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SpectralGraphTheory
SocialandTechnologicalNetworks
RikSarkar
UniversityofEdinburgh,2017.
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Project
• Proposalfeedbacktoday/tomorrow– Pleasesharewithyourteammates!
• ProjectguidelinesandIpsareuponthewebpage
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Project-teams
• Brainstorminteams.Submityourownproject• Theteamistohelpyouthinkabouttheproject,discussspecificissues
• Treatyourteammate’sprojectasanyotherbookorpaper–youcanreference/useit,butcannotclaimcredit!
• Youarefreetodiscusswithanybody.Givecreditforsignificantideas.
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Project--wriIng
• Donotkeepitfortheend!• Asyougo,putinplots,pictures,diagramsinthedocument.Youcanchange/removethemlater
• Putinsmallparagraphs,descripIonsastheyoccurtoyou–youwillnotrememberthisonthelastday.
• Rememberthethoughts,discussions,problems,ideasasyougoalong.ThiswillhelpyoutowriteaninteresIngreport.
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Project--wriIng
• Donotkeepitfortheend!• Asyougo,putinplots,pictures,diagramsinthedocument.Youcanchange/removethemlater
• Putinsmallparagraphs,descripIonsastheyoccurtoyou–youwillnotrememberthisonthelastday.
• Rememberthethoughts,discussions,problems,ideasasyougoalong.ThiswillhelpyoutowriteaninteresIngreport.
![Page 6: Spectral Graph Theory - The University of Edinburgh · 2017-10-31 · Spectral Graph Theory Social and Technological Networks Rik Sarkar University of Edinburgh, 2017. Project •](https://reader034.vdocument.in/reader034/viewer/2022042405/5f1efcd2aa4e847b3965bf7b/html5/thumbnails/6.jpg)
Topics
• Aretheretopicsyouwouldlikedisucssedinclass?LetmeknowonPiazza
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Spectralmethods
• Understandingagraphusingeigenvaluesandeigenvectorsofthematrix
• Wesaw:• Ranksofwebpages:componentsof1steigenvectorofsuitablematrix
• PagerankorHITSarealgorithmsdesignedtocomputetheeigenvector
• Today:otherwaysspectralmethodshelpinnetworkanalysis
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Laplacian
• L=D–A[Disthediagonalmatrixofdegrees]
• Aneigenvectorhasonevalueforeachnode• WeareinterestedinproperIesofthesevalues
2
664
1 �1 0 0�1 2 �1 00 �1 2 �10 0 �1 1
3
775 =
2
664
1 0 0 00 2 0 00 0 2 00 0 0 1
3
775�
2
664
0 1 0 01 0 1 00 1 0 10 0 1 0
3
775
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Laplacian
• L=D–A[Disthediagonalmatrixofdegrees]
• Symmetric.RealEigenvalues.• Rowsum=0.Singularmatrix.Atleastoneeigenvalue=0.
• PosiIvesemidefinite.Non-negaIveeigenvalues
2
664
1 �1 0 0�1 2 �1 00 �1 2 �10 0 �1 1
3
775 =
2
664
1 0 0 00 2 0 00 0 2 00 0 0 1
3
775�
2
664
0 1 0 01 0 1 00 1 0 10 0 1 0
3
775
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ApplicaIon1:Drawingagraph(Embedding)
• Problem:Computerdoesnotknowwhatagraphissupposedtolooklike
• Agraphisajumbleofedges
• Consideragridgraph:• Wewantitdrawnnicely
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Graphembedding• FindposiIonsforverIcesofagraphinlowdimension(comparedton)
• CommonobjecIve:PreservesomeproperIesofthegraphe.g.approximatedistancesbetweenverIces.Createametric– UsefulinvisualizaIon– Findingapproximatedistances– Clustering
• Usingeigenvectors– Oneeigenvectorgivesxvaluesofnodes– Othergivesy-valuesofnodes…etc
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Drawwithv[1]andv[2]
• Supposev[0],v[1],v[2]…areeigenvectors– Sortedbyincreasingeigenvalues
• PlotgraphusingX=v[1],Y=v[2]
• Producesthegrid
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IntuiIons:the1-Dcase
• Supposewetakethejtheigenvectorofachain
• Whatwouldthatlooklike?• Wearegoingtoplotthechainalongx-axis• Theyaxiswillhavethevalueofthenodeinthejtheigenvector
• Wewanttoseehowtheseriseandfall
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ObservaIons• j=0
• j=1
• j=2
• j=3
• j=19
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ForAllj
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ObservaIons
• InDim1grid:– v[1]ismonotone– v[2]isnotmonotone
• Indim2grid:– bothv[1]andv[2]aremonotoneinsuitabledirecIons
• Forlowvaluesofj:– Nearbynodeshavesimilarvalues• Usefulforembedding
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ApplicaIon2:Colouring• Colouring:AssigncolourstoverIces,suchthatneighboringverIcesdonothavesamecolour– E.g.Assignmentofradiochannelstowirelessnodes.Goodcolouringreducesinterference
• Idea:Higheigenvectorsgivedissimilarvaluestonearbynodes
• Useforcolouring!
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ApplicaIon3:Cuts/segmentaIon/clustering
• Findthesmallest‘cut’• Asmallsetofedgeswhoseremovaldisconnectsthegraph
• Clustering,communitydetecIon…
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Clustering/communitydetecIon
• v[1]tendstostretchthenarrowconnecIons:discriminatesdifferentcommuniIes
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Clustering:communitydetecIon
• MorecommuniIes• Spectralembeddingneedshigherdimensions
• Warning:itdoesnotalwaysworksocleanly
• Inthiscase,thedataisverysymmetric
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ImagesegmentaIonShi&malik’00
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Laplacianmatrix
• ImagineasmallanddifferentquanItyofheatateachnode(say,inametalmesh)
• wewriteafuncIonu:u(i)=heatati• Thisheatwillspreadthroughthemesh/graph• QuesIon:howmuchheatwilleachnodehavealerasmallamountofIme?
• “heat”canberepresentaIveoftheprobabilityofarandomwalkbeingthere
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Heatdiffusion
• Supposenodesiandjareneighbors– Howmuchheatwillflowfromitoj?
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Heatdiffusion
• Supposenodesiandjareneighbors• Howmuchheatwillflowfromitoj?• ProporIonaltothegradient:– u(i)-u(j)– thisissigned:negaIvemeansheatflowsintoi
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Heatdiffusion
• Ifihasneighborsj1,j2….• Thenheatflowingoutofiis:
=u(i)-u(j1)+u(i)-u(j2)+u(i)-u(j3)+…=degree(i)*u(i)-u(j1)-u(j2)-u(j3)-….
• HenceL=D-A
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TheheatequaIon
• ThenetheatflowoutofnodesinaImestep• ThechangeinheatdistribuIoninasmallImestep– TherateofchangeofheatdistribuIon
@u
@t= L(u)
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ThesmoothheatequaIon
• ThesmoothLaplacian:
• ThesmoothheatequaIon:
�f =@f
@t
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Heatflow
• Willeventuallyconvergetov[0]:thezerotheigenvector,witheigenvalue
• v[0]isaconstant:nomoreflow!
�0 = 0
v[0]=const
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Laplacian• ChangedimpliedbyLonanyinputvectorcanberepresentedbysumofacIonofitseigenvectors(wesawthislastImeforMMT)
• v[0]istheslowestcomponentofthechange– WithmulIplierλ0=0
• v[1]isslowestnon-zerocomponent– withmulIplierλ1
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Spectralgap• λ1–λ0
• Determinestheoverallspeedofchange• Iftheslowestcomponentv[1]changesfast– Thenoverallthevaluesmustbechangingfast– Fastdiffusion
• Iftheslowestcomponentisslow– Convergencewillbeslow
• Examples:– Expandershavelargespectralgaps– Gridsanddumbbellshavesmallgaps~1/n
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ApplicaIon4:isomorphismtesIng
• Eigenvaluesdifferentimpliesgraphsaredifferent
• Thoughnotnecessarilytheotherway
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Spectralmethods• Wideapplicabilityinsideandoutsidenetworks• Relatedtomanyfundamentalconcepts
– PCA– SVD
• Randomwalks,diffusion,heatequaIon…• ResultsaregoodmanyImes,butnotalways• RelaIvelytoproveproperIes• Inefficient:eig.computaIoncostlyonlargematrix• (Somewhat)efficientmethodsexistformorerestricted
problems– e.g.whenwewantonlyafewsmallest/largesteigenvectors