Mapping dynamical systems onto complex networks

networks

ErnestoP.Borgesa,DanielO.CajueirobandRobertoF.S.Andradec∗

a

EscolaPolit´ecnica,UniversidadeFederaldaBahia,R.AristidesNovis,2,40210-630Salvador–BA,Brazilb

DepartamentodeEconomia,UniversidadeCat´olicadeBras´ılia

70790-160Bras´ılia,Brazil

c

InstitutodeF´ısica,UniversidadeFederaldaBahia,

40210-340Salvador,Brazil

Abstract

Aproceduretocharacterizechaoticdynamicalsystemswithconceptsofcomplexnetworksispursued,inwhichadynamicalsystemismappedontoanetwork.Thenodesrepresenttheregionsofspacevisitedbythesystem,whileedgesrepresentthetransitionsbetweentheseregions.Pa-rametersusedtoquantifythepropertiesofcomplexnetworks,includingthoserelatedtohigherorderneighborhoods,areusedintheanalysis.Themethodologyistestedforthelogisticmap,focusingtheonsetofchaosandchaoticregimes.Itisfoundthatthecorrespondingnetworksshowdistinctfeatures,whichareassociatedtotheparticulartypeofdynamicsthathavegeneratedthem.

89.75.Fb:Structuresandorganizationincomplexsystems89.75.Hc:Networksandgenealogicaltrees02.10.Ox:Combinatorics;graphtheory

1Introduction

Chaoticdynamicalsystemsarecharacterizedbyseveralmeasuresthatquantifyhowirregular(despitedeterministic)thetrajectoriesare.ThesetofLyapunovexponents[1]providesameasureofthedependenceontheinitialconditionsoftheirtrajectories,whileinformationtheorymaybeusedtocharacterizeadynamicalsystemintermsoftheproductionofentropy.Actually,adynam-icalsystemwithachaoticbehaviorisregardedasarealizationofShannon’sconceptofanergodicinformationsource[2],astheKolmogorov-Sinaientropy

isequated[3]tothesumofthepositiveLyapunovexponents.Furthermea-surestodescribeachaoticsystemincludefractaldimensionsandsingularityspectra[4,5].Thisformalismappliesonlywherethesystemhas,atleast,onepositiveLyapunovexponent.However,severalauthorshavestudiedsituationsinwhichthelargestLyapunovexponentvanishes,butthedistancebetweenneighbouringtrajectoriesincreasesintimeaccordingtoapowerlaw[6,7,8,9].Thesesituationsareusuallyfoundattheonsetofchaos,whereaninfinitesimalchangeofacontrolparameterdrivesthesystemintoeitheraregularorachaoticregime.Theseinvestigationsuncoveredsomeofitsfeaturesrelatedtothesen-sitivitytotheinitialconditions[6],entropyproductionperunittime[10],multifractalgeometryoftheattractor[11],relaxationtothesystemattractor[12]andmultifractaldynamicsattheonsetofchaos[13].

Recently,theinvestigationofcomplexnetworkshassetupanewframeworkfortheanalysisofsystemswithalargenumberofdegreesoffreedom.Withinit,onehasaccesstothepropertiesofthetopologicalstructureunderlyingthemutualinteractionsamongthesystemconstituents.Thisapproachhasbeenappliedtoalargevarietyofactualsystems,rangingfromsocialinteractions,biologicalnature,informationandelectricaldistribution[14,15].

Inthiswork,wedefineamappingofdynamicalsystemsontoanetwork.Thisway,thenetworkpropertiescanbeusedtodisplaynewfeaturesforthecharacterizationofthesystem’strajectory.Thenetworknodescorrespondtocoarse-grainedregions(cells)ofthephasespacevisitedbythetrajectory.Twonodesrandsarelinkedif,duringthetimeevolution,thetrajectoryjumpsfromcellrtocells.Althoughthisnaturallyoffersaconstructionprocedureforadirectednetwork,weconsiderhereundirectednetworks.Inthisapproach,weareabletofindnovelgeometricandtopologicalpropertiesofthephasespacethroughthemeasuresthathavebeenrecentlydevelopedtocharacterizecom-plexnetworks.Thedynamicalsystem,definedonthetimedomain,ismappedintoanodedomain,whichrepresentsregionsofthephasespacevisitedbythetrajectory.Asitisbasedonthedivisionofphasespaceintoboxes,thispro-cessfollowssimilarconstructionproceduresconsideredintheevaluationofthefractaldimensionofattractorsandtheKolmogorov-Sinaientropy.

Itworthsmentioningthatsomepreviousworkshaveputtogetherideasofdynamicalsystemsandcomplexnetworks.Asweshallsee,ourapproachisratherdistinct,asthepreviouscontributionsconsidersynchronizationundertheassumptionofcertainregularityintheconnectiontopology[16,17].Theuseofanetworktorepresentthephasespaceevolutionofdiscretetimedynamicalsystemshasbeensuggestedin[18].

Werestrictourselvestotheanalysisofthequadraticmap[19]describedby

xt+1=1−ax2t(t=0,1,2,3···)

(1)

wherext∈[−1,1]anda∈[0,2].Eq.(1)leadstoavarietyofdistinctdynamical

situations,thepropertiesofwhichareexpectedtobemanifestedinthenetworkstheyoriginate.Inparticular,weexplorethoseregimesattheonsetofchaosthatproceedthroughabifurcationcascadeaswellasthoseclosetoanintermittencytransitionandalsothefulldevelopedchaos.

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2Networkcharacterization

AnindirectednetworkRisdefinedonlybythenumberNofnodesandlinksL,representedbyanassemblyofunorderedpairsSR(r,s),r,s≤N,indicatingwhichpairsofnodesaredirectlyconnected.Thisinformationprovidesafulldescriptionofthenetwork,leadingtothecomputationoftheaveragenumberoflinkspernode󰀑k󰀒,theaverageclusteringcoefficientC,themeanminimaldistanceamongthenodes󰀑d󰀒,thediameterD,theprobabilitydistributionofnodeswithklinksp(k).Othermeasures,liketheassortativitydegree[20]andthedistributionofindividualnodeclusteringcoefficientsC(k)withrespecttoitsdegreekhavealsobeenintroduced,buttheyarenotexploredhere.

RcanbedescribedbyitsadjacencymatrixM(R).Thisisnotthemostcon-ciserepresentationofanetwork,butitopensthepossibilitytotheevaluationofitsspectralpropertiesand,asrecentlyindicated,thehigherorderneighborhoodsRℓ,ℓ=1,2,...,D[21].Thisisdoneinastraightforwardway,bymeansofthesetofmatrices{Mℓ},sodefinedthat(Mℓ)r,s=1onlyiftheshortestdistancealongthenetworkbetweenthenodesrandsisℓ.Otherwise,(Mℓ)r,s=0.Al-thoughthewholeinformationonthenetworkisentailedinSR(r,s)orinM(R),eachMℓcondensesinformationonRthatisextractedfromM(R)withinthequotedframework.ThisformalismisalsoconsistentwiththerecentlyproposedproceduretoevaluatethefractaldimensionofthenetworkdF,R[22],asitnat-urallyleadstotheset{Nℓ}requiredforthisevaluation.Here,eachNℓcountsthenumberofpairsofnodeswhichareℓstepsapart.

Withinthisframework,weconsiderthateachnodeistheonlyzerothorderneighborofitself,anddefineM0=I,whereIindicatestheidentitymatrix.Also,weassumethatM1=M.SincethematrixelementsofM(R)takeonlythevalues0and1,theothermatricesMℓofthesetarerecursivelyevaluatedwiththehelpofBooleanoperations[21].Weshallmakefurtheruseofamatrixthatcondensesallinformationin{Mℓ(R)}.Asjustdiscussed,givenanytwonodesrands,itisclearthat(Mℓ)rs=1forjustonevalueofℓ.So,ifwedefineamatrix

ℓmax󰀂󰀁jMj,(2)M=

j=0

itdirectlyinformshowmanystepsapartanytwonodesinthenetworkare.Also,

󰀁tographicallydisplaythestructureitispossibletousetheinformationinM

ofanetworkwiththehelpofcolororgreyscaleplots.

Itshouldalsobementionedthatthisframeworkopensthedoorforafinercharacterizationofthenetwork,ifweconsidereachRℓasanindependentnet-work.Thus,severalofthepropertiesquotedaboveusedtocharacterizeRcanbealsoevaluatedfortheevaluationofRℓ.Thisisexploredinthenextsectionspecificallyforthethedegreedistributionandclusteringcoefficient󰀑k󰀒ℓandC(ℓ).

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3

Analgorithmtomapadynamicalsystemintoanetwork

TomapadynamicalsystemintoanetworkR,weusetheframeworkofthealgorithmintroducedin[23],toefficientlyevaluatethegeneralizedfractaldi-mensionsoffractalstructuresbytheboxcountingmethod.

Letusconsideradynamicalsystemwithmvariables.Withoutlossofgen-erality,onecanconsiderasetofpointsZ⊂ℜmconsistingofthevectorsz(i),i=1,···,T,T≫1,whichrepresentthecoordinatesofthedynamicalsystem.Thecomponentsofthesevectors,zδ(i),δ=1,···,m,areassumedtobelongtotheinterval[0,1).WedefineagraininginphasespacebydividingeachphasespaceaxisintoWequallysizeddisjointintervals,sothatthewholephaseisspannedbyasetofWmboxes.Thisrepresentsalsothemaximalpossiblenum-berofnodesinanetwork,ase.g.,inthecaseofanergodicsystem.Ofcourse,thechoiceofWdefinesthegraining,andthesizeoftheregionrepresentedbyanode.InthenextsectionweinvestigatetheeffectofWontheobtainednetworks.

Basedon[23],eachpointz(i)ofthetrajectoryismappedintoanodeofRaccordingto

m󰀂

n(i)=Wδ−1floor(Wzδ(i)),(3)

δ=1

wherefloor(x)isafunctionthatevaluatesthelargestintegerlessthanx.Actu-ally,thisisjustasimplewaytodividetheregion[0,1)minequalparts.Thus,

thenodesofthesoconstructednetworkrepresentaboxinthecoarsegrainedphasespaceofthesystem.Afterthemappingiscomplete,theboxesthatwerenotvisitedbythetrajectoryareeliminatedfromthenetwork,astheyconstitutenodeswithzerodegree(k=0),whichdonotprovideanyusefulinformationonthedynamicalsystem.Theedgesarebuiltasdescribedinthefollowingproce-dure.Letz(i)andz(i+1)betwoconsecutivepointsofthedynamicalsystem.Supposethatthesepointswerepreviouslymappedintothenodesn(r)andn(s),where0Thisprocedurecanleadtodirectedandweightednetworks;however,inthiswork,wefocusonthemostsimplesituationofundirectedandun-weightednetworks,onceourmainpurposehereistoaddresstheproblem,andshowthatwecanextractusefulinformationfromit.

4Results

Weconcentrateourinvestigationonthreedistinctregionsofa,a=ac=1.40115518909...,a∈[1.749,1.75)anda=2,whichcorrespond,respectively,tothefirstperioddoublingtransition,theregionclosetothetangentbifurcation

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totheperiod-threewindow,andthefulldevelopedchaoticstate.Representa-tivenetworkstothedistinctchaoticattractorsaregeneratedfordistinctvaluesofthegrainingW.Weonlyconsidertrajectoriesthatstartontheattractor,inordertoavoidspuriousnodes(visitedonlyonce)thatdependoninitialcondi-tions.ForafixedvalueW,thenetworkgrowsasthetrajectoryevolvesinthethephasespacewithrespecttothenumberofiterationstepst.ThereisnoaprioricriteriontodecidethetimetFafterwhichthenetworkiscomplete.InthisworkwehavefollowedhowNandLincreasewitht,foragivenW.WedefinetFasthesmallestvalueoftforwhichN(tF)=N(2tF)andL(tF)=L(2tF).FirstletusdiscusstheeffectofWonNandL.Forthepurposeofpresentingafullneighborhoodanalysisofthenetworks,wehaveselectedherevaluesofWthatleadtothemaximalnumber≈10000nodesinthenetwork.SuchchoiceforWclearlydependsona.Indeed,duetothestrategyadoptedfortheconstructionofthenetworks,NgrowswithWaccordingtoapowerlawmediatedbythefractaldimensionoftheattractordF,A.ThisisshowninFig.1a,wherewedrawpointsfora=ac,ac+10−3,1.749999and2.Foracwemeasuretheslope0.54...,whichagreeswiththeknownvalueofdF,Aoftheperioddoublingattractor.Forallothercases,theslopeis1within2%accuracy,evenfora=ac+10−3,whichliesalreadyinthechaoticregime.ThisisinaccordancewiththefactthatdF,Achangesinadiscontinuouswayata=ac.Whenacorrespondstoaperiodicsolution,thenetworkbecomesfinite,sothatNandLdonotdependonW,providedthisparameterislargeenough.

Althoughweareprimarilyinterestedinthepropertiesofthecompletenet-work,wecanalsofollowthedependencyofNandLwithrespecttot,forafixedvalueofW.AssumingN∼Lz,thisdefinesadynamicalexponentzintheearlystagesofthenetworkevolution.Theanalyzeddataindicatethatz≃1forallvaluesofa.Nevertheless,wefindthat,intheimmediatechaoticneighborhoodofa=1.75,thelaminarphasesintheintermittentregimetrapsthetrajectoryforlongintervals,demandinglargetimeofintegrationtocompletethenetwork.Nowwediscussthenetworkproperties,followingthemethodologyandpa-rametersindicatedbefore.Whenappropriate,weextendourdiscussiontoprop-ertiesofhigherorderneighborhoodsinthenetwork.Wefinddistinctnetworkstructureswhenweconsiderthechaoticregimeortheonsetofchaos.Regardingthemeanminimaldistance󰀑d󰀒anddiameterD,wefindthat,forthechaoticregime,theygrowwithrespecttoW(andN),inalogarithmicway,similarlytosmall-worldnetworks[24].Ifwewrite󰀑d󰀒=αlog10W,wefindα≃2.4and3.7,respectively,fora=2anda∈[1.749,1.749999],asillustratedintheinsetofFig.1b.AsDonlyassumesintegervalues,itispossibletoexpectasimilarbehavioronlyinanapproximateway,e.g.,equallysizedstepsinD×log10Wplots.Thus,assumingD=βlog10W,wefindβ≃4fora=2withverygoodaccuracy.Intheinterval[1.749,1.749999],wenoticedthepresenceoffluctua-tionsinthesizeofthesteps,whichincreaseswhenweapproachthethresholda=1.75.Theresultsfora=achaveacompletelydistinctbehavior:󰀑d󰀒andDincreaseaspowerlawswithrespecttoW,asillustratedinthemainpanelofFig.1b.Fora=ac+10−3,thesamedependenceprevails.Theexponents

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101054321(a)N101010210103104W10510610715(b)23〈d〉, D10210501010101101102N103104Figure1:(a)DependenceofNwithrespecttoWfora=ac(circles),a=ac+10−3(diamonds),a=1.749999(downtriangles),anda=2(uptriangles).Thethesameconventionisusedinallotherfigures.ThedistinctslopesindicatethevaluesofdF,A.(b)Powerlawdependenceof󰀑d󰀒(hollowsymbols)andD(solidsymbols)withrespecttoNfora=acanda=ac+10−3.Theinsetshowslogarithmicdependenceamongthesamequantitieswhena=1.749999anda=2.610100-1p(k)101010-2-3-4100101k102Figure2:Degreedistributionofnodesatac(circles)andinthea=2chaoticregime,(uptriangles).Behaviorclosetotangentbifurcation,a=1.75−10−4,(notshown)isquitesimilartoa=2.obtainedfor󰀑d󰀒andDare,respectively,0.67and0.73forac,and0.35and0.38fora=ac+10−3.󰀑d󰀒andDincreasesmoothlyfora=ac+10−3butata=ac,thegrowthofthetwoquantitiespresentdiscontinuitiesandsteps.ResultsinFig.1bindicatethat,unlikethepropertiesoftheattractor,reflectedbytherashchangeinthevalueofdF,A,thenetworkpropertieschangeslowlywhenthechaoticregimeisreached.Wehaveevaluatedthedistributionp(k)vs.kinalldifferentregimes.WecanseeinFig.2that,fora=ac,kdoesnotreachlargevalues(kmax≃30),sothatitisnotpossibletoidentifyapowerlawdecayinthisrange.Fora=2(andalsoa=1.749999)nodeswithlargervaluesofkcanbefound,butp(k)doesnotfolloweitherapowerlaw.

SuchdistinctbehaviorisalsoexplicitwhenweanalyzethefractaldimensionofthenetworkdF,R[22].Fig.3showsthatthea=acnetworkshaveawelldefinedscalingbehavior,whichextendsovermorethantwodecades,inaquitepreciseway.Ontheotherhand,afractaldimensionforthenetworksinthechaoticregimeisnotevident.First,thesmallvaluesofDreducestheregionofpossiblescalingbehavior.Then,weclearlyobservedeviationstotheexpectedpower-lawregime.

Regardingtheclusteringcoefficient,weobtainC≡0fora=ac,indicatingthecompleteabsenceoftrianglesinthenetwork.Fora=1.749999anda=2,

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104310N(l)102101101010l100Figure3:ClearpowerlawbehaviorforN(ℓ)×ℓwhena=ac,withdF,R=1.47.Finitesizeeffectsblurthisdependencewhenℓ≃D.Inthechaoticregime,fora=2,dF,Rcanhardlybeevaluated.Fora=ac+10−3,thepointsillustrateaslowtransitionbetweenthetworegimes.8

wefindthatCdecayswithNaccordingtoapowerlawwithexponent≈0.95,whatisnotsoclosetothevalue0.75observedfortheAlbert-Barabasiscale-freenetwork[14].However,itssmallvaluesindicatethatthenetworkhasonlyasmallnumberoftriangles.

Otherfeaturesofthenetworkmaybedrawnifweconsidertheclusteringcoefficientofhigherorderneighborhoods[21].Toobtainaclearerpictureofthisanalysisconsider,forinstance,theregularCayleytreewithℓ=2networkforthecompleteregularthreeneighborsCayleytree.Theℓ=2networkisformedbytriangles,muchliketheHusimicactusesand,correspondingly,alargevalueforC(ℓ=2).Thefollowingoddandevennumberedneighborhoodsarecharac-terized,respectively,byvaluesofC(ℓ)=0andC(ℓ)>0,wherebythevaluesofC(ℓ)forasubsetofevenneighborhoodsdecreaseinamonotonicway.Asimilarsituationisfoundforthenetworksinvestigatedhere.InFig.4awesummarizethesequenceofC(ℓ)forthethreesituationsunderinvestigation.Weseethat,fora=2anda=1.4999,theoscillatorybehaviorlastsonlyuntilℓ=5and10,respectively.AlsowenotethattheoddnumberedC(ℓ)increasesuntilitreachesvaluesashighasthoseoftheevennumberedneighborhoods.Ontheotherhand,thea=acnetworkhasC(ℓ)=0foralloddnumberedneighbor-hood.TheinsetshowsthattheC(ℓ=evennumber)decayswithℓaccordingtoapowerlaw,withexponentα≃1.

Wehavealsoanalyzedtheaveragedegree󰀑k󰀒ℓasafunctionoftheneigh-borhoodℓ.Hereagainwefindthatthebehaviorofthechaoticregimeandtheonsetofchaoshavedistinctfeatures,asexhibitedintheFig.4b.

󰀁toobtainimages,incolor(orgray)Finally,weusetheinformationinM

scale,ofthenetworkneighborhoodstructure.Theyprovideavividandeasyvisualizationoftheirdistinctpropertiesoftheattractor.Fora=2,thefirstorderneighborhoodisdistributedalongtheparaboladescribedbyther.h.s.of(1)(seeFig.5a).Itshowshowthesecondandhigherorderneighborhoodevolveaccordingtothehigherorderiteratesofthequadraticmap.However,mixingandfinitesizeeffectsstemmingfromafinitegrainingsmearsthehigherorderiterates.Thesituationisdistinctforac,whentheattractorisadF,A=0.54...dustspreadoutinthe[−1,1]interval.Onlyboxescontainingpartofthedustremaininthenetwork,sothatcontiguousnumberednodesarenotactuallyneighborsinphasespace.Theresultingimage(Fig.5b)displaysafineandintertwinedtessiturathatreflectsaverypeculiarbehaviorofthetrajectoriesattheonsetofchaos[7,8,9].

5Conclusion

Inthisworkweexploretheideaofmappingchaoticsystemsontocomplexnetworks.Networksareconstructedaccordingtoawelldefinedmethodology,andresultsforthelogisticmapindicatehowtheirpropertiescanbeassociatedtothoseoftheattractorinphasespace.Trajectoriesindistinctdynamicalregimesareexploredinordertoshowhowthemajordifferencesinphasespacearereflectedinthenetworks.Thenetworksshowseveralfeaturesofsmall-worldand

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0.60.41010100-1C0.200-2100101102102010103l2〈k〉l101010010101l102Figure4:(a)BehaviorofC(ℓ)withrespecttoℓforchaoticregimeandattheonsetofchaos.TheinsetshowsapeculiarpowerlawbehaviorC(ℓ)attheonsetofchaos.(b)Behaviorof󰀑k󰀒ℓ×ℓ.Theonsetofchaosexhibitsaslowincreasingvalueof󰀑k󰀒ℓoveralargeintervalofℓ,interruptedbyfinitesizeeffectsalreadypresentinFig.3.Inoppositiontothispicture,chaoticregimeshowsasharpincreaseof󰀑k󰀒ℓ.10󰀁fora=2(a)andFigure5:Colorscaleplots(grayscaleinpaperversion)ofM

a=ac(b).Scalerangesfromblackandblue(ℓ=0andℓ=1)tored(ℓ=D).Numberoflevelsin(a)ismuchsmaller11thanin(b).Neighborhoodstructurechangesabruptlyinthedistinctregimes.

scale-freenetworks,buttheydonotfullymatchwiththosenetworksgeneratedbythespecificalgorithmsdescribedin[24,25].TheanalysisofthenetworksintheneighborhoodofacrevealsthattheN×Wdependence,measuredbydF,A,hasasharptransitionattheonsetofchaos.So,thedistinctcharacterofthetrajectoriesinphasespaceisindeedreflectedinthenetwork.However,withexceptionofC,theresultsfortheotherindices(󰀑d󰀒,Dandp(k))changeinamuchsmootherwaywithrespecttochangesintheparametera.

Acknowledgements:ThisworkwassupportedbytheBrazilianAgenciesCNPqandFAPESB.

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