Ferromagnetic fluid as a model of social impact

InternationalJournalofModernPhysicsD󰀁

cWorldScientificPublishingCompanyFERROMAGNETICFLUIDASAMODELOFSOCIALIMPACT

PIOTRFRONCZAK,AGATAFRONCZAKANDJANUSZA.HO󰁄LYST

FacultyofPhysicsandCenterofExcellenceforComplexSystemsResearch,WarsawUniversity

ofTechnology,Koszykowa75,PL-00-662Warsaw,Polandfronczak@if.pw.edu.pl,agatka@if.pw.edu.pl,jholyst@if.pw.edu.pl

ReceivedDayMonthYearRevisedDayMonthYear

CommunicatedbyManagingEditor

Thepaperproposesanewmodelofspindynamicswhichcanbetreatedasamodelofsociologicalcouplingbetweenindividuals.Ourapproachtakesintoaccounttwodifferenthumanfeatures:gregariousnessandindividuality.Wewillshowhowtheyaffectapsycho-logicaldistancebetweenindividualsandhowthedistancechangestheopinionformationinasocialgroup.Apartfromitssociologicalaplicationsthemodeldisplaysthevarietyofotherinterestingphenomenalikeself-organizingferromagneticstateorasecondorderphasetransitionandcanbestudiedfromdifferentpointsofview,e.g.asamodelofferromagneticfluid,complexevolvingnetworkormultiplicativerandomprocess.Keywords:Isingmodel;Socialimpact;Ferromagneticfluids;Multiplicativerandompro-cesses.

1.Introduction

Interdisciplinaryresearchhasbeendrawingmuchattentioninthelastdecades.Mod-elsandmethodsdevelopedintheoreticalphysicsprovedtobefruitfulinstudyingcomplexsystems1,2,composedofrelativelysimplemutuallyinteractingelementsandcomingfromdomainsasdivergeasneuralnetworks3,diseasespreading4,pop-ulationdynamics5,etc.Buttherangeoftheinvestigationsgoesalsobeyondthenaturalsciencesandincludesproblemsfromsociologyoreconomy,likepedestrianmotionandtraffic6,migrations7,8orfinancialcrashes9.Anotherimportantsub-jectofthiskindistheprocessofopinionformationinsocialgroups.OnewayofitsquantitativedescriptionconsistsinamacroscopicapproachbasedonthemasterequationortheBoltzmann-likeequationsforglobalvariables7,10,11.Alternatively,bymakingsomesociologicallymotivatedassumptionsonthemechanismsofinter-actionsbetweenindividuals”microscopic”modelsareconstructedandinvestigatednumericallyoranalyticallybymeansofmethodsknownfromstatisticalphysics12,13

.Oneconcludesthatthevarietyoftheemergingphysicalcollectivephenomenahasmuchincommonwiththecomplexsocialprocesses.

Inparticular,Nowak,SzamrejandLatanecreatedasimplemodelbaseduponthesuccessfultheoryofsocialimpactinhumansocietesfirstintroducedbyLa-1

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tanein198114.Inthesimplestformtheirmodelcharacterizesthestrengthofthepsychologicalcouplingbetweentheindividualsbytwoqualities:persuasivenessandsupport.Theformerdescribestheabilityofoneindividualtopersuadetheotheronetochangehis/heropinion.Thelaterdescribestheabilityofoneindividualtosupporttheotheroneinhis/heropinion.Differentvariantsofthemodelwereex-plorednumerically15,16,17,andmanyoftheobservationswerethanexplainedintheframeworkofameanfieldapproach18,19,20andtheLandautheory21.

Herewewouldliketopresentaratherdifferentapproachtodescribepsycho-logicalcoupling.Insteadofpersuasivenessandsupportwewillstudytheeffectofgregariousnessandindividuality.Wewillshowhowthetwofeaturesmayaffectpsychologicaldistancebetweenindividualsandhowthedistancechangesopinionformationinthesociety.Finally,wewillshowthatourmodelcouldbemappedtoferromagneticfluidnotinEuclideanbutinasocialspace.2.Themodel

OursystemconsistsofNindividuals(membersofasocialgroup);weassumethateachofthemcanshareoneoftwooppositeopinionsonacertainsubject,denotedasσi=±1,i=1,2,...N.TheHamiltonianofthemodelreads:

󰀁

Ji,j(t)σiσj.(1)H=−

iIndividualscaninfluenceeachotherwiththestrengthJi,j(t)whichcanbeun-derstoodasaninverseoftheirdistanceinasocialspace.Theabovemeansthata

strongerimpactcorrespondstoashorterdistance.WeassumethatsocialdistancesarechangingintimeandweputonthefollowingdynamicsofthestrengthJi,j(t):

Ji,j(t+1)=Ji,j(t)(1+η−ασiσj).

(2)

Theparameterη>0isresponsibleforcontinousgrowthofthesocialstrengthandcanbeidentifiedasgregariousnessofi-thindividualwhichleadstotighteningoftieswithotherpeople.Inotherwords,peoplefromtheirnatureseekthecompanyofothers.Theparameterα>0describesanothernaturaltendencyofpeoplewhichisaneedtobedifferentthanasurroundingcrowd,i.e.itreflectstheinclinationofanindividualtodemonstratehis/herindividuality.

Forcompletenessofthemodelweassumeasaninitialconditionanypositiveval-uesofJi,j(t=0).Theconditionassuresthatduringthesystemevolutioncouplingsarealwayspositiveinthemostinterestingrangeofparametersηandα.

Now,letusconcentrateonthephasediagramforthepresentedmodel(fig.1).Itisdividedintofourdifferentregionsbythreecurves.Thecurve1isthemostobviousone.Foreverysetofparametersabovethiscurve,i.e.forη>α,couplingstrengtheswillincreasetoinfinityinexponentialway.Parametersη<α−1,limitedbythecurve3,alsomakethesystemunstable,butnowcouplingstrengthescanbecomenegative.ItmeansthatineverystepJwillchangeitssignand|J|willdivergetoinfinity.Thestableregionliesbetweentwocurves1and3.

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Ferromagneticfluidasamodelofsocialimpact

3

2.01.51unstableJ→∞2 η

1.0F0.5P0.00.00.51.03unstable|J|→∞1.52.0αFig.1.Phasediagramofthemodel(1)-(2).F-ferromagneticphase,P-paramagneticphase.Detailedexplanationinthetext.Toexplainthecurve2letusconcentrateonasinglecouplingandinvestigatethefollowingprocess:J(t)→J(t+1)→J(t+2)=J(t).Asonecanseefromeq.2,ineverystepJgrowsordecreasesbysomewelldefinedvalue.TheaboveprocessisthesimplestoneinwhichJstaysatsomefixedlevel,i.e.itgrowsandthenitdecreasestothesamevalue.LetusassumethatJ(t+1)>J(t)(theoppositecaseisanalogous).ItmeansthatJ(t+1)=J(t)(1+η+α).Then,innextstepJshoulddecrease,i.e.J(t+2)=J(t+1)(1+η−α).Fromaboveonecanobtainthefollowingcondition:

󰀂ηc=

αJi,j(t)

=

∆Ji,j(t)

α

−σiσj,(4)

thenonecanseethatαplaystheroleoftimescale.ItmeansthatforlargeαthesuccesivevaluesofJareverydistant(Jchangesveryfast)andspindynamicscannotfollowtocompensatechangesofJ.Itmanifestsitselfinlongtimeobservedparamagneticstatesinterruptedbylongtimeobservedferromagneticstates.

Insummary,theinterestingfromthesociologicalpointofviewrangeofparam-etersisηcritical<η<α≪1.

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Thedynamicsofchangesofindividual’sopinionisgivenbyasimpleMonteCarloprocedurebasedontheMetropolisalgorithm.AtemperatureTgiveninthealgorithmmaybeinterpretedasa”socialtemperature”describingdegreeofrandomnessinthebehaviorofindividuals,butalsotheiraveragevolatility.Theprocedureconsistsoftwosteps.InthefirststepweupdatestatesofNrandomlychosenindividuals.Inthesecondstepweupdatecouplingstrengthsforallnodesaccordingtoeq.(2).

Aswewillshowforawiderangeofparametersηandα,regardlessofchoosingatemperaturethesystemtendstobeinaferromagneticregime.Itmeansthatdespiteatendencytomanifestindividualitymostofindividualsinteractwiththeotherpeoplewhosharethesameopinion.3.Results

Atypicaldependenceofmagnetizationperspin|m|onsystemparametersη/αisshownatfig.2.Consideringηasanorderparameter,continous(secondorder)phasetransitionoccursforηcgivenbyeq.(3).Openpointspresentedatfig.1obtainedfromsimulationsconfirmthattheabovederivationiscorrect.

Onecanseefromthefig.4thattheabsolutevalueofthemeanmagnetizationisan(increasing)functionofη/αbutitiscompletelyindependentonthesystemtemperatureT.Thefactcanbeunderstoodasfollows.Accordingtoeq.(4)wecanwritethefollowingequationforthemeanvalueforthelogarithmofJi,j

󰀄

η1

=

dt

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Ferromagneticfluidasamodelofsocialimpact5

factorsexp(−Ji,j/T).Thustakingandvantageofthemeanfieldapproximationwethat

1

dt

󰀄

=

ηcanwriteFebruary2,20087:13WSPC/INSTRUCTIONFILEising˙jh

6P.Fronczak,J.A.Ho󰀱lystandA.Fronczak

a)|m|

1

*|m󰀀|b)|m|1*|m󰀀|0T1TC1

T0

T1TC1T2TC2

TFig.4.Schematicpictureofprocessesoccuringinthesystemduringthechangingofthetem-peratureTandη/α=const;a)equilibrium(initial)stateofthesystem;b)ThechangeofthetemperatureTfromT1toT2forcesmagnetizationcurvetoreshapetonewequilibriumconditions.

totheshapeshownatthefig.4b(obtainedbythesamemethodasbefore).Itmeansthatregardlessofachoiceofthesystemtemperaturewearealwaysbelowthecriticaltemperature,i.e.intheferromagneticstate.

Thesociologicalconclusioncouldbeasfollows:regardlessofa”socialtemper-ature”peoplealwaystrytocorrelatetheiropinionswithothers(creategroupsofinterest).Thistendencytosharethesameopinionwithotherpeople,regardlessofsomeexternalforces,makeus,people,soresistanttotrialsofdespotstomakethepeopleunorganizedanddisoriented.Ofcoursetheparameterη/αcharacterizesourown(notsocial)pointofviewwhichgivesussomeindependencyrespectingotherpeopleopinion.

Fromthepointofviewofcomplexnetworksdomain22,23itisinterestingtoconsiderthemodelasaweightednetwork,wherenodescorrespondtoindividualsandlinkshaveassignedweightsequaltocouplingstrength.Oneofnontrivialobser-vationsisadistributionofcouplingstrengthesP(J)whichispresentedatfig.5.AsonecanseeforlargetemperatureTthedistributionhasaformofpowerlawwiththeexponentγ≈0.85.

Itseemsthatthereshouldbeastrongrelationbetweentheobservedpower-lawdistributionsanddistributionsobtainedduetoamoregeneralclassofmultiplicativerandomprocesses24,25.Iffact,onecaneasilyfindsomesimilarityofeq.(2)toeq.(1)in24.Thedifferencesoccurwhenonetakesintoaccountthetemperatureanditsinfluenceondistributionsatfig.(5).Wesuspectthatthemodelstudiedbyussettlessomewherebetweentwomultiplicativerandomprocessesstudiedin24and25.Thishypothesisisstillunderinvestigationandtheresultswillbepublishedelsewhere.

Nowletusdrawattentiontosimilarityofthepresentedmodeltomagneticfluidswhicharewidelystudiedforthelastthirtyyears26.Magneticfluidsaredescribedbyinteractingmoleculeswithbothtranslationalandspindegreesoffree-dom.Theyinteractduetoweaklong-rangedexchangeinteractionsinadditiontospin-independentisotropicattractiveforces.Themostsimplephysicalparameter

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Ferromagneticfluidasamodelofsocialimpact7

101010

0

-1 η/α = 0.1 η/α = 0.4 η/α = 0.9-2P(J)10101010

-3 -4-5{T=0.01-610-410-310-2{T=0.110J

Fig.5.

Distributionofcouplingstrengthsfordifferentvaluesofparametersη/αandT.

usedinphasediagramsofmagneticfluidshasaform

󰀅

φex(󰀘r)d󰀘r

R=

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References

1.H.Haken,Synergetics.AnIntroduction(Springer-Verlag,Heidelberg,NewYork,1983);AdvancedSynergetics(Springer-Verlag,Heidelberg,NewYork,1983).

2.G.A.Cowan,D.Pines,D.Meltzer(eds.),Complexity.Metaphors,Models,andReality(Addison-Wesley,SantaFe,1994).

3.D.Amit,ModelingBrainFunction(CambridgeUniv.Press,Cambridge,1989);E.Do-many,J.L.vanHemmen,K.Schulten(eds.)ModelsofNeuralNetworks(Springer,Berlin,1995);A.Browne(ed.),Neuralnetworkanalysis,architecturesandapplica-tions(InstituteofPhysicsPublishing,Bristol,1997).

4.A.Johansen,PhysicaD78,186(1994);H.C.Tuckwell,L.Toubiana,J-F.Vibert,Phys.Rev.E57,2163(1998).

5.P.Bak,K.Sneppen,Phys.Rev.Lett.71,4083(1993);A.Pekalski,PhysicaA252,325(1998);

6.D.Helbing,Phys.Rev.E55,3735(1997);PhysicaA219,375(1995);D.Helbing,P.Molnar,Phys.Rev.E51,4282(1995).

7.W.Weidlich,G.Haag,ConceptsandModelsofQuantitativelySociology(Springer,Berlin,NewYork,1983);W.Weidlich,PhysicsReports204,1(1991).8.J.Fort,V.M´endez,Phys.Rev.Lett.82,867(1999).

9.D.Sornette,A.Johansen,PhysicaA245,1(1997);N.Vandewalle,M.Ausloos,P.Boveroux,A.Minguet,Eur.Phys.J.B4,139(1998).10.W.Weidlich,J.Math.Sociology18,267(1994).

11.D.Helbing,PhysicaA193,241(1993);J.Math.Sociology19,189(1994);D.Helbing,

QuantitativeSociodynamics(KluwerAcademic,Dordrecht,1995).12.S.Galam,PhysicaA230,174(1996);PhysicaA238,66(1997).13.D.B.Bahr,E.Passerini,J.Math.Sociology23,1(1998).14.B.Latan´e,Am.Psychologist36,343(1981).

15.R.R.Vallacher,A.Nowak(Eds.),Dynamicalsystemsinsocialpsychology(SanDiego,

AcademicPress,1994).

16.E.L.Fink,J.Communication46,4(1996);B.Latan´e,J.Communication46,13

(1996).

17.A.Nowak,J.Szamrej,B.Latan´e,Psych.Rev.97,362(1990).18.M.Lewenstein,A.Nowak,B.Latan´e,Phys.Rev.A45,763(1992).19.K.KacperskiandJ.A.Holyst,J.Stat.Phys.84,169(1996).

20.J.A.Holyst,K.KacperskiandF.Schweitzer,AnnualReviewofComput.Phys.9,

253-273(2001).21.D.Plewczy´nski,PhysicaA261,608(1998).

22.S.N.DorogovtsevandJ.F.F.Mendes,Evolutionofnetworks(OxfordUniv.Press,

2003).

23.S.BornholdtandH.G.Schuster,Handbookofgraphsandnetworks(Wiley-Vch2002).24.M.LevyandS.Solomon,Int.J.Mod.Phys.C7,595(1996).25.D.SornetteandR.Cont,J.Phys.I(France)7,431(1997).

26.P.C.HemmerandD.Imbro,Phys.Rev.A16,380(1977);J.M.Tavaresetal,Phys.

Rev.E52,1915(1995);W.FenzandR.Folk,Phys.Rev.E67,021507(2003);F.LadoandE.Lomba,Phys.Rev.Lett.80,3535(1998).