Discretized opinion dynamics of Deffuant on scale-

arXiv:cond-mat/0310243v2[cond-mat.stat-mech]30Mar2004DiscretizedopiniondynamicsofDeffuantmodelonscale-freenetworksD.Stauffer1,A.O.Sousa2andC.Schulze11InstituteforTheoreticalPhysics,CologneUniversityD-50923K¨oln,Euroland2InstituteforComputerApplications1(ICA1),UniversityofStuttgartPfaffenwaldring27,D-70569Stuttgart,Eurolande-mails:stauff[e-mail],[e-mail]delofDeffuantetalissimplifiedbyallowingformanydiscreteinsteadofinfinitelymanycontinuousopinions,onadirectedBarab´asi-Albertnetwork.Asimplescalinglawisobserved.Wethenintroducenoiseandalsouseamorerealisticnetworkandcomparetheresults.Finally,welookatamulti-layermodelrepresentingvariousagelevels,andweincludeadvertisingeffects.Keywords:MonteCarlo,sociophysics,consensus.1IntroductionComputersimulationofopiniondynamics(consensusmodels)(Axelrod1997;Deffuant2000;Deffuant2002;Weisbuch2002;Hegselmann2002;Hegselmann2004;Krause1997;Sznajd-Weron2000;Stauffer2000;Stauffer2002;Galam1990;Galam1997;Stauffer2003)isanimportantpartofsociophysics(Weidlich2000;MossdeOliveira1999;Schweitzer2003).Onechecksif,startingfromarandomdistributionofopinions(MonteCarlomethod),oneendsupwithaconsensusoradiversityoffinalopinions.Thesimulatedpeople(”agents”)arelocatedonlattices,onscale-freenetworks(Albert2002;Barab´asi2002),orformapurelytopologicalstructurewhereeverybodycanbeconnectedwitheverybody.FortheparticularcaseoftheconsensusmodelofDeffuantetal(Deffuant2000;Deffuant2002;Weisbuch2002),itwasshownthatonaBarab´asi-Albert(BA)network(Stauffer2004)(seealsoWeisbuch2004)thenumberSofdifferentsurvivingopinions(ifnocompleteconsensuswasachieved)wasanextensivequantity,i.e.itvariedproportionaltothenumberNofagents,whileitisintensive(independentofNforlargeN)wheneverybodycanbeconnectedtoeverybody(Ben-Naim2003).TheliteratureonBarab´asi-Albertnetworkscontainsmanycomparisonswithreality,e.g.forthecomputernetworksoftheInternet.Themotivationofthepresentworkistwo-fold:Wewanttohaveanunambiguouscriterionwhethertwoopinionsagreeordisagree,andthususediscreteinsteadofcontinuousvariables1fortheopinions,section2.ThenwewanttomakethemodelmorerealisticbyintroducingnoiserepresentingeventsoutsidetheopiniondynamicsofDeffuantetal,byusinginsection3amorerealisticnetwork(Davidsen2002;Holme2002;Szab´o2003)withahigherclusteringcoefficientthattheBAnetwork,byallowingforadvertisingthroughmassmedia,andbytakingintoaccountmorethanonelayerinordertoimplementanagestructure;thelasttwoeffectsaredealtwithinsection4.AnappendixgivesthebasicFortranprogram.110100100010000110100100010000Number’QSurvivingopinionsforN+mpeople,m=3,bottomtotop:N=10(line),100,1000,2500,10000(line)Figure1:NumberofdifferentsurvivingopinionsversustotalnumberQofopinions,forvariousnetworksizesN.DataforN=10andN=105areconnectedbylines.2BasicmodelandnoiseInsteadofallowingfortheopinionsanyrealnumberbetween0and1,wetakethemasdiscretenumbersq=1,2,...Q,asintheSznajdmodel(Sznajd-Weron2000;Stauffer2000;Stauffer2002).Nowitiswelldefinediftwoopinionsdifferoragree,whileforrealnumbersitdependsontheaccuracyofthesimulation.Atfirst,onlypeopledifferingby±1intheiropinioncanconvinceeachother(boundedconfidence(Hegselmann2002;Hegselmann2004;Krause1997,Deffuant2000;Deffuant2002;Weisbuch2002));thus1/Qcorrespondstotheconfidenceinterval20.0010.010.110.00010.0010.010.11101001000S/(Q-1)Q/NScalednumberSoffinalopinionsversusscalednumberQofpossibleopinions,N=10to10000Figure2:ScaledplotofthesamedataasinFig.1.Thestraightlinesindicatethe”trivial”scalinglimit:Everybodykeepsitsownopinionintherightpart,andeachopinionissharedbymanyintheleftpart.ofthepreviousmodels.Iftwoagentswithopinionsdifferingbyoneunittalktoeachother,randomlyoneofthemtakestheopinionoftheother(Axelrod1997).WeputagentsonadirectedBarab´asi-Albert(Albert2002;Barab´asi2002;Stauffer2004)network,startingwithm=3agentsconnectedwitheachotherandwiththemselves;thereafter,Nagentsareadded,eachofwhichselectsmpre-existingagentstobeconnectedwith.Theserandomlyselectedoldagentsarenotregardedasconnectedtothenewagent,i.e.theconnectionsaredirected.InthisBAnetworkthenumberofagentshavingkneighboursisknowntobeproportionalto1/k3.ThesizeofthenetworkisthenumberNofagentsonit,i.e.thepopulation.Firstweconstructthenetwork,thenwestarttheopiniondynamicsfromarandomdis-tributionofopinions.Foreachiterationwegothroughallagentsintheorderinwhichtheywereaddedtothenetwork,andeachselectsrandomlyoneofthemagentsithadchosenbeforetobeconnectedwith.Thesimulationstopsifnoagentchangedopinionduringoneiteration.(Aboutthesameresultsareobtainedfromrandominsteadofregularupdating,providedwestopiffortenconsecutiveiterationsnoopinionchanged.)31001000100001000001e+061e+071e+0810100numbersizeHistogramoffinalopinionclustersizes,nonoise,Q=10(lines),100,1000,10000,100000;N=100*QFigure3:EvidencethatclusternumbersforfixedN/Q(heretakenas100)areextensive,forlargesystemsupto10millionnodes,withNandQincreasingfrombottomtotop.Forsmallsystemstheresults(line)areverydifferent.Alldataaresummedover1000runs.Figure1showsthatforlargeNthenumberSofsurvivingfinalopinionsroughlyequalsQfornottoosmallQ;forQ=2,ontheot