Person-Situation Debate Revisited: Phase Transitions with Quenched and Annealed Disorders

14 Pages·2017·0.44 MB·English

entropy
Article
Person-Situation Debate Revisited: Phase Transitions
with Quenched and Annealed Disorders
ArkadiuszJe˛drzejewski ID andKatarzynaSznajd-Weron*
DepartmentofTheoreticalPhysics,FacultyofFundamentalProblemsofTechnology,WrocławUniversityof
ScienceandTechnology,Wrocław50370,Poland;[email protected]
* Correspondence:[email protected];Tel.:[phone]
Received:30June2017;Accepted:9August2017;Published:13August2017
Abstract: Westudytheq-votermodeldrivenbystochasticnoisearisingfromoneoutoftwotypesof
nonconformity: anticonformityorindependence. Wecomparetwoapproachesthatwereinspired
by the famous psychological controversy known as the person–situation debate. We relate the
personapproachwiththequencheddisorderandthesituationapproachwiththeannealeddisorder,
and investigate how these two approaches influence order–disorder phase transitions observed
in the q-voter model with noise. We show that under a quenched disorder, differences between
models with independence and anticonformity are weaker and only quantitative. In contrast,
annealing has a much more profound impact on the system and leads to qualitative differences
between models on a macroscopic level. Furthermore, only under an annealed disorder may
the discontinuous phase transitions appear. It seems that freezing the agents’ behavior at the
beginningofsimulation—introducingquencheddisorder—supportssecond-orderphasetransitions,
whereasallowingagentstoreversetheirattitudeintime—incorporatingannealeddisorder—supports
discontinuousones. Weshowthatanticonformityisinsensitivetothetypeofdisorder,andinall
casesitgivesthesameresult. Weprecedeourstudywithashortinsightfromstatisticalphysicsinto
annealedvs. quencheddisorderandabriefreviewofthesetwoapproachesinmodelsofopinion
dynamics.
Keywords: phasetransitions;disorder;nonlinearvotermodel;opiniondynamics
1. Introduction
What is more influential in determining a person’s behavior—personality or situation? This
questionistheessenceoftheperson–situationdebate,oneofthemostsignificantcontroversiesinthe
historyofpersonalitypsychology[1]. Inthefirstthirdofthe20thcentury,itwaspossibletobelieve
that social behavior was best understood in terms of personality traits [2]. Thus, the best way to
predictorexplainaperson’sbehaviorinaparticularsituationwouldbetoexamineapersonalitytrait
correspondingtothebehavior. However,asclaimedbyRichardE.Nisbett(oneofthemosteminent
social psychologists in the world over the past 40 years), the trait conception as the determinant
ofsocialbehaviorisincorrect. Theempiricalevidenceindicatesthatonlyveryweakpredictability
is possible using traditional trait construct [2]. Nevertheless, the person–situation debate is still
aliveamongsomepersonalitypsychologists[1]. Asphysicists,wehavenointentionofresolvingthis
controversy.However,thequestionofwhichisabetterpredictorofanindividual’sbehavior—situation
or personality—is crucial in the context of agent-based social simulations. Therefore, the debate
inspiredus3yearsagotocomparethesetwoapproaches(personvs. situation)withinarelatively
simpleyetveryinterestingmodelofopiniondynamics—namely,theq-votermodel[3,4].
Asaprimeexampleofanopinionformationmodel,theq-voterdynamicshasgainedagreatdeal
ofattentioninrecentstudies[5–11]. Asaspreadingmechanism,itfindsmanyapplicationsnotonly
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inthefieldofopiniondynamics,butalsoinmodelingthediffusionofinnovations,decision-making
processes,orexpansionofinformation[11–15]. ItwasdevelopedbyCastellanoetal. [3]asanonlinear
versionofthevotermodel[16]which,bycontrast,wasoriginallyintendedtoimitateinterspecific
competitions. Intheq-voterdynamics,interactionsresultingfromconformitygiverisetointeresting
orderingphenomena.
However, it is well known that conformity—understood as a change in behavior or belief to
accord with others—is not the only type of social response [17]. In social psychology, models of
social response concern the identification and delineation of possible responses to social pressure.
The basic goal of researchers in this field is to identify the minimum number of variables that is
neededtoadequatelydistinguishbetweenresponsessuchasconformity(positiveinfluence)andtwo
typesofnonconformity: independence(noinfluence)andanticonformity(negativeinfluence)[18].
Therefore,later,theq-votermodelwasalteredbyincorporatingtwotypesofstochasticnoisewhich
canbethoughtofasdifferentmanifestationsofnonconformity: independenceandanticonformity[5].
Bothkindsofnonconformitytendtodestroytheferromagneticorder. Therefore,themodifiedmodel
exhibitsorder–disorderphasetransitions. Interestingly,ithasoccurredthattheq-votermodelwith
independenceexhibitstricriticality;i.e.,theswitchfromcontinuoustodiscontinuousphasetransition
foracriticalvalueofq =5.Ontheotherhand,themodelwithanticonformitydisplaysonlycontinuous
phasetransitionforarbitraryvalueofq[5,6].
Originally,nonconformitywasintroducedtotheq-votermodelasastochasticdisturbance,which
meansthatwithacertainprobability,eachagentcouldbehaveasanonconformistoraconformist,
whichcorrespondstothesituation-orientedapproach. Theperson–situationdebateinspiredusto
investigatehowresultswouldchangeifweusedaperson-orientedapproach(i.e.,wheresomeagents
are permanently conformists and other nonconformists). In our previous paper, we have focused
ontheq-votermodelwithindependencewithafixedvalueofparameterdescribingthesizeofthe
group of influence q = 4 [4]. At that time, our aim was to examine what the role of the so-called
flexibilityfactor f wouldbe,whichdescribestheprobabilityofchangeincaseofindependence. We
haveshownthatthisfactorinfluencesresultsonlyincaseofasituation-orientedapproach,andforthe
person-orientedapproach,resultsdonotdependon f.
Recently, we have realized that we could overlook many interesting results by focusing only
on the q-voter model with independence and a fixed value of q = 4. Therefore, we have decided
toinvestigatebothversionsofthemodelinthiswork: modelA(conformity+anticonformity)and
modelI(conformity+independence)foranarbitraryvalueofq. Thisstudyallowsustoaddresstwo
previouslyaskedquestionsatthesametime,namely:
1. Dodifferencesbetweentwotypesofnonconformitythatarerecognizedbysocialpsychologists
ontheindividual(microscopic)levelmanifestonthesociety(macroscopic)level? Thisquestion
hasalreadybeenaddressedintwopapers,butonlywithinthesituation-orientedapproach[5,6].
2. Dodifferencesbetweentwotypesofapproach(situation-andperson-oriented)introducedon
theindividual(microscopic)levelmanifestonthesociety(macroscopic)level? Thisquestionhas
alreadybeenaddressedin[4],butonlywithintheq-votermodelwithindependencewithq =4.
From a physical point of view, the second question corresponds to the problem of
annealed(situation)vs. quenched(person)disorder. Therefore, inthenextsectionwewillbriefly
describesomeinsightsfromstatisticalphysicsintothisissue. Then,inSection3,wewillbrieflyreview
whatkindsofapproachesareusedinopiniondynamicsmodels. InSection4wewilldescribethe
modelthatwillbestudiedinthispaper,andinSection5wewillpresentanalyticalresultsforour
model. Wewillconcludeinthelastsection.
2. QuenchedandAnnealedDisorder
Disorderisaninherentpartofanyrealsystem. Itintroducesinhomogeneitiesintothemodels
imitatingexternalfluctuations,structuralchanges,orimpuritieswhicharepresentintherealworld.
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Developing a fundamental understanding of disorders is, therefore, one of the essential aspects
in modern statistical physics. They play especially important role in modeling complex systems
by appearing in the network structures, for instance, changing the link arrangements or node
degrees [19–21]. Disorders may also modify the character or the strength of interactions [22,23].
Regardless of their origin and nature, we can distinguish two types of disturbances based on the
time-scaleoftheirappearance[23–25]. Disorderissaidtobequenchedwhenitdoesnotevolvein
time. Itispermanentlyfrozenthroughtheentireperiodofobservation,sothedisorderfluctuation
time-scaleisconsiderablylongerthantherateofchangesofdynamicalvariableswhichdescribethe
system. Insuchacase,theimpuritiesarenotallowedtomove,thenetworkstructureisfixed,and
theinteractionsareestablished. Therefore, inordertostudythethermodynamicpropertiesofthe
model,thepartitionfunctionisdeterminedforthespecificdisorderrealization,andonlythestate
functionsareaveragedoverdistinctdistributionsofdisturbances. Acompletelydifferentapproach
shouldbeappliedwhenbothtime-scalesarecomparable. Then,therandomnessconnectedwiththe
disorderisnotfixed,anditvacillatesintime. Ifthisisthecase,wetalkabouttheannealeddisorder,
andnow,thedistributionofdisturbanceshastobetakenintoaccountalreadyinthecalculationofthe
partitionfunction.
Ingeneral,thesetwospecifictimeregimesgivecompletelydifferentresults.Theannealeddisorder
connectedwiththenetworkstructureintroducesadditionalfluctuationsintothesystemwhichmay
lowerthetransitiontemperatureasin[23,26], orevencauseso-calledrobustcriticalbehavior[26].
Furthermore,whenweconsiderscale-freenetworkswithpowerlawtailsinthedegreedistribution,
theannealedsystemexhibitsanomalousfinite-sizescalingincontrasttothequenchedone[19]. Onthe
other hand, the same type of disorder but this time associated with the character and strength of
interactionsmayincreasethecriticaltemperature[27]sothatthefinalsystemismorestable. Recently,
ithasbeenshownthatsomedynamicsaremoresensitivetothechangesofdisturbancekind.Thiseffect
isespeciallypronouncedintheIsingmodelwithMetropolisdynamics,wherethegraphannealingcan
evenswitchthetypeofphasetransitionfromcontinuoustodiscontinuous[28].
3. Personvs. SituationApproachinModelsofBinaryOpinionDynamics
Richard E. Nisbett claims that behavior can be predicted much better from the social setting
than from personality traits. Indeed, in the strong version of his position, personality is only an
illusion—albeitaquitepowerfulone[2]. Ononehand,hecallstheperson-orientedapproach,after
Lewin,anaivepsychology. Ontheother,heexplainswhytheideaofpersonaltraitsasadominant
predictorofbehaviorissoattractiveandnatural. Hecomparesnaivepsychologytoancientphysics,
when the behavior of objects was understood exclusively in terms of object itself: A stone sinks
whenplacedinwaterbecauseithaspropertyof“gravity”;apieceofwoodfloatsbecauseithasthe
propertyof“levity”[2]. Infact,thisexampleisnotveryconvincingforaphysicist,andfollowingthis
argument,wecouldarguethatprobablybothpersonalityandsituationareimportant. Itistruethat
thegravitationalandbuoyantforcesactonbothitems,butstoneandwoodbehavedifferentlybecause
ofanimportantpropertyofanymaterial,namelydensity(i.e.,its“personaltrait”). Anyway,associal
psychologistsclaim,theempiricalevidenceindicatesthatpersonalityisaveryweakpredictorofan
individual’sbehavior,andthesituationcancompletelydominateoverthepersonaltraits[2,17].
The question of which approach is more fundamental and better describes real systems is
particularlyimportantforsocialagent-basedmodels,suchasmodelsofopiniondynamics. Frequently,
these models originate from statistical physics. For instance, the Ising model with ferromagnetic
interactions in an external magnetic field were used to interpret collective social phenomena and
decision-makingprocesses[29,30]. Asintheq-votermodel,anopinionwasmappedintoatwo-state
variable,buttheinteractionsweregivenbythefunctionmeasuringthedegreeofconvergenceinthe
societywhichbringstomindtheHamiltonianfunction. Instead,intheq-votermodel,theinteractions
aredefinedbythemicroscopicdynamicalrulesdescribedinthenextsection.Moreover,intheproposed
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model,theindividualbiasconnectedwithpersonalbelievesandexperienceswasintroducedbya
quenchedlocalfield,andtheannealedapproachwasnotconsidered.
Nowadays,insociophysics,wecanfindmodelsofopiniondynamicswhichincorporateboth
approaches. Forexample,Galamhasintroducedacontrarianbehaviortohismodelinthespiritof
annealeddisorder[31]. Thismeansthatineachtimestep,afixedfractionaoftheagentsisrandomly
chosentobecontrarians(i.e.,anindividualthatistossedasacontrarianinonetimestepmayactasa
non-contrarianinthenexttimestep)[32]. Afterwards,differentmodificationshavebeenproposed
toGalam’smodel,includingavariableintimefractionofcontrariansorone-sidedcontrarians,the
antagonisticbehaviorofwhichisrestrictedonlytotheglobalmajorityopinion,butallofthemshowed
up in the annealed form [33]. Stauffer and Martins introduced another—in their opinion, more
realistic—quenchedapproach,andshowedthatresultsforthesetwoapproachesaredifferentforthe
originalGalam’smodel[32]. Intheirapproach,afractionaofthepopulationisrandomlyselectedas
contrariansatthebeginningofthesimulation,andtheykeepthistraituntiltheendofthesimulation.
As the order parameter they have chosen (as usual in this kind of model) magnetization; i.e., the
numberofindividualswithpositiveopinionminusthatofnegativeopinion,dividedbythetotalsize
ofthepopulation. Ithasbeenshownthatcontinuousorder–disorderphasetransitionisobserved
withinbothapproaches,butthecriticalfractionofcontrariansismuchhigherforthequenchedversion.
In a similar spirit, Schneider has introduced three different types of behavior in addition to
conformity to the Sznajd model [34]: unconvincibles (people who cannot be convinced at all),
opportunists (agents who adopt the opinion of the majority of all others, instead of neighbors),
andcontrarians(inhisapproach,theseagentscannotbeconvincedbytheirneighborhood,butthey
alwaysadopttheoppositeopinionofthemajorityofallotherpeople). Hehasinvestigatedthesethree
typesofbehaviorwithinquenchedorannealedapproaches. Similarly,asforGalam’smodel,only
quantitativedifferencesbetweenthesetwoapproacheshavebeenseen. Ingeneral,differenceshave
beenobservedintheexittime(i.e.,timeneededtoreachanabsorbingstate)—evolutionwasfasterfor
theannealedapproach. OneshouldnoticethatthecontrarianbehaviorinvestigatedbySchneiderisof
amean-fieldtype,contrarytoGalam’sidea,whoassumedthatcontrarianstaketheoppositeopinion
totheneighborhoodandnottoallotherpeople. Theimportanceoftheinfluencerangeinopinion
dynamicswaslaterstudiedbyJavaroneinbothquenchedandannealedapproaches[35]. Basedonthe
majorityrule,heintroducedconformistsandcontrarianstohismodel. Inonescenario,theinteractions
haveonlylocaldimension(i.e.,thenonconformistbehaviorisrestrictedonlytothesocialcircleofeach
agent),whereasinthesecondscenariothisbehaviorisrelatedtothewholepopulation. Onceagain,
thequantitativedifferencesbetweentwodisorderswereobserved—theannealedmodelturnsoutto
exhibithighervaluesofthemagnetization.
Just as Stauffer and Martins argued that the quenched approach is more realistic, Lama et al.
claimedthatintroducingagivenfractionofcontrariansintheinitialstatewasartificial. Therefore,
Lamaetal. introducedtheprobabilityofbeingcontrarians,asaparameter,totheSznajdmodel[36].
This means that with certain probability each agent can behave like a contrarian, and with
complementaryprobability,itbehaveslikeaconformist,whichcorrespondstotheannealedapproach.
GalamandJacobshavenoticedthatinadditiontocontrarianbehavior,anothertypeofbehavior
isalsoquitecurrentinrealopiniondynamics;namely,inflexibleattitude. Theyhavealsoassumed
thattheinflexibleattitudeisaquenchedindividualstate: inflexibleagentskeeptheiropinionalways
unchanged[37].Inflexiblesandunconvinciblesareinfactthesametypeofbehavior,whichcorresponds
toindependence. Anothertypeofindependentagenthasbeenintroducedwithinthevotermodelby
Mobiliaunderthename“zealot”[38]. Originally,thezealotisanindividualwhofavorsoneopinion,
andthus,itisallowedtochangeitsstatewithoutregardtoitsneighbors. Recently,theideaofinflexible
zealots,whichneverchangetheiropinion,hasalsobeeninvestigatedwithintheq-votermodel[7,8].
Zealotshavebeenintroducedwithinthequenchedapproach,contrarytoindependencewhichhas
beenintroducedtotheq-votermodelasanannealeddisorder[5,6]. Unfortunately,zealotrybehavior
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cannotbecompareddirectlywithindependencebecausezealotsintheq-votermodelareinflexible,
whereasunderindependenceanagentcanstillchangeitsopinionbutindependentlyofothers.
Applications of opinion dynamics also range over the evolutionary game theory. Lately, the
significanceofconformistshasbeenanalyzedinthespatialpublicgoodsgame,butonlyinthespirit
ofquencheddisordersothatindividualscouldnotchangetheirattitude[39]. Itturnsoutthatsocial
influencemaycausedifferentsystembehavior,leadingtofullcooperationorbistableequilibria.
Theabovebriefreviewshowsthatindeeddifferentapproachesareused,andforsomephysicists
the annealed approach seems to be more realistic, whereas for others quenched disorder is better,
whichrecallstheperson–situationdebateamongpsychologists. However,itseemsthatrealityismore
complex,andpersonallong-termtraitsinteractwiththeimmediatesituationtoinfluenceindividuals’
behavior. AsnotedbyGoldstein: eventhemostviolentindividualisnotperpetuallyviolent[40]. It
hasalsobeenshownempiricallythatsituationalvariablesaremoreimportantinpredictingimmediate
behaviorinspecificsituations,whereaspersonaltraitsdescribethelong-termaveragebehavior[41].
Takingthesefindingsintoaccount,onecouldspeculatethatindescribingopinionsrelatedtoimportant
fundamentalissuessuchasreligiononeshouldusethequenchedapproach. Probablyforthesame
reason,thequenchedapproachwouldbemoresuitabletodescribepoliticalopinions. Ontheother
hand,inmodelingdifferentmarketingphenomenalikediffusionofinnovation,theannealedapproach
wouldbemorerealistic. Apartfromwhichapproachismorerealistic,physicistsareinterestedina
comparison between quenched and annealed approaches [4,32,34]. In all described examples, the
differences between approaches were only quantitative: visible in the shift of the critical value or
intheexittime. Here,wewillshowthatasystematicstudyoftheq-votermodelwithtwotypesof
nonconformityforarbitraryqrevealsmoreprofounddifferences.
4. ModelDescription
Withintheq-votermodel,weconsiderNagentsplacedonacompletegraph,sothateachnodeof
anetworkisassociatedwithonlyoneautonomousvotercharacterizedbyabinaryvariables = ±1,
i
i ∈ {1,...,N}. Thechosentopologyimpliesthatallindividualsareneighborsandcaninteractmutually.
Although such structures may be too simple and unrealistic to model large social networks, they
are decent representations of interactions within small groups [14]. Additionally, they allow for
analytical treatment based on the mean-field approach, and are therefore often incorporated into
analyses [3,5,8]. More complex structures are also investigated as underlying frameworks for the
q-voterdynamics[10,42],butforsimplicity,weconsideronlyafullyconnectedgraph. Interactions
in the system are driven by three factors interpreted as different social responses: conformity,
independence,andanticonformity[43]. Onlythefirsthasorderingproperties,whereastheothertwo
tendtodisturbtheorder,andthustheyareregardedasstochasticnoise. Inpreviouspapers[5,6],two
distinctmodelswereconsidered. Bothofthemincludeconformityasorderingforce,buttheroleof
disorderisonceenactedbyindependenceandoncebyanticonformity. Inbothapproaches,inevery
elementarytimestep,wechooseatrandomanagentwhichisreconsideringitsopinionandagroupof
influencecomprisedofqofitsnearestneighbors. Thegroupiscalledq-panelandtriestoexertsocial
pressureonthechosenvoter. Withprobability1−p,theagentconformstothispressureandembraces
thesameopinionoftheq-panel,butonlyifthegroupisunanimous(i.e.,allqindividualsareinthe
samestate). Otherwise,theopinionofthechosenvoterremainsunchanged. Withcomplementary
probability p,ontheotherhand,itactsasanonconformist,andtherefore,twodifferentscenariosmay
happendependingonthespecificdisorderinginteractionsincorporatedinthestudiedq-votermodel:
• Model I (Independence + Conformity): the voter autonomously decides whether to change
its viewpoint, and it does not rely on its peers’ opinions. In the simulation, it is realized by
changingtheagent’sstatetotheoppositeonewithprobability1/2. Thissocialresponserefersto
independence[43].
• ModelA(Anticonformity+Conformity): thechosenagentdisagreeswiththegroup. Itdoesnot
submittosocialnorms,anditteasinglyadoptstheoppositeopiniontotheq-panel’s,provided
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thatthegroupofinfluenceisunanimous. Ifitisnot,thechosenagentpreservestheviewpointit
hadbefore. Thissocialresponsereferstoanticonformity[43].
It was shown that these two different types of noncomformity—identified at microscopic
level—lead to distinctive macroscopic behavior of the system. From the point of view of social
science,itmeansthatdifferencesbetweenresponsestosocialpressurerecognizedbypsychologistsat
theindividuallevelmaymanifestonalargescaleofwholesocieties[6,43]. However,inmoststudies
andsimulations,theagent’sbehaviorsweredeterminedbysituationalfactorsandwerenotconnected
withpersonaltraits. So,theparameterpwasthesameforallagents,andthesamevotercouldbeeither
conformistornonconformistindifferenttimesteps. Thisapproachintroducesanannealeddisorder
tothesystem,sinceindividualsmaychangetheirbehaviorduringonesimulationandactonceas
conformists and once as nonconformists. The quenched disorder, on the other hand, corresponds
to the situation in which before the simulation we determine whether an agent is conformist or
nonconformist,anditsattitudedoesnotchangeintime. Inordertoachievethatandadditionallymake
bothapproachescomparable,atthebeginningofsimulation,werandomlyassignthetypeofbehavior
toallagents. Withprobability p,agivenagentbecomesanonconformist,andwiththecomplementary
probability1−p, itturnsintoaconformist. Afterthisassignment,theagent’sattitudestaysfixed.
Lateron,themechanismbehindtheopinionchangeisthesameasintheannealedmodel. Inevery
elementarytimestep,wechooseanagentatrandom,anddependingonthisassignedattitude,the
agentdecideswhethertochangeitsopinion.
Bothdescribedvariantsoftheq-voterdynamics—modelIwithindependenceandmodelAwith
anticonformity—have already been studied in terms of annealed disorder [5,6]. In this work, we
analyzethequenchedapproachtothesemodels,whichhasnotbeeninvestigatedbeforeexceptforthe
particularcasewhereq = 4andonlyformodelI[4]. Additionally,werecallthepreviousresultsin
ordertocomparebothquenchedandannealedapproaches.
Tomaintainthereviewcharacterofprevioussectionsandforclarityofpresentation,wedescribe
belowalgorithmsforallmodelsconsideredinthiswork.
4.1. ModelI(Independence+Conformity)withintheAnnealedApproach
Initialization(settimet =0):foreachnodeofagraphi =1,...,Nchoosearandomnumberfrom
uniformdistributionr ∼U(0,1). Ifr < c(0)thenassigntothenodeastateS(0) =1(↑);otherwise,
i i i
Si(0) = −1(↓). Thismeansthatinitiallytheconcentrationofup-spinsisequaltoc(0) = N↑(0)/N,
whereN↑(0)isanumberofup-spinsatt =0. Usuallyoneoftwotypesofinitialstateisused: ordered
(c(0) =1)orrandom(c(0) =1/2). Afterinitialization,thesystemevolvesaccordingtothefollowing
algorithm:
1. Chooserandomlyanduniformlyifromtheset{1,2,...,N}andgoto2.
2. Choosearandomnumberfromuniformdistributionr ∼U(0,1)andgoto3.
3. Ifr < pthenS(t+∆t) = −S(t)withprobability1/2,andwiththesameprobability,S remains
i i i
unchanged(independence),sett = t+∆tandgoto1;otherwise,goto4.
4. Choose randomly a group of q distinct neighbors of the chosen i-th node: n ,n ,...,n ∈
1 2 q
{1,2,...,N}andgoto5.
5. Ifallqneighborsareinthesamestate,thechosenspinS takesthesamestateasitsqneighbors
i
(conformity);i.e.,S(t+∆t) = S (t),sett = t+∆tandgoto1.
i nq
Asusual,∆t =1/N,whichmeansthatthetimeunitconsistsofNelementaryupdates.
4.2. ModelA(Anticonformity+Conformity)withintheAnnealedApproach
InitializationisthesameasinModelIwithintheannealedapproach,describedinSection4.1.
Afterinitialization,thesystemevolvesaccordingtothefollowingalgorithm:
1. Chooserandomlyanduniformlyi ∈ {1,2,...,N}aswellasqdistinctneighborsofthei-thnode:
n ,n ,...,n ∈ {1,2,...,N}andgoto2.
1 2 q
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2. Ifallqneighborsareinthesamestate,theywillinfluenceS(t)andthusgoto3;otherwise,set
i
t = t+∆tandgoto1.
3. Choosearandomnumberfromuniformdistributionr ∼U(0,1)andgoto4.
4. Ifr < p,thenS(t+∆t) = −S (t)(anticonformity);otherwise,S(t+∆t) = S (t)(conformity),
i nq i nq
sett = t+∆tandgoto1.
4.3. ModelI(Independence+Conformity)withintheQuenchedApproach
Initialization(settimet =0): foreachnodeofagraphi =1,...,Nchoosetworandomnumbers
r,z ∼ U(0,1). Ifr < c(0),thenassigntothenodeastateS(0) = 1(↑);otherwise, S(0) = −1(↓).
i i i i i
Additionally, set the type of the node, which will be constant in time: if z < p then spin S is
i i
independent;otherwise,itisaconformist. Afterinitialization,thesystemevolvesaccordingtothe
followingalgorithm:
1. Chooserandomlyanduniformlyifromtheset{1,2,...,N}andgoto2.
2. Ifthechosenspinisindependent,thenS(t+∆t) = −S(t)withprobability1/2,andwiththe
i i
sameprobabilityitremainsunchanged;sett = t+∆tandgoto1. Otherwise,goto3.
3. Choose randomly a group of q distinct neighbors of the chosen i-th node: n ,n ,...,n ∈
1 2 q
{1,2,...,N}andgoto4.
4. If all q neighbors are in the same state, the chosen spin S takes the same state as its q
i
neighbors(i.e.,S(t+∆t) = S (t)),sett = t+∆tandgoto1.
i nq
4.4. ModelA(Anticonformity+Conformity)withintheQuenchedApproach
Initialization is almost the same as in Model I within the quenched approach, described in
Section4.3. Theonlydifferenceisthatinsteadofindependentspinswehaveanticonformists. Thus,
ifz < pthenthespinS isananticonformist;otherwise,itisaconformist. Afterinitialization,the
i i
systemevolvesaccordingtothefollowingalgorithm:
1. Chooserandomlyanduniformlyi ∈ {1,2,...,N}aswellasqdistinctneighborsofthei-thnode:
n ,n ,...,n ∈ {1,2,...,N}andgoto2.
1 2 q
2. Ifallqneighborsareinthesamestate,theywillinfluenceS(t)andthusgoto3;otherwise,set
i
t = t+∆tandgoto1.
3. IfS isaconformistthenS(t+∆t) = S (t);otherwise,S(t+∆t) = −S (t). Sett = t+∆tand
i i nq i nq
goto1.
5. Results
Inordertoquantitativelydescribethebehaviorofthesystem,wedefineanup-spinconcentration
asafractionofvoterswithapositiveopinion(i.e., si = 1). Ifwedenoteby N↑ thetotalnumberof
agentsinthestates =1,thentheup-spinconcentrationhasthefollowingform:
i
N
↑
c = , (1)
N
anditmightbeinterpretedasaprobabilityoffindingavoterwithapositiveopinioninthesystem.
Sinceweconsiderarandomsequentialupdatingscheme,ineveryelementarytimestep,onlya
singleagentcanchangeitsopinion. Therearethreepossiblescenarios. Thetotalamountofup-spins
N↑ mayincreaseby1,decreaseby1,orremainunchanged. Hence,aflipcorrespondstoachangein
theconcentrationcby1/N,andthefollowingtransitionratescanbewritten:
(cid:18) (cid:19)
1
γ+ =P c → c+ ,
N
(2)
(cid:18) (cid:19)
1
γ− =P c → c− .
N
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Therefore,thetimeevolutionofthesystemisgovernedbytherateequation
1
c(t+∆t) = c(t)+ (γ+−γ−). (3)
N
Basedontheaboveformalism,theannealedversionoftheq-votermodelwithindependenceand
anticonformitywasstudiedin[5,6]. Herein,weadoptthesamemethodologyinordertodescribethe
modelwithquenchedagentattitudes,whichreflectsthepersonapproach.
Atthebeginning,wedivideoursocietycomprisedof N agentsintotwogroups. Thefirstone
incorporates exclusively nonconformists, and its size is N = pN. The second one gathers only
1
conformists and consists of N = (1− p)N agents. Thanks to the above partition, we still have
2
thesameprobability poffindinganonconformistasintheannealedapproach,whichmakesboth
scenarioscomparable. Furthermore,wecanwritedownexplicitlychancesforaflipineachgroup,and
consequently,findstationaryvaluesofup-spinconcentrationsfromthefollowingcondition:
γ+−γ− =0. (4)
In order to do so, let us consider separately the q-voter model with independence and
anticonformity.
5.1. ModelI:Independence+Conformity
Forthegroupofindependentagents:
1
γ+ = (1−c ),
1 2 1
(5)
1
γ− = c ,
1 2 1
where c refers to the up-spin concentration among independent agents. From Equation (4) (i.e.,
1
equalizingaboveequations),wegetthestationaryconcentrationofindependentagentswithapositive
opinion c = 1/2. For the second group comprised of only conformists, we write down similar
1
formulas:
γ+ = (1−c )cq,
2 2
(6)
γ− = c (1−c)q.
2 2
Thistime,weobtainthestationaryconcentrationofconformistswithapositiveopinion:
cq
c = . (7)
2 cq+(1−c)q
Having c and c , we are able to determine the overall up-spin concentration by using the
1 2
followingformula
c = pc +(1−p)c . (8)
1 2
CombiningEquation(8)withbothconcentrations,wehavethefinalresult
c(1−c)q−(1−c)cq
p =2 . (9)
(1−c)q−cq
ThecorrespondingformulatoEquation(9)fortheannealedapproachobtainedin[5]hasthe
followingform
c(1−c)q−(1−c)cq
p = . (10)
c(1−c)q−(1−c)cq+1/2−c
Entropy2017,19,415 9of14
Inordertobettercomprehenddifferencesbetweentheaboveresults(i.e.,Equations(9)and(10)),
weplotphasediagramswhichrepresenttheimpactofthemodelparameter ponthestationaryvalues
oftheup-spinconcentration. Figure1apresentsphasediagramsobtainedfortheannealedq-voter
modelwithindependence,obtainedfromEquation(10),whereasFigure1bshowsphasediagrams
forthequenchedmodel,obtainedfromEquation(9). Themoststrikingdifferenceisthechangeof
transitiontypefromdiscontinuousintheannealed(situation)approach,tocontinuousinthequenched
(person) approach, for q ≥ 6. It seems that freezing the agent behaviors at the beginning of the
simulation,introducingquencheddisorder,supportssecond-orderphasetransitions,whereasallowing
agentstoreversetheirattitudeintime,incorporatingannealeddisorder,supportsdiscontinuousones.
Figure 2a additionally presents the decomposition of the overall up-spin concentration into the
concentrationofadoptedindividualsamongindependentagentsc andamongconformistsc .
1 2
1 1 1
(a) (b) (c)
0.8 0.8 0.8
q=3
0.6 0.6 0.6 q=4
c c c q=5
0.4 0.4 0.4 q=6
q=7
0.2 0.2 0.2
0 0 0
0 0.1 0.2 0.3 0.4 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
p p p
Figure1. Phasediagramsfortheq-votermodelwithindependenceandanticonformitywithinannealed
andquenchedapproaches.Darkerlinescorrespondtolargergroupsofinfluence(i.e.,highervaluesof
q).(a)Annealedmodelwithindependence,Equation(10);(b)Quenchedmodelwithindependence,
Equation (9); and (c) Model with anticonformity where annealed and quenched approaches give
exactlythesameresults. Notethatforthequencheddisorder, wealwayshavecontinuousphase
transitionsregardlessofthestochasticdrivingtype. Annealeddisorder,ontheotherhand,reveals
notonlyquantitativebutalsoqualitativedifferencesbetweenindependenceandanticonformity.For
independence,transitiontypechangesfromcontinuousq ≤ 5todiscontinuousq ≥ 6,whereasfor
anticonformityalltransitionsarecontinuous.
1 1
c1
c2
0.8 0.8
0.6 0.6
c c
0.4 0.4
0.2 0.2
(a) (b)
0 0
0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8
p p
Figure2. Phasediagramsforthequenchedmodeldrivenbystochasticnoise:(a)Independenceand
(b)Anticonformity.Blacklinescorrespondtotheoverallup-spinconcentrations.Theupperpartsof
thediagrams—thethickblacklines—weredecomposedintotwocoloredcurveswhichrefertothe
up-spinconcentrationsamongspecificsocialgroups,sothatEquation(8)isfulfilled:bluelinesstandfor
conformistsandredonesfor(a)independentindividualsand(b)anticonformists.Thedecomposition
ofthelowerpartsofthediagramsconstitutesamirrorimagealongc=1/2.
Entropy2017,19,415 10of14
5.2. ModelA:Anticonformity+Conformity
Asimilaranalysiscanbedonefortheq-votermodelwithanticonformity. Onceagain,wedivide
our community into two groups. The first one consists of N = pN agents, and comprises only
1
anticonformists. Thesecondone,aspreviously,combinesconformistsinthenumberofN = (1−p)N.
2
Thetransitionrateshavetheforms
γ+ = (1−c )(1−c)q,
1 1 (11)
γ− = c cq,
1 1
andtheequationsleadtothefollowingstationaryvalueofup-spinconcentration
(1−c)q
c = . (12)
1 (1−c)q+cq
Inthecaseofconformity,wehavethesameformulasderivedbeforeEquations(6)and(7). Using
Equation(8),weareabletoobtainthefinalresultforthestationaryvalueofup-spinconcentration
c(1−c)q−(1−c)cq
p = . (13)
(1−c)q−cq
Theassociatedformulaforthesameq-votermodelwithanticonformitybutwithintheannealed
approachisidenticalandderivedin[5].Therefore,theq-votermodelwithanticonformityisinsensitive
to switching between types of disorder, in contrast to the model with independence. The phase
diagramsforthisquencheddisorderarepresentedinFigure1candinFigure2b; wecanseeonce
againthedecompositionoftheoverallup-spinconcentrationintoc andc . Notethatinthecaseof
1 2
quenching,onlycontinuousphasetransitionsarepresentinthesystem,regardlessofthestochastic
drivingtype. Thus,thedifferencesbetweenmodelswithindependenceandanticonformityareonly
quantitative. Themodelwithindependenceexhibitshigherstationaryvaluesofup-spinconcentration
than the model with anticonformity at the same levels of parameter p; compare Figure 1b,c. It is
worthnotingthatthestationaryvaluesofup-spinconcentrationsareequalforbothquenchedmodels
withindependenceandanticonformityundertheconditionthatthelevelofindependenceistwiceas
largeasthelevelofanticonformity;compareEquations(9)and(13). Evenso,Figure3showsthatthe
trajectoriesinaphasespacearedifferentforthesecases. Additionally,thesedifferencesarealsovisible
inthecorrespondingtimeevolutionspresentedinFigure4.
1 1
(a) (b)
0.8 0.8
0.6 0.6
2 2
c c
0.4 0.4
0.2 0.2
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
c1 c1
Figure3. Phaseportraitsfortheq-votermodelwithquencheddisorderandthegroupofinfluence
comprisedofq=4agents.Thecasewith(a)independenceatlevelp=0.5and(b)anticonformityat
levelp=0.25.Arrowsizeindicatesthestrengthoftheflowinthephasespace.Redcurvesillustrate
trajectoriesofasystemstartingfromtheinitialpoint(c ,c )=(1,0.5).Reddotsrefertostablepoints.
1 2