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TemporalSeriesAnalysisApproachtoSpectraofComplexNetworksHuijieYang♣,FangcuiZhao,LongyuQi,BeilaiHuSchoolofPhysics,NankaiUniversity,Tianjin300071,ChinaAbstractThespacingofnearestlevelsofthespectrumofacomplexnetworkcanberegardedasatimeseries.JointuseofMulti-fractalDetrendedFluctuationApproach(MF-DFA)andDiffusionEntropy(DE)isemployedtoextractcharacteristicsfromthistimeseries.FortheWS(WattsandStrogatz)small-worldmodel,thereexistacriticalpointatrewiringprobability32.0=rp.Foranetworkgeneratedintherange32.00rp,thecorrelationexponentisintherangeof64.1~0.1.Abovethiscriticalpoint,allthenetworksbehavesimilarwiththatat1=rp.FortheERmodel,thetimeseriesbehaveslikeFBM(fractionalBrownianmotion)noiseatNpER/1=.FortheGRN(growingrandomnetwork)model,thevaluesofthelong-rangecorrelationexponentareintherangeof83.0~74.0.FormostoftheGRNnetworksthePDFofaconstructedtimeseriesobeysaGaussianform.InthejointuseofMF-DFAandDE,theshufflingprocedureinDEisessentialtoobtainareliableresult.PACSnumber(s):89.75.-k,05.45.-a,02.60.-x♣Correspondingauthor,E-mailaddress:huijieyangn@eyou.com_______________________________________________________________________________[1].Twoclassesofmodels,calledthesmall-worldgraphsandthescale-freenetworks,areproposedtocapturetheclusteringandthepower-lawdegreedistributionpresentinmanyrealnetworks,respectively[2-5].However,mostanalyseshavebeenconfinedtocapturethestaticstructuralproperties,e.g.,degreedistribution,shortestconnectingpaths,clusteringcoefficients,etc.Capturingtheglobalcharacteristicsofcomplexnetworksisanessentialroleatpresenttime.Anotherproblemisthelackofsuitabletechniques,whichleavesalargegapinourcapturingthebasicpropertiescomprehensivelyandunderstandingnetworkstheoretically.Thus,anotherimportantroleistouseconceptsortechniquesdevelopedinotherfieldstocharacterizecomplexnetworks.Itisdemonstratedinextensiveliteraturethatthepropertiesofgraphsandtheassociatedadjacencymatricesarewellcharacterizedbyspectralmethods.Investigationsonspectrumcanprovideglobalmeasuresofthenetworkproperties[6-16].Actually,analyzingspectrumisoneofthemostimportanttoolstounderstandcomprehensivelythedynamicalprocessesinacomplexquantum-mechanicalsystem[17-21].Inrecentliterature,itispointedoutthatjointuseofvariance-baseddetectorsandtheDE(diffusionentropy)analysisisapowerfultooltocapturethescalinginvarianceembeddedinatimeseries[22].Inthispaper,regardingthespacingofnearestlevelsofaspectrumasatimeseries,wetrytodetecttheself-similarstructuresandlong-rangecorrelationsembeddedinthespectrumoftheadjacencymatricesofcomplexnetworksbymeansofjointuseofDEandMF-DFA(multifractaldetrendedfluctuationapproach).II.METHODSAcomplexnetworkGcanberepresentedbyitsadjacencymatrix)(GA.Foranundirectedcomplexnetwork)(GAshouldbearealsymmetricmatrix:1==jiijAA,ifnodesiandjareconnected,or0,ifthesetwonodesarenotconnected.Themainalgebraictoolthatwewillusefortheanalysisofcomplexnetworkswillbethespectrum,i.e.,thesetofeigenvalesofthecomplexnetwork’sadjacencymatrix,calledthespectrumofthecomplexnetwork.Denotingthisspectrumas{}NEEEE,,,210,wecanconstructatimeserieswiththeintervalsbetweentwosuccessiveeigenvalesas,中国科技论文在线_______________________________________________________________________________{}{}NNNkEEEEEEEENkE----=+=Δ-011201,,,1,2,1(1)TheMF-DFAmethod[23-25]isusedtomeasurethelong-rangecorrelation.Theoriginspectrum{}NEEEE,,,210canbeemployedastheprofileoftheconstructedtimeseries.Connectingthestartingandtheendofthisprofile,wecanobtainallpossiblesegmentswithlengthl,{}NmEEElmmm,2,1,0),,(11=-++.Fiteachsegmentwithar-orderpolynomialfunction.Thefittingresultcanberegardedasthelocaltrendsofallthesegments.Takingthelocaltrendsoutfromthecorrespondingsegments,ifthereexistlong-rangecorrelationthevariancewillobeyapower-law,thatis,),(/12/0211),,(),,())((1)1(21),,,(rqqqNmlsmFsmlqrlvqrlVsEElNqrlvα∝=-⋅⋅+===-+(2)WheremFEisthefittingresultforthem’thsegment.If5.0),2(=rα,thereisnocorrelationandthesignalisanuncorrelatedsignal(whitenoise);if5.0),2(rα,thesignalisanti-correlated;if5.0),2(rα,thereisapositivecorrelationinthesignalIftheanalyzedsignalbehaveslikeBrowniannoise,wehave5.1),2(=rα.Itshouldbenotedthatoverlappingwindowsareusedinthispaperinsteadofthenon-overlappingprocedureindividingtheprofileintosegments[26].TheconceptofDE[22,27-29]isalsousedtofindself-similarstructures.Connectingthestartingandtheendoftheinitiallyconstructedtimeseries,wecanobtainasetofdelayregistervectorsas,{}11201,,----nnEEEEEE{}nnEEEEEE---+12312,,{}21010,,-----nnNEEEEEE(3)Consideringeachvectorasatrajectoryofaparticleindurationofntimeunits,alltheabovevectorscanberegardedasadiffusionprocessforasystemwith1+Nparticles.Accordingly,foreachtimedenotedwithnwecanreckonthedistributionofthedisplacementsofalltheparticles中国科技论文在线_______________________________________________________________________________