签订房屋租赁合同的十大注意事项.ppt

Lecture5 Networkcentrality SlidesaremodifiedfromLadaAdamic 太原房产网 MeasuresandMetrics Knowingthestructureofanetwork wecancalculatevarioususefulquantitiesormeasuresthatcaptureparticularfeaturesofthenetworktopology basisofmostofsuchmeasuresarefromsocialnetworkanalysisSofar Degreedistribution Averagepathlength DensityCentralityDegree Eigenvector Katz PageRank Hubs Closeness Betweenness SeveralothergraphmetricsClusteringcoefficient Assortativity Modularity Characterizingnetworks Whoismostcentral networkcentrality Whichnodesaremost central Definitionof central variesbycontext purposeLocalmeasure degreeRelativetorestofnetwork closeness betweenness eigenvector Bonacichpowercentrality Katz PageRank Howevenlyiscentralitydistributedamongnodes Centralization hubsandautthorities centrality who simportantbasedontheirnetworkposition indegree Ineachofthefollowingnetworks XhashighercentralitythanYaccordingtoaparticularmeasure outdegree betweenness closeness Outline DegreecentralityCentralizationBetweennesscentralityClosenesscentralityEigenvectorcentralityBonacichpowercentralityKatzcentralityPageRankHubsandAuthorities Hewhohasmanyfriendsismostimportant degreecentrality undirected Whenisthenumberofconnectionsthebestcentralitymeasure peoplewhowilldofavorsforyoupeopleyoucantalkto influenceset informationaccess influenceofanarticleintermsofcitations usingin degree degree normalizeddegreecentrality dividebythemax possible i e N 1 Prestigeindirectedsocialnetworks when prestige maybetherightwordadmirationinfluencegift givingtrustdirectionalityespeciallyimportantininstanceswheretiesmaynotbereciprocated e g diningpartnerschoicenetwork when prestige maynotbetherightwordgivesadviceto canreversedirection givesordersto lendsmoneyto dislikesdistrusts Extensionsofundirecteddegreecentrality prestige degreecentralityindegreecentralityapaperthatiscitedbymanyothershashighprestigeapersonnominatedbymanyothersforarewardhashighprestige Freeman sgeneralformulaforcentralization canuseothermetrics e g ginicoefficientorstandarddeviation centralization howequalarethenodes Howmuchvariationisthereinthecentralityscoresamongthenodes maximumvalueinthenetwork degreecentralizationexamples CD 0 167 CD 0 167 CD 1 0 degreecentralizationexamples examplefinancialtradingnetworks highcentralization onenodetradingwithmanyothers lowcentralization tradesaremoreevenlydistributed whendegreeisn teverything Inwhatwaysdoesdegreefailtocapturecentralityinthefollowinggraphs abilitytobrokerbetweengroupslikelihoodthatinformationoriginatinganywhereinthenetworkreachesyou Outline DegreecentralityCentralizationBetweennesscentralityClosenesscentrality betweenness anothercentralitymeasure intuition howmanypairsofindividualswouldhavetogothroughyouinordertoreachoneanotherintheminimumnumberofhops whohashigherbetweenness XorY X Y Wheregjk thenumberofgeodesicsconnectingj k andgjk thenumberthatactoriison Usuallynormalizedby numberofpairsofverticesexcludingthevertexitself betweennesscentrality definition betweennessofvertexi pathsbetweenjandkthatpassthroughi allpathsbetweenjandk directedgraph N 1 N 2 betweennessontoynetworks non normalizedversion A B C E D AliesbetweennotwootherverticesBliesbetweenAand3othervertices C D andECliesbetween4pairsofvertices A D A E B D B E notethattherearenoalternatepathsforthesepairstotake soCgetsfullcredit betweennessontoynetworks non normalizedversion betweennessontoynetworks non normalizedversion broker Nodesaresizedbydegree andcoloredbybetweenness example Canyouspotnodeswithhighbetweennessbutrelativelylowdegree Whatabouthighdegreebutrelativelylowbetweenness betweennessontoynetworks non normalizedversion A B C E D whydoCandDeachhavebetweenness1 Theyarebothonshortestpathsforpairs A E and B E andsomustsharecredit 1CanyoufigureoutwhyBhasbetweenness3 5whileEhasbetweenness0 5 Alternativebetweennesscomputations SlightvariationsingeodesicpathcomputationsinclusionofselfinthecomputationsFlowbetweennessBasedontheideaofmaximumflowedge independentpathselectioneffectstheresultsMaynotincludegeodesicpathsRandom walkbetweennessBasedontheideaofrandomwalksUsuallyyieldsrankingsimilartogeodesicbetweennessManyotheralternativedefinitionsexistbasedondiffusion transmissionorflowalongnetworkedges Extendingbetweennesscentralitytodirectednetworks Wenowconsiderthefractionofalldirectedpathsbetweenanytwoverticesthatpassthroughanode Onlymodification whennormalizing wehave N 1 N 2 insteadof N 1 N 2 2 becausewehavetwiceasmanyorderedpairsasunorderedpairs betweennessofvertexi pathsbetweenjandkthatpassthroughi allpathsbetweenjandk Directedgeodesics Anodedoesnotnecessarilylieonageodesicfromjtokifitliesonageodesicfromktoj k j Outline DegreecentralityCentralizationBetweennesscentralityClosenesscentrality closeness anothercentralitymeasure Whatifit snotsoimportanttohavemanydirectfriends Orbe between othersButonestillwantstobeinthe middle ofthings nottoofarfromthecenter Closenessisbasedonthelengthoftheaverageshortestpathbetweenavertexandallverticesinthegraph ClosenessCentrality NormalizedClosenessCentrality closenesscentrality definition dependsoninversedistancetoothervertices closenesscentrality toyexample A B C E D closenesscentrality moretoyexamples degreenumberofconnectionsdenotedbysizeclosenesslengthofshortestpathtoallothersdenotedbycolor howcloselydodegreeandbetweennesscorrespondtocloseness Closenesscentrality ValuestendtospanarathersmalldynamicrangetypicaldistanceincreaseslogarithmicallywithnetworksizeInatypicalnetworktheclosenesscentralityCmightspanafactoroffiveorlessItisdifficulttodistinguishbetweencentralandlesscentralverticesasmallchangeinnetworkmightconsiderablyaffectthecentralityorderAlternativecomputationsexistbuttheyhavethe