签订房屋租赁合同的十大注意事项

Lecture5:NetworkcentralitySlidesaremodifiedfromLadaAdamicMeasuresandMetricsKnowingthestructureofanetwork,wecancalculatevarioususefulquantitiesormeasuresthatcaptureparticularfeaturesofthenetworktopology..basisofmostofsuchmeasuresarefromsocialnetworkanalysisSofar,,Degreedistribution,Averagepathlength,DensityCentralityDegree,,Eigenvector,Katz,PageRank,Hubs,Closeness,Betweenness,…….SeveralothergraphmetricsClusteringcoefficient,Assortativity,Modularity,……2Characterizingnetworks:Whoismostcentral????3networkcentralityWhichnodesaremost‘‘central’??Definitionof‘central’variesbycontext/purposeLocalmeasure:degreeRelativetorestofnetwork::closeness,,betweenness,eigenvector((Bonacichpowercentrality),,Katz,,PageRank,……Howevenlyiscentralitydistributedamongnodes??Centralization,hubsandautthorities,……4centrality:who’’simportantbasedontheirnetworkpositionindegreeIneachofthefollowingnetworks,,XhashighercentralitythanYaccordingtoaparticularmeasureoutdegreebetweennesscloseness5OutlineDegreecentralityCentralizationBetweennesscentralityClosenesscentralityEigenvectorcentralityBonacichpowercentralityKatzcentralityPageRankHubsandAuthorities6Hewhohasmanyfriendsismostimportant.degreecentrality((undirected)Whenisthenumberofconnectionsthebestcentralitymeasure?peoplewhowilldofavorsforyoupeopleyoucantalkto(influenceset,informationaccess,,…))influenceofanarticleintermsofcitations((usingin-degree)7degree::normalizeddegreecentralitydividebythemax.possible,,i..e.((N--1)8Prestigeindirectedsocialnetworkswhen‘prestige’maybetherightwordadmirationinfluencegift-givingtrustdirectionalityespeciallyimportantininstanceswheretiesmaynotbereciprocated((e.g.diningpartnerschoicenetwork))when‘prestige’maynotbetherightwordgivesadviceto((canreversedirection)givesordersto((-””--)lendsmoneyto((-””-))dislikesdistrusts9Extensionsofundirecteddegreecentrality--prestigedegreecentralityindegreecentralityapaperthatiscitedbymanyothershashighprestigeapersonnominatedbymanyothersforarewardhashighprestige10Freeman’sgeneralformulaforcentralization:(canuseothermetrics,e.g..ginicoefficientorstandarddeviation)centralization:howequalarethenodes??Howmuchvariationisthereinthecentralityscoresamongthenodes?maximumvalueinthenetwork11degreecentralizationexamplesCD=0..167CD=0..167CD=1..012degreecentralizationexamplesexamplefinancialtradingnetworkshighcentralization:onenodetradingwithmanyotherslowcentralization:tradesaremoreevenlydistributed13whendegreeisn’’teverythingInwhatwaysdoesdegreefailtocapturecentralityinthefollowinggraphs??abilitytobrokerbetweengroupslikelihoodthatinformationoriginatinganywhereinthenetworkreachesyou……14OutlineDegreecentralityCentralizationBetweennesscentralityClosenesscentrality15betweenness:anothercentralitymeasureintuition:howmanypairsofindividualswouldhavetogothroughyouinordertoreachoneanotherintheminimumnumberofhops??whohashigherbetweenness,XorY?XY16Wheregjk=thenumberofgeodesicsconnectingj-k,andgjk=thenumberthatactoriison.Usuallynormalizedby::numberofpairsofverticesexcludingthevertexitselfbetweennesscentrality:definition17betweennessofvertexipathsbetweenjandkthatpassthroughiallpathsbetweenjandkdirectedgraph:((N--1)**(N--2)betweennessontoynetworksnon--normalizedversion:ABCEDAliesbetweennotwootherverticesBliesbetweenAand3othervertices:C,D,andECliesbetween4pairsofvertices(A,D)),(A,E)),(B,D)),(B,E))notethattherearenoalternatepathsforthesepairstotake,soCgetsfullcredit18betweennessontoynetworksnon--normalizedversion:19betweennessontoynetworksnon--normalizedversion:20brokerNodesaresizedbydegree,andcoloredbybetweenness.exampleCanyouspotnodeswithhighbetweennessbutrelativelylowdegree??Whatabouthighdegreebutrelativelylowbetweenness?21betweennessontoynetworksnon--normalizedversion:ABCEDwhydoCandDeachhavebetweenness1?Theyarebothonshortestpathsforpairs((A,E),and(B,E)),andsomustsharecredit::½+½==1CanyoufigureoutwhyBhasbetweenness3..5whileEhasbetweenness0.5??22AlternativebetweennesscomputationsSlightvariationsingeodesicpathcomputationsinclusionofselfinthecomputationsFlowbetweennessBasedontheideaofmaximumflowedge-independentpathselectioneffectstheresultsMaynotincludegeodesicpathsRandom--walkbetweennessBasedontheideaofrandomwalksUsuallyyieldsrankingsimilartogeodesicbetweennessManyotheralternativedefinitionsexistbasedondiffusion,transmissionorflowalongnetworkedges23ExtendingbetweennesscentralitytodirectednetworksWenowconsiderthefractionofalldirectedpathsbetweenanytwoverticesthatpassthroughanodeOnlymodification:whennormalizing,wehave(N-1)*((N-2)insteadof(N-1))*(N-2))/2,,becausewehavetwiceasmanyorderedpairsasunorderedpairsbetweennessofvertexipathsbetweenjandkthatpassthroughiallpathsbetweenjandk24DirectedgeodesicsAnodedoesnotnecessarilylieonageodesicfromjtokifitliesonageodesicfromktojkj25OutlineDegreecentralityCentralizationBetweennesscentralityClosenesscentrality26closeness::anothercentralitymeasureWhatifit’snotsoimportanttohavemanydirectfriends??Orbe““between”othersButonestillwantstobeinthe““middle”ofthings,,nottoofarfromthecenter27ClosenessisbasedonthelengthoftheaverageshortestpathbetweenavertexandallverticesinthegraphClosenessCentrality:NormalizedClosenessCentralityclosenesscentrality:definition28dependsoninversedistancetootherverticesclosenesscentrality:toyexampleABCED29closenesscentrality:moretoyexamples30degreenumberofconnectionsdenotedbysizeclosenesslengthofshortestpathtoallothersdenotedbycolorhowcloselydodegreeandbetweennesscorrespondtocloseness?31ClosenesscentralityValuestendt