pt

• e.g. label di�usion or label propagation Pt+1 TPt

4. Iteration:

November 9, 2015 Background 914

EXCESSDEGREEDISTRIBUTION• What'sthedifferencebetweendegreedistributionandexcessdegreedistribution?Whatdoesonetellusthattheotherdoesn't?

• Whyisitnecessarywhenwecancalculateitfromdegreedistribution?

• Canwegooverexcessdegreedistribution(inHW2)andwhyitdiffersfromrandomversusrealworldgraphs?Conceptually,whatexactlydoesthismeasure?

à degreedistributiontellsuswhatthenodedegreeisforarandomlychosennode

à excessdegreedistributiontellsuswhatthenodedegree(-1)isforanendnodeforarandomlychosenedge

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EXCESSDEGREEDISTRIBUTIONà excessdegreedistribution:

à mean =

=avg.degreeoftheneighborofanodeà Result:avg.degreeoftheneighborofanode> avg.degreeofanode

Yourfriendshavemorefriendsthanyou.Yourcollaboratorshavemorecollaboratorsthanyou.

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configurationmodelestimate

)𝑘𝑞,,

= )𝑘𝑘𝑝,𝑘

,

= 𝑘/

𝑘

Note:Calculationsusingsimplifiednetworkmodels(suchastheconfigurationmodel)cangiveyouafeel for• thetypesofeffectsonemightexpecttosee• generaldirectionsofchangesinquantitiesButtheyusuallydonot givequantitatively accuratepredictions.

DEGREEDISTRIBUTION

• IsthedegreedistributionofarandomgraphreallyNormallydistributed?ItseemstometobePoissonDistributed.à Yes(forsmall𝑛)

17DEMO:ComplexNetworks

DEGREEDISTRIBUTION

• IsthedegreedistributionofarandomgraphreallyNormallydistributed?ItseemstometobePoissonDistributed.à BinomialdistributionPoissonDistribution

𝑛 → ∞

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InthelimitthePoissondistribution isagoodapproximationofaBinomialdistribution.

DEGREEDISTRIBUTION

• IsthedegreedistributionofarandomgraphreallyNormallydistributed?ItseemstometobePoissonDistributed.à Binomialdistribution,notNormalDistributionà Numberofsuccessesinasequenceof𝑛 − 1

independentyes/noexperiments

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RANDOMGRAPHMODELS• Whyarerandomgraphssignificanttostudy?Howdothey

helpusandwhyareweeveninterestedinsimulatingsocialnetworks?Aren’trealworldnetworksmoreinteresting?à Yes,butyouneedtobeabletocompare thestatisticsyou

computedfromthegraphrepresentingtherealnetworktoanullmodeltobeabletointerpret them

• Whyisaveragepathlengthimportant?Whymustweapproximaterealworldaveragepathlengthsinourmodels?à youwanttomeasureifyournetworkisasmallworldà crucialforinformationfloworwhenstudyingspreadofdisease)à wedonot approximatetheavg.pathlengthinrealworld

networks

• Canyouexplainphasetransition?à inphysicsorintherandomgraphmodel?

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DIRECTEDGRAPHS• Allofourworksofarhascoveredundirectedgraphs.Aretherereallifesituationswheregraphsshouldbedirected,andhowwouldourmethodsofstudyingthesegraphschange?Forexample,howwouldtheclusteringcoefficientbecalculatedinadirectedgraph?

à socialnetworksaretypicallyundirectedà examplesfordirectedgraphs(cf.NCM Ch 2.4)• informationlinkagegraphs(webgraph)• dependencynetwork(flowchart)• foodwebs(cf.DSCN Ch 1)

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OTHERQUESTIONS• Whatistheprobabilityalocalbridgeexistsinarandomgraphofnnodesandmedges?à goodquestion(I’llmakeitaBONUSquestiononthenexthomework…)

• Canyouexplainmorewhatthepointofusinglog-logscaletoplotthedegreeprobabilityis?Iunderstanditmakesarealworldnetworkclearertosee.Butdoesn’titalsoaddstotheburdenofvisualinterpretation?Ijustfindsimpledistributionplotismoreintuitivetoread.à log-logplotsshowpowerlawdistributions

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WHAT’SONTHEEXAMQUESTIONS

• whatisalistofmeasurementsweshouldknowfortheexam(ie betweennesscentrality,excessdegreedistribution,harmoniccentrality)à yes,thoseareexamrelevantandeverythingelsewecoveredinclassoronthehomeworkproblems

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LOGISTICSQUESTIONS

• Canwegetaccesstothevideorecordings?à No

• Canwegetanoutlineforthecoursebecauseeverythingseemsverydisconnectedatthemoment?à CourseCalendar&Roadmap

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