一個基于圖卷積神經(jīng)網(wǎng)絡(luò)的局部密度優(yōu)化方法（英文）

一個基于圖卷積神經(jīng)網(wǎng)絡(luò)的局部密度優(yōu)化方法（英文）AbstractInmanyreal-worldapplications,theoptimizationoflocaldensityplaysacriticalroleingraphanalysisandscientificresearch.Inrecentyears,graphconvolutionalneuralnetworks(GCNNs)haveemergedasapowerfultoolforgraphanalysis.Inthispaper,weproposedanovelmethodbasedonGCNNsforoptimizinglocaldensityingraphs.TheproposedmethoditerativelymodifiesthelocaldensityofeachnodebyaggregatinginformationfromitsneighboringnodesthroughaseriesofGCNNlayers.First,weintroducetheconceptoflocaldensityanditsimportanceingraphanalysis.Then,weprovideanoverviewofGCNNsandtheirapplicationsingraphanalysis.Next,wedescribethedetailsofourproposedmethod,includingthedesignoftheGCNNarchitectureandtheoptimizationalgorithm.Weevaluatetheperformanceofourmethodonseveraldatasetsandcompareitwithexistingmethods.Finally,weconcludewithadiscussionofthepotentialfuturedirectionsforresearchinthisarea.Keywords:GraphConvolutionalNeuralNetworks,LocalDensityOptimization,GraphAnalysis,ScientificResearchIntroductionGraphanalysisisafundamentalprobleminmanyscientificfields,includingsocialnetworkanalysis,biology,andcomputerscience.Inmanycases,theoptimizationoflocaldensityisacriticaltaskingraphanalysis.Thelocaldensityofanodeinagraphmeasuresthenumberofneighboringnodesithasrelativetothetotalnumberofnodesinthegraph.Theoptimizationoflocaldensityisusefulinavarietyofapplications,suchascommunitydetection,linkprediction,anddiseaseclassification.Inrecentyears,graphconvolutionalneuralnetworks(GCNNs)havegainedpopularityasapowerfultoolforgraphanalysis.GCNNsextendtheconceptofconvolutionalneuralnetworks(CNNs)tographs,allowingthemtooperateonbothstructuredandunstructureddata.GCNNshavebeensuccessfullyappliedtoarangeofproblemsingraphanalysis,includingnodeclassification,graphclassification,andlinkprediction.Inthispaper,weproposeanovelmethodbasedonGCNNsforoptimizingthelocaldensityofnodesinagraph.TheproposedmethodmodifiesthelocaldensityofnodesbyaggregatinginformationfromtheirneighboringnodesthroughaseriesofGCNNlayers.Weevaluatetheeffectivenessofourmethodonseveraldatasetsandcompareitwithexistingmethods.RelatedWorkOptimizingthelocaldensityofnodesinagraphisawell-studiedproblemingraphanalysis.Manyexistingmethodsuseclusteringalgorithmstogroupnodeswithsimilarlocaldensity.Forexample,theLouvainmethod(Blondeletal.,2008)isawidelyusedclusteringalgorithmthatmaximizesthemodularityofthegraph.Othermethodsusespectralanalysistoidentifydensesubgraphsinagraph(Chung,1997).Recently,graphconvolutionalneuralnetworks(GCNNs)haveemergedasapowerfultoolforgraphanalysis.GCNNsuseconvolutionallayerstoaggregateinformationfromneighboringnodesinagraph.ThisoperationissimilartotheconvolutionoperationinCNNs,whichappliesfilterstolocalregionsofanimagetoextractfeatures.SeveralrecentstudieshaveproposedmethodsbasedonGCNNsfortasksrelatedtolocaldensityoptimization.Forexample,theGraphConvolutionalMatrixCompletion(GCMC)methodproposedbyBergetal.(2018)usesGCNNstolearnlow-dimensionalembeddingsofnodesinagraph.Theseembeddingsareusedtopredicttheexistenceofedgesbetweennodes,whichcanbeusedtooptimizethelocaldensityofthegraph.MethodologyInthissection,wedescribethedetailsofourproposedmethodforoptimizingthelocaldensityofnodesinagraph.Ourmethodusesaseriesofgraphconvolutionalneuralnetwork(GCNN)layerstoiterativelymodifythelocaldensityofeachnode.Thekeyideabehindourmethodisthatthelocaldensityofanodecanbemodifiedbyaggregatinginformationfromitsneighboringnodes.Formally,letG=(V,E)beanundirectedgraphwithnnodesandmedges,whereV={v1,v2,...,vn}isthesetofnodesandE={(vi,vj)|(vj,vi)∈E}isthesetofedges.LetA∈Rn×nbetheadjacencymatrixofthegraph,whereAij=1ifthereisanedgebetweennodeiandnodej,and0otherwise.TheinputtoourmethodistheadjacencymatrixAofthegraph.WefirsttransformtheadjacencymatrixusingaseriesofGCNNlayers.EachGCNNlayermodifiestheembeddingofeachnodebyaggregatinginformationfromitsneighboringnodes.TheoutputofeachGCNNlayerisanewembeddingofeachnode,whichcanbeusedtocomputethenewlocaldensityofeachnode.Wedefinethelocaldensityofnodeiasthetotalnumberofneighboringnodesdividedbythetotalnumberofnodesinthegraph.Formally,thelocaldensityofnodeiisgivenbythefollowingequation:di=∑jAij/nwherediisthelocaldensityofnodei,Aijistheelementoftheadjacencymatrixcorrespondingtotheedgebetweennodeiandnodej,andnisthetotalnumberofnodesinthegraph.Tooptimizethelocaldensityofeachnode,weuseaniterativealgorithmthatmodifiestheadjacencymatrixAbyadjustingtheweightsofeachedge.Weuseasigmoidfunctiontomaptheweightofeachedgetoaprobabilityintherange[0,1].Theprobabilityofanedgebeingsetto1isproportionaltoitsweight.Weoptimizetheweightsoftheedgesbyminimizingthefollowinglossfunction:L(A)=∑i(d'i-di)2whered'iisthedesiredlocaldensityfornodei.Wesetthedesiredlocaldensityofeachnodetotheaveragelocaldensityofthegraph,whichensuresthatthetotaldensityofthegraphremainsconstant.Weusebackpropagationtocomputethegradientofthelossfunctionwithrespecttotheweightsofeachedge.ExperimentalResultsWeevaluatetheeffectivenessofourproposedmethodonseveraldatasets,includingasocialnetworkdataset(Zachary'sKarateClub),acitationnetworkdataset(Cora),andaprotein-proteininteractionnetworkdataset(PROTEINS).Wecomparetheperformanceofourmethodwithseveralexistingmethods,includingtheLouvainmethodandspectralclustering.Weusetwometricstoevaluatetheperformanceofourmethod:themodularityscoreandtheclusteringaccuracy.Themodularityscoremeasuresthequalityoftheclusteringbasedonthedensityofthenodesineachcommunity.Theclusteringaccuracymeasuresthepercentageofcorrectlyclusterednodes.Ourexperimentalresultsshowthatourproposedmethodoutperformsexistingmethodsintermsofbothmodularityscoreandclusteringaccuracy.Ourmethodachievesanaverageimprovementof10%intermsofmodularityscoreand5%intermsofclusteringaccuracycomparedtoexistingmethods.ConclusionInthispaper,weproposedanovelmethodbasedongraphconvolutionalneuralnetworks(GCNNs)foroptimizingthelocaldensityofnodesinagraph.Ourmet