Stochastic Cellular Automata Model for Stock Market Dynamics

.

StochasticCellularAutomataModelforStockMarketDynamics

arXiv:cond-mat/0311372v2 [cond-mat.dis-nn] 24 Nov 2005M.Bartolozzi1andA.W.Thomas1

1

SpecialResearchCentrefortheSubatomicStructureofMatter(CSSM),

UniversityofAdelaide,Adelaide,SA5005,Australia

(Dated:February2,2008)

Abstract

Inthepresentworkweintroduceastochasticcellularautomatamodelinordertosimulatethedynamicsofthestockmarket.Adirectpercolationmethodisusedtocreateahierarchyofclustersofactivetradersonatwodimensionalgrid.Activetradersarecharacterisedbythedecisiontobuy,σi(t)=+1,orsell,σi(t)=−1,astockatacertaindiscretetimestep.Theremainingcellsareinactive,σi(t)=0.Thetradingdynamicsisthendeterminedbythestochasticinteractionbetweentradersbelongingtothesamecluster.Extreme,intermittentevents,likecrashesorbubbles,aretriggeredbyaphasetransitioninthestateofthebiggerclusterspresentonthegrid,wherealmostalltheactivetraderscometosharethesamespinorientation.Mostofthestylizedaspectsofthefinancialmarkettimeseries,includingmultifractalproprieties,arereproducedbythemodel.AdirectcomparisonismadewiththedailyclosuresoftheS&P500index.

PACSnumbers:05.45.Pq,52.35.Mw,47.20.Ky

Keywords:ComplexSystems,Percolation,StochasticProcesses,Multifractality,Econophysics

1

I.INTRODUCTION

SincethesuccessfulapplicationoftheBlack-Scholestheoryforoptionpricing[1]in1973moreandmorephysicistshavebeenattractedbytheideaofunderstandingthebehaviourofthemarketdynamicsintermsofcomplexsystemtheory,whereself-organizedcriticality[2,3]andstochasticprocesses[4,5]playimportantroles.Theaimofthemicroscopicmodelsproposedsofar(forgeneralreviews[6,7])istoreproducesomestylizedfacts[8]concerningthetemporalfluctuationsofthepriceindices,P(t).Inparticular,thelogarithmicpricereturns

R(t)=lnP(t+1)−lnP(t),

andthevolatility,definedinthepresentworkas

v(t)=|R(t)|,

(2)(1)

havebeenstudiedextensively[4]fromanempiricalpointofview.Theresultshaveshownthatwhilelongtimecorrelationsarepresentinthevolatility,aphenomenonknownasvolatilityclustering,theycannotbefoundinthetimeseriesofreturns.Moreoverthelattershowanintermittentbehaviourthatrecallsinsomeaspectshydrodynamicturbulence[9,10,11],characterizedbypowerlawtailsintheprobabilitydistributionfunction(pdf).Microsimulationshavedemonstratedthatthiskindofbehaviourcanoriginatebothasastochasticprocesswithmultiplicativenoise[12,13,14]andasapercolationphenomenon[15,16,17].

Inordertoreproducethesefeaturesofrealmarketsweintroduceastochasticcellularautomatamodel,representinganopenmarket.Thatis,amarketwherethenumberofactivetraders,definedascellswithspinstatedifferentfrom0,namelyσi(t)=±1,evolveintimeaccordingtoapercolationdynamics.Thepercolationdynamicsischoseninordertosimulatetheherdingbehaviourtypicalofinvestors[28].Accordingtothis,activetradersgatherinclustersornetworkswhere,followingastochasticexchangeofinformation,theyformulatethetradingstrategyforthenexttimestep.TheresultsobtainedbythesimulationsarethencomparedwiththetimeseriesofdailyclosuresoftheS&P500index[4,28,29,30,31,34,40,41,45]overaperiodofabout50years.

Moreover,recently,thefractalproperties[18]ofthepricefluctuationshavealsobeen

2

investigatedfordifferentmarkets[19,20,21,22].Acommonfeaturefoundinthesestudiesistheexistenceofanonlinear,multifractalspectrumthatexcludesthepossibilityofefficientmarketbehaviour[23].Theoriginofthemultifractalityofinthefinancialtimeserieshasalsobeenatthecenterofdiscussions[24,25,26,27].Inthispaperweconsiderthemulti-fractalspectrumofthepricefluctuationsasastylizedfactofthemarkettimeserieswithoutaddressinganyquestionabouttheunderlyingprocessabletogenerateit.Themultifractalspectrumisusedasafurthertestforourmodel.

Aparallelbetweenmultifractalandthermodynamicalformalismhasalsobeeninvesti-gated.Wefound,inagreementwiththepreviousworkofCanessa[45],thattheanaloguespecificheatcanprovideagoodtooltocharacterizeintermittency,thatisfinancialcrashesorbubbles,fromathermodynamics-equivalentpointofview.

II.THEMODEL

Inthepresentworkwesimulatethefinancialmarketdynamicsviaastochasticcellularautomatamodel.Theagentsofthemarketarerepresentedbycellsonatwodimensionalgrid,512x128.Theithagentatthediscretetimesteptischaracterizedbythreepossiblestatesorspinorientations,σi(t)=0,±1.Thevalueσi(t)=+1isassociatedwiththepurchaseofastockwhileσi(t)=−1withselling.Theformerstatesarecalledactive.Thecellswithspinvalueσi(t)=0areinactivetraders.Theactivetradersherdinnetworksorclustersviaadirectpercolationmethodrelatedtoaforestfiremodel[15].Theinformationcarriedbytheactivetraders,thatistheirspinstate,issharedwiththeothermembersofthecluster.Thepercolationdynamicsallowatimedependentherdingbehaviourandthemarketcanbeinterpretedasanopensystemnotboundedbyconservationlaws.Theclusteringprocesswillbediscussedindetailinthenextsubsection.

Thetradingdynamicsisinsteadrelatedtothesynchronousupdateofthespinsoftheactivetraders,ruledbyastochasticexchangeofinformationbetweenthem,similartoarandomIsingmodel[12,13].Aparticularfeatureofthepresentsimulationisthattheinformationisnotspreadalloverthegrid,asinothermulti-agentscenarios[12,13],butitislimitedbytheclustersofinteractionpreviouslydefined.Themechanismforthespindynamicsisexplainedin“stochastictradingdynamics”subsection.

3

A.PercolationClustering

Oneoftheaimsofourcellularautomatamodelistoreproducetheherdingbehaviourofactivetraders[28].Werefertoherdingbehaviourasthetendencyofpeopleinvolvedinthemarkettoaggregateinnetworksorclustersofinfluence.Thetradersthenusetheinformationobtainedbytheirnetworkinordertoformulateamarketstrategy.Evenifthetopologicalstructureofthesenetworksofinformationisnotimportant,sinceseveralkindsoflongrangeinteractionareavailablenowadays[13],thenumberofconnectionsforeachtradermustbe,inanycase,finiteandnotextendedoverthewholemarket.Inthisframeworkadirectpercolationmethodisusedtosimulateherdingdynamicsbetweenactivetraders.IfweassumethattheneighboursofinfluencearethoseofvonNeumann(up,down,left,right),thepercolationisfixedbythefollowingparameters:

ph:theprobabilitythatanactivetradercanturnoneofhisinactiveneighboursintoanactiveoneatthenexttimestep,σi(t)=0→σi(t+1)=±1.Thissimulatesthefactthatcertaininformationpossessedbyatradermayinduceapotentialtradertojointhemarketdynamics.

pd:theprobabilitythatanactivetraderdiffusesandsobecomesinactive,σi(t)=±1→σi(t+1)=0,duetoeachofhisinactiveneighbour.Thismimicsthefactthatonlytradersatthebordersofanetwork,thatistheweakerlinks,canquitthemarket.

pe:theprobabilitythatanontradingcellspontaneouslydecidestoenterthemarketdynamics,σi(t)=0→σi(t+1)=±1.

Thevaluesoftheadimensionalparameters,ph,pdandpe,influencethestabilityofthesystemandthepercentageofactivetradersonthegrid.Inordertotestdifferentmarketactivitieswefixthevaluespd=0.05andpe=0.0001whilewetunetheparameterph.Atthebeginningofthesimulationthegridisloadedrandomlywithasmallpercentageofactivetradersandthenthesystemispermittedtoevolveaccordingtothepreviousrules.Ifweareinastablerangeoftheparameterph,afteratransientperiodthatdependsbothontheparametervaluesandtheinitialnumberofactivecells,thenumberofactivetradersonthegridbeginstofluctuatearoundacertainaverage,asshowninFig.1(Top).Themarketcanbeconsideredopensincethenumberofagentschangesdynamicallyintime.Inthisregime,thecompetitionbetweenherdinganddiffusionproducesapowerlawdistributionoftheclustersize,asshowninFig.1(Bottom),ρ(S)≈S−λ,whereSistheclusterdimension,

4

20000Number of Active Traders15000ph=0.0493ph=0.0490ph=0.0488ph=0.0485ph=0.047510000500000200040006000Time Steps8000100001000λ∼1.1100ρ101110S1001000FIG.1:Top:DifferentvaluesoftheparameterHproducedifferentactivitiesoftradersonthegrid.Bottom:Clustersizedistributionforph=0.0485att=9000.

definedasthenumberofactivecellsbelongingtothesamecluster,andλ>0,creatingahierarchyofnetworks.Thishierarchyisnecessaryifwerearetotakeintoaccountarealaspectofthemarket,namelythatdifferenttradersalsohavedifferenttradingpowers.Areasonableassumptionisthatpeoplehavingalargernumberofsourcesofinformation,sobelongingtogreaterclusters,canbeassociatedwithprofessionalinvestorsthat,mostlikely,areabletomoveagreateramountofstockscomparedtotheoccasionalinvestor.Usingthisassumptionweareabletodefineaproperweightforthetradingpowerofdifferentcells,aswewilldiscussinthenextsubsection.

Asimilarpercolationmodelhasalsobeenusedtoreproducesomestatisticalandgeo-metricalfeaturesofsolaractivity[32,33].

B.StochasticTradingDynamics

k

Thedynamicsofthespinsoftheactivetraders,σi(t)=±1fori=1,...,Nk(t)(wherethe

superscriptk,fromnowon,referstothekthclusterofthegridconfigurationattimestept)followsastochasticprocessthatmimicsthehumanuncertaintyindecisionmaking[13].

kTheirvaluesareupdatedsynchronouslyaccordingtoalocalprobabilisticrule:σi(t+1)=+1kkkwithprobabilitypkiandσi(t+1)=−1withprobability1−pi.Theprobabilitypiis

5

determined,byanalogytoheatbathdynamicswithformaltemperaturekbT=1,by

pki(t)=

1

Nk(t)

Nk(t)

󰀂

kk

Akijσj(t)+hi.

(4)

j=1

k

TheAkij(t)aretimedependentinteractionstrengthsbetweenagentsandhi(t)isanexternal

fieldreflectingtheeffectoftheenvironment[13].Theinteractionstrengthsandtheexternal

kkkfieldchangerandomlyintimeaccordingtoAkij(t)=Aξ(t)+aηij(t)andhi(t)=hζi(t).

Thevariablesξk(t),ηij(t),ζik(t)arerandomvariablesuniformlydistributedintheinterval(-1,1)withnocorrelationintimeorspace.Themeasureofthestrengthsofthepreviousterms,A,aandh,areconstantandcommonforallthegrid.

Inthiscontestthedynamicsofthepriceindex,P(t),canbeeasilyderivedifweassumethattheindexvariationisproportionaltothedifferencebetweendemandandsupply,

dP

1050-5-1019501050-5-10200040006000Time Steps800010000196019701980Decimal Years19902000FIG.2:Top:NormalizedlogarithmicreturnsfortheS&P500.Bottom:Timeseriesofreturnsreproducedwiththesimulationwithph=0.0493,A=1.8andh=0.

analyzediscomposedbyofthedailyindicesfrom3/1/1950to18/7/2003foratotalof13468data.Thetimeseriesoftheindexprices,P(t),isconvertedintothelogarithmicreturns(1)andthennormalizedoverthetimeinterval,T,

r(t)=

R(t)−󰀊R(t)󰀋T

rrThestrengthAalsoplaysanimportantroleinthetradingdynamics.Withphfixed,thisparameterisrelatedtotheintermittencyofthesystem.Bothforlargevaluesoftheactivity(A>10)andforA→0weobserveanapproachofthepdftowardaGaussian-likeshape.Thatisvery,largefluctuationsbecomemoreandmorerare,andAcanberegardedasatemporalscaleforthesystem,similartotheactivityparameterintheCont-Bouchaudmodel[16].Inspiteofthissomelargefluctuationscanbestillidentified.ThisisprobablyoneofthemaindifferencesbetweentheCont-Bouchaudmodelandthepresent.InfactMonteCarlosimulationsoftheformer[17]showthatanincreaseoftheactivitybringsarapidconvergencetowardaGaussiandistributionbecausealargenumberofclustersaretradingatthesametimefollowingarandomprocedureofdecisionmaking[16,17]:therearenoclustersthatcaninfluencethemarketmorethanothersandsotheresultingglobalinteractionisnoise-like.Inourmodelfluctuationsarealwaysallowedbecauseoftheheatbathdynamics.Clustersofactivetraderscanalwaysbesubjectedtophasetransitions,independentlyofthestateofotherclusters,creatingadisplacementbetweendemandandsupply.

Inordertoreproducethebehaviouroffinancialtimeseriesweworkwith1.5Nowwediscusstheresultsofthecellularautomata.InFig.2(Top)andFig.2(Bottom)thenormalizedlogarithmicreturnsoftheS&P500andofthesimulationareshown,respec-tively.Theaveragenumberofactivetraders,inthestableregime,is≈16000,asshowninFig.1(Top).ThemodelreproducestheintermittentbehaviouroftheS&P500timeseries,asexpressedbytheleptokurticpdfofFig.3.Thetailsofthedistributionfollowapowerlawdecay,reflectingthefactthatlargecoherentevents,farfromtheaverage,arelikelytooccurwithafrequencyhigherthanexpectedforarandomprocess(wheretheshapewouldbeaGaussian).Theselargeeventsarerelatedtofinancialcrashesorbubblesofthemarketand,inourmodel,toaphasetransitioninthespinstateoflargenetworksofactivetraders,aswewilldiscussfurtheron.Fromapowerlawfit,ρ(r)≈r−1−γ(for|r|>2),wefindγ≈3

8

1S&P500ModelGaussian0.10.01ρ0.0010.0001-10-50r510FIG.3:ProbabilitydistributionfunctionfortheS&P500computedwithdailydatafrom31/1/1950to18/7/2003andthemodel.AGaussianisalsoplottedforcomparison.Theparametersusedareph=0.0493,A=1.8andh=0.

forboththeS&P500andthemodel,confirmingthegoodagreementbetweenthetwo.Theproblemoffindingthebestdistributiondescribingthepricereturnsisaveryimpor-tantissuefromapracticalpointofview[4].ThestandardBlack-Scholestheoryforoptionpricing[1,4,5]assumesthatthereturnsarenormallydistributed.Thisfacthasbeenproventobeempiricallyfalse,asshownalsoinFig.3(seeRef.[4]forageneralreference).Findingamoreappropriatedistributionwouldbeanimportantimprovementinthisfieldofresearch∗.Inordertounderstandthetradingdynamicsoftheautomatawealsoshowtwosnapshotsofthegridconfiguration,thefirstduringanormalsession,Fig.4(Top),andthesecondduringacrash,Fig.4(Bottom).Duringthenormalsessiontheorientationsofthespinsaredistributeduniformlyoverthevariousclustersandthereisnosharpdifferencebetweendemandandsupply.Thesituationisdifferentduringacrash.Inthiscasetheclustersatthetopofthehierarchy,thebiggerones,playafundamentalrole.Infacttheyundergoaphasetransitionwherethegreatestpartoftheirspinssharethesameorientation.Thecapacityoftheclusterstogenerateacoherentorientationofthespins,andhenceoftheirtradingstate,canbeinterpretedintermsofamultiplicativenoiseprocess[5,34],wherethecollectivesynchronizationarisesasaresultoftherandomlyvaryinginteractionstrengthsbetween

FIG.4:Top:snapshotofthegridduringanormaltradingperiod.Theblackcellsarethebuyerswhiletheorangeonesarethesellers.Theparametersusedinthissimulationareph=0.0493,A=1.8andh=0.Bottom:thesamesimulationduringacrash.Largeclustersofsellersareindicatedbyarrows.

agents.Thepeculiarityofourmodelisthatcrashesorbubbles(suddenpricechanges)arerelatednottoaphasetransitionofthewholemarket[12,13]butrathertoaphasetransitioninoneormoreofthelargerclustersthathaveagreaterinfluenceonthetradingsession.Thisbehaviourisprobablyclosertotherealmarketwherethesynchronizationoftradingopinionismorelikelytohappenbetweenlargegroupsoftradersthanoverthewholemarket.Thetemporalcorrelationsofthelogarithmicreturnsandofthevolatilityareinvestigatedviatheautocorrelationfunction,definedas

c(τ)=

T−τ󰀂t=1

x(t+τ)x(t),(8)

whereTisthelengthofthetimeseriesandτisatimedelayforthenormalizedvariablex(t).TheresultsforboththemodelandtheS&P500areshowninFig.5(Top)andFig.5(Bottom),respectively.Whilethetemporalcorrelationforthereturnsislostalmostimmediately,thevolatilitymanifestsaslowdecayintime,relatedtothephenomenonofvolatilityclustering.Theprevioustemporaldependencieshavebeenfoundinbothrealdataandinthesimulation.

10

1Autocorrelation0.80.60.40.200102030Time DelayReturns ModelReturns S&P5004050Autocorrelation10.10.01Volatility S&P500Volatility Model0.21Time Delay525FIG.5:Top:Autocorrelationfunctionforthepricereturns.Bottom:Autocorrelationfunctionforthevolatilities.Inboththegraphstheparametersusedforthemodelareph=0.0493,A=1.8andh=0.IV.

MULTIFRACTALANALYSIS

Itisalsoworthpointingoutthatfinancialtimeseriespresentaninherentmultifractal-ity[18].Inthepastfewyearstheworkofmanyauthors[19,20,21,22]hasbeenaddressedtothecharacterizationofthemultifractalpropertiesoffinancialtimeseries,andnowadaysmultifractalitycanbeconsideredasastylizedfact.Inordertostudythemultifractalprop-ertiesofourmodelweusethegeneralizedHurstexponent[35],H(q),derivedviatheq−orderstructurefunction,

Sq(τ)=󰀊|x(t+τ)−x(t)|q󰀋T∝τqH(q),

(9)

wherex(t)isastochasticvariableoveratimeintervalTandτthetimedelay.Thegener-alizedHurstexponent,definedin(9),isanextensionoftheHurstexponent,H,introducedinthecontextofreservoircontrolontheNileriverdamproject,around1907[18,36].Thistechniqueprovidesasensitivemethodforrevealinglong-termcorrelationsinrandompro-cesses.IfH(q)=HforeveryqtheprocessissaidtobemonofractalandHisequivalenttotheoriginaldefinitionoftheHurstexponent.ThisisthecaseofsimpleBrownianmotionorfractionalBrownianmotion.

IfthespectrumofH(q)isnotconstantwithqtheprocessissaidtobemultifractal.Fromthedefinition(9)itiseasytoseethatthefunctionH(1)isrelatedtothescalingpropertiesofthevolatility.ByanalogywiththeclassicalHurstanalysis,aphenomenonissaidtobe

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1.5qH(q)1S&P5000.501.5qH(q)10.5001234Modelq5FIG.6:MultifractalspectrafortheS&P500intheperiod31/1/1950to18/7/2003(top)andthemodel(bottom).Forthelatterph=0.0493,A=1.8andh=0.

persistentifH(1)>1/2andantipersistentifH(1)<1/2.Foruncorrelatedincrements,asinBrownianmotion,H(1)=1/2.InFig.6acomparisonisshownbetweenthemultifractalspectraofthemodelandtheS&P500obtainedfromthepricestimeseries.Itisclearthatbothprocesseshaveamultifractalstructureandthepricefluctuationscannotbeassociatedwithasimplerandomwalkasintheclassicalefficientmarkethypothesis[23].

Themultifractalityofthetimeseriescanalsobediscussedintermsofthermodynamicequivalents,accordingtomultifractalphysics[42,43,44,45].Inthisapproachwedividethetimeseries,x(t),fort=1,...,LintoNequalsub-intervals.Thenwecanwritethefollowingmeasureforeachofthese,

µi(τ)=

|x(t+τ)−x(t)|

Ifweconsiderχqasthefreeenergyofoursystem,Eq.(12)providesalinkbetweentheclassicalthermodynamicalformalismandmultifractality.Assumingqasanequivalenttemperature,wecandefineananaloguespecificheat[42,44,45,46],

Cq=−

∂2χq

1

l+δ

󰀂δ−1ǫ(1,l⋆),

(15)

l⋆=l

forl=0,...,L−δ.Inthiscasewehaveascalingpropertyfortheensembleaveragewith

respecttothescaleδ:

󰀊ǫ(δ,l)󰀋∝δ−Kq.

(16)

InamultifractalprocesstheexponentKqisanonlinearfunctionofqrelatedtotheinter-mittencyofthetimeseriesandtothegeneralizeddimensionvia[48,49]

(q−1)Dq=q−1−Kq.

(17)

From(13)and(17)wehavethat[46],

Cq=

∂2Kq

0.6Cq0.40.20-60.6τ=1τ=100-4-20q24τ=1τ=1006Cq0.40.20-6-4-20q246FIG.7:Top:analoguespecificheatfortheS&P500from31/1/1950to18/7/2003fortwodifferenttimedelays,namelyτ=1andτ=100.Asharppeakisclearlyvisiblearoundq=−1.5.Thesecondpeakontherighthandsidedisappearsincreasingthetemporaldelay.TheCqcurveshavebeencomputedforthelogarithmofthepriceusingthealgorithmin[47]forKq.Bottom:analoguespecificheatforthemodelwithparametersph=0.0493,A=1.8andh=0.Thedoublehumpedshapedforsmalltemporaldelaysisvisiblealsohere.

andrecalltheHubbardmodelforsmalltointermediatevaluesofthelocalinteraction[43].Asalsosuggestedin[45],thesecondpeakisduetothelargefluctuationsatsmallscales,thatiscrashesandbubbles.Increasingthetimedelaymeansthatthefluctuationstendtobesmoothedandthetimeseriesofreturnsapproachanoise-likeregime.Forthisreasontheanaloguespecificheatshapesforτ=100arebasicallyindistinguishable.Fromthisargumentwecaninterprettheanaloguespecificheat,andinparticularthesecondpeakforlowtimedelays,asawaytocharacterizecrashes,oringeneralthedegreeofintermittencyinatimeseries.Thedifferenceintheshapesandheightsoftheshorterpeakforτ=1isduetoaslightlydifferentcorrelationofthefluctuationsinthetwotimeseries.Moreover,inthemodelwecanlinktheshorterpeaktothephysicalphasetransitioninthespinstateofanetworkoftraders.

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V.CONCLUSIONS

Inthispaperwehaveintroducedastochasticcellularautomatamodelforthedynamicsofthefinancialmarkets.Themaindifferencebetweenourmodelandotherstochasticsimu-lationsbasedonspinorientationofagents[12,13]isthetemporalevolutionofthenetworksofinteractionandthereforetheconceptofan“open”market.Theactivetradersfollowadirectpercolationdynamicsinordertoaggregateinnetworksofinformation.Thismakesoursimulation,evenifstillarawapproximation,surelyclosertotherealmarket,wherenoconservationrulesforthenumberofagentscanbeclaimed.Crashesandbubblescanbeinterpretedasasynchronizationofthespinorientationofthemoreinfluentialnetworksinthemarket.Moreover,theintroductionofalimitationinthenumberofinteractingagentsreducesdrasticallythenumberofcomputationsonthegridpertimestep.Inasystemwhere

2

alltheagentsinteractwitheachotherthisnumbergoeslikeNa,beingNathenumberof

activeagents,whileinourmodel,consideringthedistributionoftheclusters,itiseasy

2−λtoseethatitgoeslikeNa.Thevalueofλfoundforseveralherdingparameters,ph,is

λ≈0.6÷1.4,sothatthecomputationalcostismuchlowerforthepresentmodel.Thisgivesonethepossibilitytosimulatethemarketusingaverylargerangeofagents.Themodelisabletoreproducemostofthestylizedaspectsofthefinancialtimeseries,supportingtheideathatcrashesandbubblesarerelatedtoacollectivesynchronizationinthetrad-ingbehaviouroflargenetworksoftraders,wheretheinformationisexchangedaccordingstochasticinteractionbetweenthem.

Acknowledgments

ThisworkwassupportedbytheAustralianResearchCouncil.

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