Understanding influence power of opinion leaders in ecommerce networks: An opinion dynamics theory perspective

In this paper, from the perspective of opinion dynamics theory, we investigate the interaction mechanism of a group of autonomous agents in an ecommerce community (or social network), and the influence power of opinion leaders during the formation of group opinion. Đề tài Hoàn thiện công tác quản trị nhân sự tại Công ty TNHH Mộc Khải Tuyên được nghiên cứu nhằm giúp công ty TNHH Mộc Khải Tuyên làm rõ được thực trạng công tác quản trị nhân sự trong công ty như thế nào từ đó đề ra các giải pháp giúp công ty hoàn thiện công tác quản trị nhân sự tốt hơn trong thời gian tới.

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ContentslistsavailableatScienceDirect

InformationSciences

journalhomepage:www.elsevier.com/locate/ins

Understandinginﬂuencepowerofopinionleadersin

e-commercenetworks:Anopiniondynamicstheory

perspective

YiyiZhaoa,GangKoua,YiPengb,∗,YangChena

a School of Business Administration, Southwestern University of Finance and Economics, No.555, Liutai Ave, Wenjiang Zone, Chengdu,

611130, China

b School of Management and Economics, University of Electronic Science and Technology of China, No 2006, Xiyuan Ave, West Hi-Tech

Zone, Chengdu, 611731, China articleinfoArticle history: Received 16 March 2017 Revised 12 October 2017 Accepted 13 October 2017 Available online 16 October 2017 Keywords:E-commerce network Opinion dynamics Bounded conﬁdence rule Opinion leader subgroup Herd behavior

abstract

Inthispaper,fromtheperspectiveofopiniondynamicstheory,weinvestigatetheinter-actionmechanism ofa groupof autonomousagents inan e-commerce community (orsocialnetwork),andtheinfluencepowerofopinionleadersduringtheformationofgroupopinion.Accordingtotheopinion’supdatemannerandinfluence,thispaperdividessocialagentswithinasocialnetworkintotwosubgroups:opinionleadersandopinionfollowers.Then,weestablishanewboundedconfidence-baseddynamicmodel foropinionleadersandfollowerstosimulatetheopinionevolutionofthegroupofagents.Throughnumeri-calsimulations,wefurtherinvestigatetheevolutionmechanismofgroupopinion,andtherelationshipbetweenthe influencepower ofopinionleadersandthree factors:thepro-portionofthe opinionleadersubgroups,theconfidencelevelsofopinionfollowers, andthedegrees oftrusttoward opinionleaders Thesimulation resultsshow that,in ordertomaximizetheinfluencepowerine-commerce,enhancingopinionleaders’credibilityiscrucial.

1.Introduction

TherapiddevelopmentofInternettechnologyandWeb2.0hasstimulatedthegrowthofcustomer-centerede-commerce,whichhasrecentlyreceivedincreasedattentionintheﬁeldsofbusinessapplications,businessstrategies,anduserbehavior[34] Within the e-commerce environment, agents accesssocial knowledge and share experiences peer-to-peer (P2P) orthroughwordofmouth(WOM),andthenmaketheirowndecisions.Insuchacollectivedecision-makingprocess,opinionsplayafundamentalrolesincetheycandeeplyinteractwitheachother[3].

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Inthe propagationprocessofpublic opinion,opinion leadershavea profoundimpact ontheopinion formationofor-dinaryagents.Inthefieldofbusinessandmarketing,Rogers[23] showedthatasmallgroupofinfluentialopinion leadersdetermines theutility ratioof agiven innovation.Comparedto thespreadof publicopinion ina social networkwithoutopinion leaders,opinionstendto propagatefasterina socialnetworkwithopinion leaders[17].Inaddition,a numberofzealotopinionleaderswithdefiniteobjectiveswereemployedin[22] togeneratemomentumandinfluencevoters’decision-makingbehaviors, whileAmblardandDeffuant[2] andDeffuantetal.[6] appliedboundedconfidencetheorytoconstructopiniondynamicsmodelstoanalyzetheinfluenceofopinionleadersinsocial networks.Theresultsrevealedthat,aslongastheconfidencelevelsofordinaryagentsinasocialgrouparesufficientlyhigh,eveniftheinitialopinionsoftheordinaryagentsaredissimilar tothoseoftheopinion leaders,the opinionleadersareeventually abletoguidetheordinaryagentsto accept their desiredopinions Consideringthat, insome cases, opinion leaderscannot always help spread the desiredopinion,Afshar andAsadpour[1] extendedthetraditionalDeffuant–Weisbuchmodelandbuiltan informedagentsmodel.Accordingtothismodel,informedagentsarecommonagentspossessingdesiredinformation.Theyinitiallypretendtohaveopinionssimilartothoseofothers,andgraduallychangetheiropinionstowardthedesiredinformationthroughintentionalinteractions.

Ine-commercenetworks,theroleofopinionleadersismainlyreflectedintwoaspects:influencingconsumers’decisionoutcomesanddispersalofopinionsbywordofmouth(WOM).AccordingtoChaudhryandIrshad[4],averageconsumerswilloftenconsider theopinionsofopinion leadersintheir purchasedecision-makingprocesses.Moreover, itwaspointedoutin[28] thatthedegreeofdiscrepancyinopinionleaders’impactonthepurchasedecisionsofaverageconsumersismainlycausedbydifferencesbetweentheculturalbackgroundandproductfocusofbothopinionleadersandaverageconsumers.Villanuevaetal.[30] believedthatopinionleaderscouldprovideproductinformationandadviceforpurchasedecisionstoother consumersthrough frequentWOM communication, andthereby affectthe attitudes,beliefs,andbehaviorsof otherconsumers.Samson[24] pointedoutthatopinionleaderswithahigherconfidencelevelaremorewillingtobecomeWOMcommunicators, and can enhance consumers’ purchase intentions through positive WOMcommunication Further, WOMcommunication focuseson the process of “opinionleader→WOMcommunication→consumerbehavior”; however,moststudies tendto dividethe processinto twostages (opinionleadersandWOMcommunication, andWOMcommunicationandconsumerbehaviors)andanalyzethe relationshipbetweenthetwo variablesinvolved ineachstage, respectively.Liuetal.[17] usedatime-varyinghypergraphtomodelonlinesocialnetworks,andadomain-awareapproachtoidentifyeffec-tiveopinionleaders.Asfarasweknow,thereisnoquantitativeresearchthatfocusesontheevolutionofopinioninteraction(consumerbehavior)asadirectrelationshipbetweenopinionleadersandopinionfollowers(consumers).Inordertounder-standtheimpactofopinionleadersonopinionfollowers,itisnecessarytoconsiderthemechanismofopinioninteractionbetweenopinionleadersandopinionfollowers.

Thedisseminationprocessofpublicopinionisacomplexsystemofco-evolutionofopinionsandnetworks,andinvolvesmanyvariables,suchasnetworkstructure,thenumberofagentsinvolved,anddescriptionofopinions.Besides,itisdiﬃcultforprobability-orstatistics-based mathematicalmodels todescribethe dynamicevolution ofcollectiveopinions.Opiniondynamics models focus on the interaction mechanism between opinions,and assume that agentswill decide their ownopinionsbasedonthose oftheiropinion neighborsinthenetwork.Onthat account,an opiniondynamicsmodelismoresuitableforthestudyoftheopiniondisseminationmechanismonuserrelationship-basedsocialmediaplatforms.

Themainpurposesofthispaperare(1)toexploreboththeinfluenceofopinionleadersonthedecision-makingprocessandopinionformationofordinaryagentsduringthedisseminationofpublicopinioninsocialnetworks;and(2)toprovideatheoreticalbasis,aswellassuggestfeasiblemeasures,forenterprises,undertakings,andgovernmentdepartmentstodesignappropriate measurestoguideandcontrolpublicopinion.Inthispaper,opinionleadershavedefinitetarget opinionsandare interestedin influencing theupdating process of opinion followers’opinions Inaddition, the opinion leadersin thispaperarenotnecessarilyleadersbasedonsocialclass,norfromofficialorganizations,andmaynotevenpossessobservableleadershipqualities.Infact,opinionleadersonsocialmediaplatformscouldbeordinaryagentsthathavetheabilitytoexertsignificantinfluenceonothers’opinionsduetotheirprofessionalbackgroundorfamiliaritywithspecificevents.

Thepapercontributestoliteraturebybuildinganewopiniondynamicsmodelforasocialgroupwithtwoopinionleadersubgroupswithoppositetargetopinions,basedontheboundedconfidenceprinciple.Itanalyzestherelationshipsbetweentheinfluencepoweroftheopinionleadersandsomefactors,suchastheproportionoftheopinionleaders, theconfidencelevelsofthefollowers,andthetrustdegreesofthefollowerstowardtheopinionleaders.

The rest of this paper is organized as follows.Section 2 provides a background on bounded confidence opinion dy-namics andsocial network theory Section 3 firstgives a motivation examplefor opinion leaders, and then buildsnovelbounded confidence opinion dynamics models for opinion followers, positive and negative opinion leaders, respectively.Section4 presentssomequantitativeresultsthroughcomputersimulationstostudytheinfluencepowerofopinionleadersandtheevolutionofthegroupopinion.Section5 concludesthepaper.

2.Background

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2.1.Boundedconﬁdenceopiniondynamics

Opiniondynamicsisanimportantclassofsocialdynamicsthatstudiestheformationanddisseminationofpublicopiniononsocialnetworks.Basedonthemethodsby whichopinionsaremanifested,opinion dynamicscan beclassifiedaseitherdiscreteorcontinuous.Sincethe1960s,researchershaveproposedaseriesofmodelswiththecontinuousupdatingofopin-ionsinordertostudythesufficientconditionsforagroupofexpertstoreachaconsensus[26].Thesewerepioneerstudiesthat modeledandanalyzed the evolutionofcontinuous opinions.In2002,Krause andHegselmann [8] andDeffuantandWeisbuch [32] proposed, respectively, two boundedconfidence-based opinion dynamics models—the Hegselmann–Krause(HK)andDeffuant–Weisbuch(DW)models.Specifically,theoriginalHKmodelisdescribedbyxi (t+1)=j: |x i(t)−xj(t)|≤εai j xj (t)j: |x i(t)−xj(t)|≤εai j , (2-1)whiletheoriginalDWmodelisdescribedbyxi (t+1)=xi (t)+μxj (t)− xi (t)xj (t+1)=xj (t)+μxi (t)− xj (t) (2-2)

forthecase|xi (t)− xj (t)|≤ε,wherexi (t)istheopinionofagenti,ε istheconfidencelevel,ai j istheinteractionweightofagent j onagenti,andμistheconvergenceparameter.ThemaindifferencebetweenthesetwomodelsisthattheDWmodeladoptsanasynchronousopinionupdatingprocess,whiletheHKmodeladoptsasynchronousupdating process.Theboundedconfidencerulesuggeststhateachagenthashis/her/itsownconfidencerangewhentrustingothers;onlywhenthedifferencesbetweentheopinionsofotheragentsandhis/her/itsownarenotgreaterthanaspecificthresholdorconfidencelevel,willhe/she/itshareandexchangeopinionwiththoseagents.

In a bounded confidence model, the confidence level and initial opinion of an agent usually determine the opinionneighborsthat he/she/it is likelyto communicate withat differenttime instants There are three possiblefinal states ofcollectiveopinionssimulatedbyaboundedconfidencemodel—consensus,opinionpolarization,andopinionfragmentation—thatare closelyrelatedto theconfidencelevels,initial opinions,andsome convergenceparameters.Ontheone hand,re-searcherscommit themselves to consensus measures or optimalconsensus under some specific conditions Forexample,someconsensusmeasurealgorithmswereproposedin[7] and[35],foronline-offlinesocialnetworksandlarge-scalegroupdecision-making,respectively.Forgroupdecision-making,[13,14] builtsomeinterestingoptimalconsensusmodelsbasedonminimumcostandmaximalreturn.Ontheotherhand,manyresearchersdevotethemselvestostudyingtherelationshipsbetweenthefinalopinionpattern(notjustconsensus)andinfluencefactors.Forexample,forboundedconfidencemodels,Lorenz[18,19] categorizedagentsfroma socialgroupintohigh- andlow-confidencesubgroups,andproposed respectivelyheterogeneousHKandDW models.Inthe modifiedmodels,agents within thesamesubgroup havethesame confidencelevels,whilethose indifferentsubgroupshavedifferentconfidencelevels Inadditionto consideringtheheterogeneity ofagents,Lorenz[20] also examined the evolutionary mechanism ofpublic opinions in a dynamic social network, where theinter-agentinfluencechangesaccordingtoaMarkovchain.MirtabatabaeiandBullo[21] alsoconsideredaheterogeneousHKmodelwithatime-varyingcommunicationnetwork,whichassumesan equilibriumexistsinthecollectiveopiniondynam-ics.Theythenappliedthenonlinearsystemtheorytoanalyzetheconvergenceofthecollectiveopinions.FortheErdos-Renyisocialnetworktopology,basedonthetraditionalHKmodel,SuandLiu[27] examinedthecoevolutionofopinionsandtheinterconnectionnetwork,andobtainedsomeresultsonconsensusorfragmentationforgroupopinions.TheoriginalHKandDWmodelswereappliedin[31] toinvestigateonlineconsumerreviewsine-commercenetworks,andtoanalyzesomein-fluencefactorsintheopinionevolution.Veryrecently,foraheterogeneoussocialnetwork,ahorizontalandverticaldivisionprincipleforagents hasbeenproposed foragents,andmulti-levelheterogeneous HKmodels andleader-followeropiniondynamicsmodels have beenconstructed, to systematically analyzethe impact of differentfactors on thespread of pub-licopinion [11–16,37–39].Motivatedby [16],atime-varyingconfidence levelupdate rulewasproposed in[36],based onin-degreesandout-degreesofagentstoextendtheoriginalHKmodel.

2.2.Socialnetworktopology

Thedisseminationprocessofpublicopiniononsocialnetworksisessentiallyacoevolutionofopinionsandtheassociatednetworktopology.Inthissubsection,wegiveanetworkdescriptionoftherelationshipsamongagents.

Inthisstudy,weassume thatopinion leadershavedeﬁnitetarget opinionsand,thus,are notaffectedby theopinionsofopinion followers.Theopinionleadersonlyexchangeopinionswithother opinionleadersinthesamesubgroup.More-over,opinionleadersplaydominantrolesintheformationofcollectiveopinions,andareconcerned withguidingopinionfollowers.

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Fig 1 Opinion exchange in a network with two opinion leader subgroups at a ﬁxed time

opinions,thoseintheupperrightcorner(markedwithatriangleshapeandlabeled7-11)areopinionleaderswithnegativetargetopinions,andthoseremaining(markedwithacircleshapeandlabeled12-20)areopinionfollowers.Opinionleadersfromthetwosubgroups havetheir ownexplicittarget opinions;opinionfollowers,however,donothaveaspecific targetopinion,andonlyexchangetheiropinionswithagentswithintheirconfidenceranges,includingopinionleaders.Iftheopin-ionleadersinthesamesubgrouphavethesametargetopinionsandconfidencelevels,thentheinteractionsamongleaderswithin thesameleader subgroupsaresymmetrical.Consideringthediversityanduniversalityofopinion followers,we as-sumethattheir confidencelevelsare heterogeneousandsatisfy auniformdistributionwithintheinterval [0,1].Basedontheseassumptions,theopinionexchangebetweentheopinionfollowersmaynotrespectthesamebi-directionalsymmetri-cal modeasthat oftheopinion leaders.Whentheopinion differenceoftwo opinionfollowersislessthantheconfidencelevelofoneoftheagents,anexchangeofopinionsmaybeasymmetrical,suchasthatbetweenagents12and16,15and16,and14and17.Atagiventimet,onlywhentheopiniondifferencebetweenanytwogivenopinionfollowersisnotgreaterthantheconfidencelevelsofboth agents,cantherebeasymmetricalexchange— suchasthatbetweenagents13and14,12and15,and15and18.Sincethispaperassumesthattheopinionsoftheopinionfollowersdonotinfluencethoseoftheopinionleaders,andtheopinionleadersfromdifferentleadersubgroupsdonotcommunicateduetotheirdissimilartargetopinions,therearenoedges betweenthetwoopinionleader subgroups.Theedges betweentheopinionleadergroup andopinionfollowersubgrouparedirectedtowardtheopinionfollowergroup.Specifically,opinionleaders1and7,aswell as3 and9,belong totwo differentsubgroups; hence,they do not havean opinion exchange.However, the opinion leaders4and5 (fromthepositive opinion group)havean influenceonthe opinionsoftheopinion followers12 and13,respec-tively,andtheopinion leaders7,9,and11haveaninfluenceontheopinionsoftheopinion followers13and14; yet,theaforementionedopinionfollowerscannotaffecttheopinionupdatingofthecorrespondingopinionleaders.

3.Modelingtheinﬂuencepowerofopinionleaders

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Anewgraph-basedcomprehensivereputationmodelandahybridtrust-basedrecommendersystemwerebuiltin[33] and[40], respectively, to improve the role of opinion leaders in social commerce Additionally,an online survey experimentwasconductedin[29] toshow thatopinion leaders’recommendationscould increasethetrustofordinalagentsonsomeparticularmedia.

3.1.Amotivationexample

Inthispaper,the opinion leadersare definedastheagentsthat have definite,unwaveringtarget opinionsThe agentsexceptopinionleadersinasocialnetworkarecalledopinionfollowers.Theleadersarenotaffectedbytheopinionsoftheopinionfollowersduringtheopinion updateprocesses.Theyhavetheintentiontoinfluencethe opinionsofothers.Addi-tionally,theopinionleadersinthispaperarenotrequiredtobeimportantagentsinanyofficialorganizationorinstitution,andcanbeordinaryagentswhohaveasignificantamountofinformationabouttheobjectoreventinthepublicopinion.Theagents,exceptopinionleadersinasocialnetwork,arecalledopinionfollowers.Forexample,peer-to-peer(P2P)lendingisa classicalsocial networkserviceinthe fieldof finance.Itis well knownthat,withthe quickdevelopmentofinternetande-commerce,manyP2Plendingcompanies—suchasZopa,Prosper,LendingClubFundingCircle,andRateSetter—provideonlineinvestmentplatformstoattractlendersandinvestorstoidentifyandpurchaseloans.Allofthecountriesintheworldhavehighexpectations forP2P lendinginfinancial innovation; however, inChina, a P2P lendingcompany calledEzubaolaunchedinJuly2014andwassubsequentlyshutdowninFebruary2016becauseitwasaccusedofaPonzischeme.Conse-quently,about900,000customersandtheir50billion RenminbiwereinvolvedintheEzubaocase Twoclassesofopinionleadersessentiallyinfluencedtheopinionsofagents:(1)Ezubao,aswellassomeassociatedmediaandexperts;and(2)thethird-partyP2Pratingorganizations.InordertopersuademoreagentstouseEzubao,thecompanymadesignificantinvest-mentsintoadvertisementsincertainwell-knownmedia,suchasChinaCentralTelevision,localtelevisionstations,metrosinbigcities,andexperts’popularization.Moreover,thethird-partyP2P ratingorganizationsoftenpublishedsomeadvanced-riskwarningsuggestionsandreportedthatEzubaowasratedasLevelC.Thus,whenopinionleadersholdpolarizedorevenfragmentalopinions,itisinterestingtoinvestigatehowtheopinionsevolveforopinionleadersubgroupsaswellasforthewholegroup.Thequestionssurroundingthekeyfactorsassociatedwiththeopinionleaders’influnecepowerremaintobefurtherexplored.

3.2.Boundedconﬁdence-basedopiniondynamics

We now construct a newmodel to analyze the influence power of opinion leaders, based on the framework of theboundedconfidencetheory.Wefurtherrevealtheevolutionarymechanismofgroupopinionsundertheinfluenceofmulti-pleopinionleadersubgroups.Inreality,therearemorethantwoopinionleadersubgroupsinagivensocialnetworkgroup.Thedifferencesinthetargetopinionsheld bythesesubgroupsare notnecessarilysubstantiallylarge Inordertosimplifytheanalysisprocess,withoutlossofgenerality,thispaperassumesasituationwhereonlytwosubgroupsofopinionleadersexistina givensocial network.Each leadersubgroup hasits owntarget opinion Moreover,the leader subgroupsare re-ferredtoaspositiveandnegativeifthetargetopinionsarecompletelypositiveandnegative,respectively.Byestablishingamodelofinfluencepowerforopinionleaders,thispapersystematicallyinvestigatestherelationshipsbetweentheinfluencepowerofopinion leadersandsome associatedfactors,suchastheproportionofopinionleader subgroups,theconfidencelevelsofopinionfollowers,andtheirdegreesoftrusttowardtheopinionleaders.

Suppose there is a social network with N agents,among whom N1 is opinion followers, N2 is opinion leaders withapositivetarget opinion,N3 is opinionleaderswithanegativetargetopinion, andN1+N2+N3=N.Then,when aneventoccurs,agents’initialopinionsoftheeventmaybediverse;speciﬁcally,agentscanholdanopinionsomewherebetweenthecompletelypositiveandcompletelynegativeopinionsoftheevent.When theopinionsattimet ofall agentsare denotedbyxi (t),withoutlossofgenerality,thecompletelypositiveandcompletelynegativeopinionsoftheeventare,respectively,deﬁnedby xi (t)=1or0.5andxi (t)=−1or−0.5,fori=1, ,N.Weassume thattheinitialopinionsofallagentsxi (0)

obeyauniformdistributionwithin theinterval[−1,1]or[−0.5,0.5].Atanytimet,theopinionofagentisatisﬁesxi (t)∈[−1,1]or[−0.5,0.5].Forconvenienceofdescription,wedenoteX(t)=col(x1(t),x2(t), ,xN (t))∈RNasthevectorofthecollectiveopinions,andX(0)astheinitialopinionproﬁleattimet=0.

Anopinion-updatingmodelisproposedfortheopinionfollowersasfollows:xF i (t+1)=(1−αi −βi )NF 1i (t)N 1j=1ai j (t)xj (t)+αi 1NP i (t)N 1+N 2j=N 1+1ai j (t)xj (t)+βi 1NNi (t)N j=N 1+N 2+1ai j (t)xj (t),(3-1)wherethe updating weight ai j (t)=1, xi (t)− xj (t)≤εF i 0,otherwise , i=1, ,N1, j=1, ,N,and εF

i representsthe conﬁdence

leveloftheopinionfolloweri.NF i (t)=N 1

j=1

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ofopinionfolloweriattimet,andNi P (t)= N 2

j=N 1+1

ai j (t)andNi N (t)= N 3

j=N 1+N 2+1

ai j (t)arethenumbersofopinionleaderswhoare theopinion neighborsofagenti fromthe positiveandnegativeleader subgroups,respectively αi , βi ,and1−αi −βi

are thedegreesof trustassignedto the positiveopinion leader subgroup,negativeopinion leader subgroup,andopinionfollowersubgroup,respectively,andαi,βi,1−αi−βi∈[0,1].Whenαi=0orβi=0,opinionfollowericompletelydistruststhe positiveornegativeopinion leaders, respectively Contrarily, ifαi =1 orβi =1,then opinion followeri iscompletelyinﬂuencedbytheopinionleadersfromthepositiveornegativeopinionsubgroups,respectively.

Accordingtotheirdefinitionandcharacteristics,opinion leaderstendtohavearelativelycomprehensiverangeofinfor-mationonthesameevent,whencomparedtotheopinionfollowers.Inaddition,theirtargetopinionsareveryspecific,andthey onlyexchangeopinionswiththeopinionleadersthatmeettheir confidencelevelsinthesamesubgroup.Inordertoachieveacommongoal,theconfidencelevelsbetweenopinionleadersinthesamesubgrouparerelativelyhigh.Thus,arel-ativelymoderatevalueεP

i =εN

i =0.25isassignedastheconﬁdencelevelfortheopinionleaders.Then,theopinion-updatingmodelofopinionleaderswiththepositivetargetopinioncanbedescribedasfollows:xP i (t+1)=(1− wi )NP 1i (t)N 2j=N 1+1ai j (t)xj (t)+wi d,i=N1+ 1, ,N1+N2, (3-2)whereai j (t)=1, xi (t)− xj (t)≤εP i 0,otherwise ,εPi istheconﬁdencelevel,andNPi (t)= N 2j=N 1+1

ai j (t)isthenumberofneighborsofpositiveopinionleaderi.Thevariabled isthevalueofthetargetopinionofthepositiveopinionleadersubgroup,whichisaconstantthatfallsintheinterval[0,1];wi and1− wi aretheinﬂuenceweightsofthetargetopiniond andofotherpositive

opinion leadersthat satisfy theconditionxi (t)− xj (t)≤εP

i on thepositive leader i,respectively. Further,forsimplicity,

thevaluesofwiareassumedtobethesame,thatis,wi=wj.

Similarly, theopinion-updatingmodeloftheopinion leaderswiththe negativetarget opinioncan bedescribedasfol-lows:xN i (t+1)=(1− zi )NP 1i (t)N j=N 1+N 2+1ai j (t)xj (t)+zi g,i=N1+N2+ 1, ,N, (3-3)whereai j (t)=1, xi (t)− xj (t)≤εNi 0,otherwise ;NN i (t)= N j=N 1+N 2+1

ai j (t)representsthenumberofneighborsofnegativeleaderi;g

isavalueofthenegativetargetopinion,whichisaconstantbetween[−1,0];and i and1− zi arerespectivelytheinﬂuence

weightsofthetargetopinionsandothernegativeopinion leadersonthenegativeleaderi.Inaddition,thevaluesof i areassumedtobethesame.

Byestablishingtheopiniondynamicsmodelsoftheopinionfollowersandleaders,wedeﬁnetheinﬂuencepoweroftheopinionleadersastheratiobetweenthenumberoftheopinionfollowerseventuallyledbytheopinionleadersandthatofopinionfollowersattheinitialstage,whichcanbedescribedasfollows:

η=Nf

NF , (3-4)

whereNf isthenumberoftheopinionfollowersthathavesimilaropinionsasthoseoftheopinionleaders.Thenextsectionisdevoted tothe analysisoftherelationship betweentheinﬂuencepower oftheleadersandthefractionof theopinionleaders,theopinionfollowers’conﬁdencelevels,andthedegreesoftrusttowardtheopinionleaders.

4.Simulationresultsandquantitativeanalysis

4.1.Datapreparation

BasedonModels(3-1),(3-2),and(3-3),acomputersimulationmethodisadoptedtoinvestigatetheinﬂuencepoweroftheopinionleadersandtheevolutionofthecollectiveopinions.

As shownin [38], the evolution of the collective opinions tends to become relatively stable when the network sizereaches2000nodes.This indicatesthat a furtherincrease in thenetwork sizewill havenosignificant impact onthe in-fluencepoweroftheopinionleadersandtheopinionevolutionprocess ofthewholegroup.Therefore,thispaperassumesthat thesize ofthe considerednetwork isN=2000.Forall ofthecomputer experiments,the MonteCarlo simulation isconducted1000times.Unlessotherwisespecified,thefollowingassumptionsareappliedtoalloftheexperiments:

(1)thesizeofthesocialnetworkisN=2000;

(2)theinitialopinionsoftheopinionfollowersandpositiveandnegativeopinionleadersallobeyauniformdistribution;(3)theconﬁdence levelsofboth thepositive andnegativeopinionleader groupsare,respectively, εP

i =εN

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62b8ff4 cfb7b2 b498 1e6 c5e54 de90e 94c0130 0f8e 5158 f5 b0044 471 f83d013 f7 ab27a6e2 e738a2 f0 c256 bd5 9b74 d78e 78c6c1 597b758e5 5e7 f9d6 025 c9f2685a Table 1 Number distribution of the inﬂuenced opinion followers versus the proportion of the positive opinion leaders Distribution PP [ −0.5, −0.4] [ −0.4, −0.3] [ −0.3, −0.2] [ −0.2, −0.1] [ −0.1,0] [0,0.1] [0.1,0.2] [0.2,0.3] [0.3,0.4] [0.4,0.5] 0.0 0 05 0 0 291 0 1582 0 26 0 0 0 0.0010 0 0 204 0 1615 0 79 0 0 0 0.0050 0 0 172 0 1659 0 59 0 0 0 0.0100 0 0 103 0 1694 0 83 0 0 0 0.0500 0 0 89 0 1632 0 78 0 0 0 0.10 0 0 0 0 91 0 1516 0 93 0 0 0 0.1500 0 0 71 0 1436 0 92 0 0 0 0.20 0 0 0 0 62 0 1346 0 92 0 0 0 0.2500 0 0 77 0 1240 0 83 0 0 0 0.30 0 0 0 0 53 0 1154 0 92 0 0 0 0.3500 0 0 38 0 1065 0 97 0 0 0 0.40 0 0 0 0 39 0 972 0 88 0 0 0 0.4500 0 0 36 0 888 0 76 0 0 0 0.50 0 0 0 0 38 0 788 0 74 0 0 0 0.5500 0 0 26 0 712 0 61 0 0 0 0.60 0 0 0 0 25 0 618 0 57 0 0 0 0.6500 0 0 30 0 525 0 45 0 0 0 0.70 0 0 0 0 18 0 431 0 51 0 0 0 0.7500 0 0 20 0 348 0 32 0 0 0 0.80 0 0 0 0 19 0 250 0 30 0 0 0 0.8500 0 0 8 0 176 0 16 0 0 0 0.90 0 0 0 0 5 0 84 0 10 0 0 0 0.9400 0 0 2 0 16 0 2 0 0 0

(4)thedegreesof trust oftheopinion followers towardthe positiveand negativeopinionleaders satisfy αi +βi =0.8,whilethosebetweenopinionfollowerssatisfy1−αi −βi =0.2;

(5)thetargetopinionofthepositiveopinionleadersisd=0.5,andthatofthenegativeopinionleadersisg=−0.5;and(6)theinﬂuenceweightsofthetargetopinionsonboththepositiveandnegativeopinionleadersubgroupsare0.5,that

is,w= =0.5.

4.2.Proportionofopinionleaders

Inthe followingsimulation experiment, theproportionofnegative opinionleaders isset asPN =5%.Thus, ina socialnetworkwith2000nodes,the numberofnegativeopinion leadersisconstantly 100.The degreesoftrust oftheopinionfollowers toward the positive and negative opinion leaders are respectively set asαi=βi=0.4, and those between theopinion followers are 0.2 Without loss ofgenerality, we alter the proportion of positive opinion leaders to observe theimpactofsuchchangesontheevolutionofthecollectiveopinionsandinﬂuencepoweroftheopinionleaders.

Inthis experiment, the proportionPP ofpositive opinion leaders is initiallyset as0 and, accordingly, that ofopinionfollowersis setasPF =0.9500.As theproportionofpositive opinion leadersPP increasesto0.0005,0.001, 0.005,0.0100,andamaximumof 0.9400,theproportionofopinion followersin thesamesocial network isreducedto0.9495,0.9490,0.9450,0.9400,andeventually0.0100,respectively.

Fig.2 illustrates theevolution ofthecollectiveopinions ofthethreesubgroups whentheproportionofpositive opin-ionleaders changes from0.0005 to 0.94 The red, blue, andblack solid lines represent the evolution trajectoriesof theopinionsofthepositiveopinionleaders,opinionfollowers,andnegativeopinionleaders,respectively,overtime.Regardlessofthe proportionof the positive opinion leaders(0.0005,0.5000,0.8000, oreven 0.9400) or theopinion leaders’ initialopinions,both the positive and negativeopinion leaders can swiftly(in less than sixtime steps)converge to thetargetopinion ofthe corresponding subgroup.However, the opinion evolution of theopinion followers is relatively morecom-plexandrequiresmoretimetoreachastablestate.Regardlessofwhichsubgroup(positiveornegative)isatanadvantageinsize, the ﬁnal opinions offollowers are divided into three clustersat thevalues of −0.2,0, and0.2 Furthermore, nofollower’sopinion converges tothe target opinions (0.5and−0.5), suggesting that,in a social networkwith two oppos-ing opinion leadergroups, noneofthe opinion leadersappears tohave absoluteinﬂuencepower on theopinionsof thefollowers.

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Fig. 2 Evolution of collective opinions with different fractions of positive opinion leaders (a) P P = 0 0 0 05 ; (b) P P = 0 050 0 ; (c) P P = 0 50 0 0 ; (d) P P = 0 80 0 0 ;

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Fig 3 Relationship between inﬂuence power and the proportion of positive opinion leaders

[−0.5,−0.4].Unliketheresultsin[38],wherethereisonlyoneopinionleadergroupintheentiresocialnetwork,theﬁnalopinionsoftheopinionfollowerstendtolie intwo opinion-intervals:thetarget opinionandcompletelyopposingopinionofopinionleaders.

Fig 3 directly demonstrates that, with increases in the proportion of positive opinion leaders, the proportion of in-fluenced opinion followers in the three opinion subintervals [−0.3,−0.2], [−0.1,0], and [0.1,0.2] does not represent amonotonic increasing/decreasingor linear correlation, butmanifests a certain degree of fluctuation at a certain level Inaddition,whentheproportionofpositiveopinionleadersreachestheminimumormaximumvalues,thefluctuationoftheinfluencepowerinthethreesubintervalsbecomesrelativelyobvious.

We observefrom thesimulation experiment that the final opinionsof the followerstend to form threeclusters Thedistribution ofthose influenced by the opinion leaders has the following features First, the largest cluster is located inthemiddle ofthe opinion interval When the opinion interval is [−0.5,0.5], thelargest opinion cluster is located inthesubinterval[−0.1,0].Second,exceptforthelargestopinion clusterinthemiddleopinion subinterval,thefinal opinionsoftheremainingopinionfollowersformthetwosubintervals[−0.3,−0.2]and[0.1,0.2].Third,astheproportionofthepositiveleadersincreases,thatof theinfluencedopinion followers inthesubinterval[0.1,0.2] tendsto increase atthebeginning.However,theproportioninthesubinterval[0.1,0.2]showsafluctuationatthefinalstage.Theproportionoftheinfluencedopinionfollowersinthesubinterval[−0.3,−0.2]presentsadropfollowedbyarisingtrend.Thechangesintheproportionofopinionfollowersinthemiddleopinioncluster[−0.1,0]showasimilarpatternasthatinthesubinterval[0.1,0.2].

Thesimulationexperimentrevealsthat,whentherearetwoopinionleadersubgroupsinasocialnetwork,theinfluencepower ofeither opinion leader subgroup is restricted,even if the fraction ofthe opinion leadersis large enough Thus,it is necessary to investigate the reasons for the restriction in the exertion of the opinion leaders’ influence power ontheopinionfollowers.Thenext subsectionfurtherexamines theimpactoftheopinionfollowers’confidencelevels ontheopinion leaders’ influence power in order to investigate whetherheterogeneity and low confidence levels are the mainconstraintsoftheopinionleaders’influencepower.

4.3.Conﬁdencelevelsofopinionfollowers

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62b8ff4 cfb7b2 b498 1e6 c5e54 de90e 94c0130 0f8e 5158 f5 b0044 471 f83d013 f7 ab27a6e2 e738a2 f0 c256 bd5 9b74 d78e 78c6c1 597b758e5 5e7 f9d6 025 c9f2685a Table 2 Number distribution of the inﬂuenced opinion followers with heterogeneous conﬁdence levels Distribution εFi [ −0.5, −0.4] [ −0.4, −0.3] [ −0.3, −0.2] [ −0.2, −0.1] [ −0.1,0] [0,0.1] [0.1,0.2] [0.2,0.3] [0.3,0.4] [0.4,0.5] [0,1] 0 0 89 0 1632 0 78 0 0 0 [0.1,1] 0 0 95 0 1598 0 106 0 0 0 [0.2,1] 0 0 114 0 1577 0 108 0 0 0 [0.3,1] 0 0 117 0 1550 0 132 0 0 0 [0.4,1] 0 0 5 0 1733 0 61 0 0 0 [0.5,1] 0 0 0 0 1799 0 0 0 0 0 [0.6,1] 0 0 0 0 1799 0 0 0 0 0 [0.7,1] 0 0 0 0 1799 0 0 0 0 0 [0.8,1] 0 0 0 0 1799 0 0 0 0 0 [0.9,1] 0 0 0 0 1799 0 0 0 0 0

we continuously reduce the rangeofthe confidencelevels of theopinion followers.At the sametime, we increase theirconfidencelevelsinordertoexaminetheimpactoftheopinionfollowers’confidencelevels ontheinfluencepoweroftheopinionleaders,aswellasexploretheevolutionofthecollectiveopinions.

InFig.4 andTable2,thenumbersofthepositiveandnegativeopinionleadersareidentical,andtheopinionfollowershaveequaldegreesoftrusttowardthepositiveandnegativeopinionleaders.Astherangeoftheconfidencelevelsoftheopinion followers shrinks and confidence levels simultaneously increase, both the positive and negative opinion leadersreachtheir targetopinionsquickly.Furthermore,fromTable2,theopinionevolution oftheopinion followerspresentsthefollowingcharacteristics.First,withtheincreaseinconfidencelevels,thenumberofthefinalopinionclustersoftheopinionfollowersdecreases,andtheirfinalopinionstendtobecomemoreaggregated.Second,theconvergencespeedoftheopinionfollowers’opinionsisacceleratedwiththeincrease oftheir confidencelevels.Third,regardlessofthefinal opinionsoftheopinion followersbeingdivided intoone orthreeclusters,theseclustersare symmetricallydistributedwithin theopinioninterval;noobviousbiastowardeitheropinionleadersubgroupisobserved.Fourth,theincreaseinconfidencelevelscannotdistinctlyimprovetheinfluencepower.

Inordertoexplicitlydemonstratetherelationshipbetweentheconfidencelevelsoftheopinionfollowersandtheopinionleaders’influencepower,wereducethelengthoftheconfidencelevelrangeoftheopinionfollowersineachexperiment—forexample,[0.1,1],[0.2,1],[0.3,1],… [0.9,1]—toanalyzethechangesinthenumberdistributionoftheinfluencedopinionfollowers.Whentheconfidencelevelrangeisreducedfrom[0,1]to[0.1,1],[0.2,1],and[0.3,1],thenumberoftheinflu-encedagentsintheopinionsubinterval[−0.1,0]gradually decreases,whilethoseintheopinion subintervals[−0.3,−0.2]and[0.1,0.2]showan increasingtrend.Astheconfidencelevelrangeisfurtherreducedto[0.4,1],drastic changesinthenumberoftheinfluencedopinionfollowersareobservedamongthethreeopinionclusters.Theagents,whosefinalopinionsbelongto[−0.3,−0.2]and[0.1,0.2]forlongerconfidencelevelranges,tendtoconvergetothemiddleopinionsubinterval.Whentherangereaches[0.5,1],allopinionfollowersreachconsensusinthemiddleopinionsubinterval[−0.1,0].Thisre-sultshowsthat,aftertheconfidencelevelsoftheopinionfollowershaveincreasedto[0.5,1],thecollectiveopinionsoftheopinion followerswillreach aconsensus atthe compromiseopinion 0.The resultsalsoindicatethat, ina socialnetworkwith equal degreesof trust toward the positive and negativeopinion leader subgroups, increasing the confidence levelsof theopinion followersto a certain degree is conduciveto the influence powerof the opinion leaders.However, whenthe confidence levels ofthe opinion followers surpass a certain threshold,the opinion leaders’ influence power will notincrease Hence, the situation inwhich the opinion followers’opinions are completely dominatedby one of the opinionleaders’groupsisunlikelytooccur.Whentheopinionfollowers’confidencelevelsaresufficientlylarge,theyappeartohavemoredifficulties inmakinga decisionbetweentheopinionsofthepositive andnegativeopinion leaders.Asa result,theinfluence power ofthe positive and negativeopinion leadersoffset one another Moreover, due toherd mentality,moreagentstendtoabandontheopinionleaderstheyinitiallyfollowedandjointhelargestopiniongroup.

Intheaboveexperiment,thepositiveandnegativeopinionleadersubgroupshavethesamepropagation.Moreover,theopinionfollowershaveequaldegreesoftrustintheopinionleaders.Thiswell-matchedcasemaybethereasonforrestrict-ingtheexertionoftheinfluencepoweroftheopinionleaders.Inordertoinvestigatetheimpactoftheopinionfollowers’confidence levels onthe influence powerof theopinion leaders, under theconditionof two leader subgroups not beingwell matched,wealtertheopinion followers’degreesoftrusttowardthepositiveopinionleadersαi from0.4to0.6.This

issothatthepositiveopinionleadersmayhaveastrongerinﬂuenceontheagents.Correspondingly,theirdegreesoftrusttowardthenegativeopinion leadersare reducedtoβi =0.2.We thenrepeatthe processoftheprevious simulation,withtheotherparametersunchanged.

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Table 3

Number distribution of the influenced opinion followers with heterogeneous confidence levels Distribution final opinions εFi [ −0.5, −0.4] [ −0.4, −0.3] [ −0.3, −0.2] [ −0.2, −0.1] [ −0.1,0] [0,0.1] [0.1,0.2] [0.2,0.3] [0.3,0.4] [0.4,0.5] [0,1] 0 0 0 0 497 0 420 746 136 0 [0.1,1] 0 0 0 0 352 21 0 474 952 0 [0.2,1] 0 0 0 0 123 29 0 567 1080 0 [0.3,1] 0 0 0 0 0 32 0 618 1149 0 [0.4,1] 0 0 0 0 0 0 0 721 1078 0 [0.5,1] 0 0 0 0 0 0 0 832 967 0 [0.6,1] 0 0 0 0 0 0 0 1102 697 0 [0.7,1] 0 0 0 0 0 0 0 1460 339 0 [0.8,1] 0 0 0 0 0 0 1799 0 0 0 [0.9,1] 0 0 0 0 0 0 1799 0 0 0

opinionfollowersbegintoshifttowardthetargetopinionofthepositiveopinionleaders,noneofthemactuallyreachesthetargetopinionof0.5;rather,theiropinionsbecomeconsistentwithintheopinionsubinterval[0.1,0.2].Theseresultsshowthattheincreaseintheopinionfollowers’degreesoftrusttowardthepositiveopinionleaderscanstrengthentheinﬂuencepowerofthepositiveopinionleaders.However,whenthedegreesoftrustareassignedasαi =0.6,theinﬂuencepowerofthepositiveopinionleadersstillcannotbefullyexploited.

Table 3 presents the number distribution of the opinion followers when reducing the confidence level range, undertheconditionthat theopinion followershavedifferentdegreesoftrust towardthepositive andnegativeopinion leaders.ComparedtoTable2,whentheopinionfollowershavelargerdegreesoftrusttowardthepositiveopinionleadersandtheseleadersbecomemoreinfluential,thenumberdistributionoftheinfluencedagentspresentssomesubstantialchanges.Firstly,thepatternoftheirfinal opinionsbeingdistributedinthethreesubintervals([−0.1,0],[−0.3,−0.2],[0.1,0.2])disappears.Thefinalopinionsofthefollowersskewrighttoward0.5.Specifically,inadditionto[−0.1,0],theopinionsofthefollowersaredistributedinthesubintervals[0,0.1],[0.1,0.2],[0.2,0.3],and[0.3,0.4],totherightofthemiddleopinion0.Secondly,withthe narrowing of the confidence levelrange, some of the opinion followers, whose opinionsare distributed in thecenterinterval,beginto movetowardthe opinionintervals[0.2,0.3] and[0.3,0.4];theseare closertothetarget opinionofthepositiveopinionleaders.Whentheconfidencelevelrangeisreducedto[0.4,1],allopinionfollowersareaggregatedinthesubintervals[0.2,0.3]and[0.3,0.4].Itisworth notingthataftertheconfidencelevels εF

i havereached[0.3,1],thenumberof the influenced opinion followers in the subinterval [0.2,0.3] begins to increase, while that of the influencedopinionfollowersinthesubinterval[0.3,0.4]correspondinglydecreases.Aftertheconfidencelevelrangeisnarrowedtothesubinterval[0.7,1],allfollowersreachaconsensusinthesubinterval[0.1,0.2].Comparedtowhenthepositiveandnegativeopinion leadersare well matched,the final collective opinionsin thisexperimentare closerto thetarget opinion of thepositiveopinionleaders.Lastly,comparedtowhenthepositiveandnegativeopinionleadersarewellmatched,theopinionfollowersneedagreateroverallconfidencelevelinordertoreachconsensus.Specifically,whentheopinionfollowershaveequaldegreesoftrusttowardtheopinionleadersandtheconfidencelevelssatisfyεF

i ∈[0.5,1],theopinionfollowersreachconsensus However, when the opinion followers have unequal degreesof trust, the conﬁdence levels should reach εF

i ∈[0.8,1],inorderforthefollowerstocometoaﬁnalconsensus.

Onthataccount,ina socialnetworkthat consistsofmorethan oneopinion leadergroup,theconfidence levelsoftheopinionfollowersarenolongerthedecisivefactorthataffectstheinfluencepoweroftheopinionleaders.However,regard-lessofwhetherthepositive andnegativeopinion leadersubgroups arematched orunmatched, increasing theconfidencelevelsofthefollowersishelpfulinreachingaconsensus.Furthermore,althoughtheincreaseinthedegreesoftrusttowardpositiveopinion leaderscould enhancetheir influenceonthe opinionfollowers,the influenceisstill notatits maximumpotential.Theseresultsshowthatinasocialnetworkwithmultipleopinionleadersubgroups,theinfluencepowerofeitheropinionleader subgrouptends tobe restrictedbythe othersubgroups—particularlywhen thesesubgroupsholdopposingtargetopinions.Inaddition,itbecomesincreasinglymoredifficultfortheopinionfollowerstomakedecisionsanddeterminetheiropinionneighborswhenmultipleleadersubgroupsexistinthesocialnetwork.

4.4.Trustdegreestowardopinionleaders

Althoughtheprevioussectionrevealedthatenhancingthedegreesoftrustoftheopinionfollowerstowardthepositiveopinionleadersisconducivetotheinﬂuencepoweroftheseleaders,therelationshipbetweentheopinionfollowers’degreesoftrust toward the leadersandthe inﬂuence power ofthese leadersremains to some extent unclear In order tocheckwhethertherelationshipispositivelycorrelated,weagainconductaseriesofsimulationexperiments.

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62b8ff4 cfb7b2 b498 1e6 c5e54 de90e 94c0130 0f8e 5158 f5 b0044 471 f83d013 f7 ab27a6e2 e738a2 f0 c256 bd5 9b74 d78e 78c6c1 597b758e5 5e7 f9d6 025 c9f2685a Table 4 Number distribution of the inﬂuenced opinion followers with different degrees of trust Distribution αi [ −0.5, −0.4] [ −0.4, −0.3] [ −0.3, −0.2] [ −0.2, −0.1] [ −0.1,0] [0,0.1] [0.1,0.2] [0.2,0.3] [0.3,0.4] [0.4,0.5] 0.40 0 0 83 0 1609 0 107 0 0 0 0.41 0 0 63 0 1518 0 218 0 0 0 0.42 0 0 65 0 1420 0 314 0 0 0 0.43 0 0 54 0 1421 0 324 0 0 0 0.44 0 0 39 0 584 800 376 0 0 0 0.45 0 0 36 0 547 768 448 0 0 0 0.50 0 0 0 3 516 681 0 599 0 0 0.60 0 0 0 0 497 0 420 746 136 0 0.70 0 0 0 0 430 0 0 0 1369 0 0.80 0 0 0 0 367 0 0 0 0 1432

oftrusttowardthepositiveopinionleader subgroupαi ,andreducethosetowardthenegativeopinion leadersubgroup,to

investigatetherelationshipbetweenthetrustdegreeαi andtheinﬂuencepowerofthepositiveopinionleaders.

Fig.6 illustratestheevolution ofthecollective opinionswhenthedegreesoftrustoftheopinionfollowerstowardthepositiveopinionleaderstakedifferentvalues.Weﬁndthatboththepositiveandnegativeopinionleaders’opinionsrelativelyswiftly(approximatelysixtimesteps)convergetothetargetopinions0.5and−0.5,respectively,whiletheﬁnalopinionsofthefollowersarerelativelyfragmentalandtakelongertoreachthesteadystates.Whenthedegreesoftrustoftheopinionfollowerstowardthepositiveandnegativeopinionleadersarethesame(αi=βi=0.4),theopinionsoftheopinionfollowerssymmetricallyformthreeopinionclusterscenteredontheintermediateopinion0(Fig.6(a)).Whenthedegreesoftrustoftheopinionfollowerstowardthepositiveopinionleadersαi increasefrom0.4to0.43,0.5,0.6,0.7,and0.8,thethreeopinion

clustersbegintoshifttowardthetargetopinion0.5ofthepositiveopinionleaders.Thisindicatesthatthepositiveopinionleadersinﬂuencemoreopinionfollowers.Further,whenthesedegreesoftrusttowardtheleadersreachthemaximumvalue

αi =0.8,some oftheopinionfollowers reachconsensus onthetargetopinion ofthepositive opinionleaders, whereastheopinions of those remaining are concentrated atthe center opinion 0 (Fig 6(f)) Note that none ofthe opinions of anyfollowerconvergestothetargetopinion−0.5ofthenegativesubgroup.

Table 4 presents the detailed relationship between the degrees of trust toward the positive opinion leadersand thenumberdistributionoftheopinionfollowersinthe10opinionsubintervals.

ThefollowingcharacteristicscanbeobservedfromTable4 above.First,thenumbersdistributedinthethreesubintervals[−0.3,−0.2],[−0.1,0],and[0.1,0.2]havechanged.Withtheincrease intheopinionfollowers’degreesoftrusttowardthepositiveleaders,alargerproportionofopinionfollowersleavetheopinionsubintervals[−0.3,−0.2]and[−0.1,0],andjoinin [0.1,0.2], which is closerto the target opinion of the positive opinion leaders Evenif there is a subtle increase, forexample,0.01ofαi from0.4to0.41,0.42,0.43,and0.45,thereisasubstantialincreaseinthenumberofopinionfollowers

insubinterval[0.1,0.2], andthenumbersin theother twointervalstend tocorrespondingly decrease.When thedegreesoftrust reachαi ≥ 0.5,thenumberofopinion followersin theinterval [−0.3,−0.2] isreducedto zero.Second, therearechangesinthedistributionofthenumberofopinionfollowersacrossthe10opinionsubintervals.Withtheenhancementoftheopinionfollowers’degreesoftrusttowardthepositiveopinionleaders,theiropinionsshowatendencytomovetowardtheopinion subintervalthatisclosetothetargetopinion ofthepositiveopinion leaders(0.5) Whenαi >0.5,apartfromsome opinion followersthat remain inthe interval[−0.1,0],theremaining agents allconvergeto thesubinterval that isclosertothe targetopinion ofthepositive opinionleaders.Whenαi =0.6,theopinion followersare distributedbetween[−0.1,0],[0.1,0.2],[0.2,0.3],and[0.3,0.4].Third, comparedtothe situationinwhichthere isonlyone group ofopinionleadersinthenetwork,theinﬂuencepowerofopinionleadersseems tobeweakenedinanetworkwithmultipleopinionleadersubgroupswhentheopinionfollowers’degreesoftrusttowardtheopinionleadersarethesame.Forexample,whenthe degrees of trust of the opinion followers toward the positive opinion leaders are αi =0.6 and αi =0.7, no opinionfollowerfallsintotherangeofthetargetopinionofthepositiveopinionleader[0.4,0.5].Fourth,whentheopinionfollowers’degreesof trusttoward thepositive opinion leaders areαi =0.8,although there are still negativeopinion leadersinthesocialnetwork, thedegreesofthetrusttowardthenegativeopinionleadersbecome βi =0.Thus,themodel(3-1) canberevisedasfollows:xF i (t+1)=(1−αi )NF 1i (t)N 1j=1ai j (t)xj (t)+αi 1NP i (t)N 1+N 2j=N 1+1ai j (t)xj (t). (4-1)

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4.5.Lessonslearned

Fromtheaboveanalysis,wenotethat:(1)Theproposed models(3-1)–(3-3)areverygeneralboundedconfidencemod-els, which can be reduced to the model (4-1) with one subgroup ofopinion leaders, or the HKmodel without opinionleadersundersomeconditions.(2)Theabovesimulationexperimentfullydemonstratesthat,whenotherconditionsremainunchanged,andthedegreesofthetrustofopinionfollowerstowardthepositiveopinionleadersincrease,theinfluenceofpositiveopinionleadersislikelytogrow.Onthataccount,inasocialnetworkwithmultipleopinionleadersubgroups,im-provingopinionfollowers’degreesoftrusttowardopinionleadersisanessentialapproachtoenhancetheinfluencepoweroftheseopinionleaders.(3)Infuture,realopiniondatacould becollectedfromsocialnetworkstovalidate theproposedmodels(3-1)–(3-3).Inrecentyears,withtherapiddevelopmentofInternettechnology,peoplecaneasilyusetools,includ-ingScribe,Chukwa,Kafka,andFlume,toacquiredataonsocialmedia platformsthroughtechniquessuchaswebcrawlersandapplicationprogramminginterfaces.

5.Conclusions

Thispaperconsidered asocial networkwithmultipleopinion leadersubgroups Itestablished averygeneralboundedconfidence-basedopiniondynamicsmodelforopinionleadersandfollowers,whentheopinionleadersubgroupspossesseddifferenttargetopinions.Wethenutilizedacomputersimulationtechniquetoinvestigatetherelationshipbetweenthepro-portionofopinionleaders,confidencelevelsofopinionfollowers,anddegreesoftrustofopinionfollowerstowardtheopin-ionleaders.Theresultsprovidedaquantitativeanalysisforthecollectivedecision-makingofasocialgroupine-commercenetworks Insummary,throughthecomparative analysisofthethreefactors,thedegreesoftrustofopinionfollowersto-wardopinionleadershaveamoreimportanteffectontheinfluencepowerofopinion leaders.Thus,inordertomaximizethepropagationeffectine-commerce,enhancingopinionleaders’credibilityisacrucialprecondition.

Wenotedthat opiniondynamicsresearch generallyusescomputer simulationmethodstoinvestigatetheopinion evo-lutionmechanismfordifferentinﬂuencefactors.Whengroupopinionsevolveinane-commerceenvironment,wemayusesometools,includingScribe,Chukwa,Kafka,andFlume,toacquireopiniondataonsocialmediaplatforms.Futureresearchliesinusingtheacquireddatatotestthedegreeofapproximationbetweenmathematicalmodelsandtheactualprocessesofopiniondisseminationonsocialmedia platforms.Thus,itwouldhelptocontinuouslyimprovethemathematicalmodel,aswellasdeepentheunderstandingoftheprincipleofevolutionofpublicopinion.

Acknowledgments

ThisresearchhasbeenpartiallysupportedbygrantsfromtheNationalNaturalScienceFoundationofChina (#71725001,#71325001,#71503206,#71471149 and#71433001), and the Major project ofthe National Social Science Foundation ofChina(#15ZDB153).

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