When ego-involved and non ego-involved agents lead to polarization despite a large consensus

Transcription

1 When ego-involved and non ego-involved agent lead to polarization depite a large conenu S. Huet To cite thi verion: S. Huet. When ego-involved and non ego-involved agent lead to polarization depite a large conenu. Social Siulation Conference 2016 (SSC 2016), Sep 2016, Roe, Italy. 2016, < <hal > HAL Id: hal Subitted on 29 Jun 2017 HAL i a ulti-diciplinary open acce archive for the depoit and dieination of cientific reearch docuent, whether they are publihed or not. The docuent ay coe fro teaching and reearch intitution in France or abroad, or fro public or private reearch center. L archive ouverte pluridiciplinaire HAL, et detinée au dépôt et à la diffuion de docuent cientifique de niveau recherche, publié ou non, éanant de établieent d eneigneent et de recherche françai ou étranger, de laboratoire public ou privé.

2 When ego-involved and non ego-involved agent lead to polarization depite a large conenu Sylvie HUET ylvie.huet@irtea.fr LAPSCO, UBP, Cleront-Ferrand, France / Irtea, LISC, Aubière, France Abtract. Huet & Deffuant [33] howed polariation on inor iue can be reproduced by a dynaic inpired fro laboratory experient [29] conidering a population of elf-involved agent on a ain iue. Thi paper tudie the ipact of a ixed population of elf-involved and not elf-involved agent. The dynaic of elfinvolved agent i baed on [33] eaning agent are attracted on every iue if they are cloe onto the ain iue but reject their peer opinion if they are only cloe onto econdary iue. The dynaic of non elf-involved agent i a iple 2D bounded confidence odel [11]. In thi paper, we focu on the level of polariation of agent when they are ajorly gathered into one cluter. We oberve the eergence of variou type of extree inor cluter. They can lead the population to a polariation on inor iue, but alo on fundaental one. A very long tranitory tate of ingroup polariation on econdary iue ha been alo exhibited. Soe of thee tate are pecific to ixed population. Keyword. Attraction, rejection, negative influence, involveent, bounded confidence odel, polarization, extree inority effect, in-group polariation Polariation i claically defined a an extreiation of the attitude after oe dicuion in a group copared to the attitude initially held by people of thi group. It i claically eaured by the average opinion before and after a dicuion. When the average opinion ha increaed in abolute value, the dicuion i aued having led to polariation. Thi i an individual a well a a collective proce. Egoinvolveent refer ore generally to the involveent of the elf or peronal involveent. Ego-involving topic were thoe that have intrinic iportance and peronal eaning.. Iportant or involving iue were thoe that had elfrelevance. [1]. Muzafer Sherif epecially outlined the role of ego-involveent in peruaion:... Regardle of the dicrepancy of the poition preented, we predict that the ore the peron i involved in the iue (the ore iportant it i to hi), the le uceptible he will be to hort-ter attept to change hi attitude [2]. In thi paper, inpired by [3] and the ocial identity theorie [4-6], we aue a trong egoinvolveent explain polariation. We explore our hypothei uing a iulation agent-baed odel inpired fro ocial pychology tudie and conidering agent with two different level of involveent in the dicued iue. In thi conference paper, ince the ot unexpected type of polariation i a polariation of the whole

3 population, we focu on the paraeter ubpace where agent are ajorly gathered into one cluter to tudy if it polarie. We dicovered variou type of eerging extree inor cluter. They can lead to a polariation on inor iue, but alo on fundaental one. A very long tranitory tate of ingroup polariation on econdary iue ha been exhibited alo. The following preent a non exhautive tate-of-theart before an overview of the odel. The three next ection give detail onto the reult and their explanation. A lat part dicue what we learn. 1 The tate-of-the-art For long polariation ha been neglected by cientit. Indeed, a large body of tudie dedicated to how individual change their attitude have at firt exhibited the tendency of people to be attracted to each other [7, 8], to confor Ach [9]. Thee dynaic of attraction lead to a conenu of people gathered in a ingle group due to a ajority effect puhing everyone to a centred poition. Therefore, with uch a doinant dynaic, it i difficult to explain the diverity of opinion that i generally oberved. Siulation odel actually propoed different approache, uing binary attitude, pecific network of interaction and variou weight aociated to the agent [10-12]. Another approach i inpired by the ocial coparion theory [13], and the ocial judgeent theory [14], auing that the attraction decreae or even vanihe when the agent attitude are very contrated. Several coputer odel are baed on thi idea, and the iulation how that the agent tend to gather in group of hoogeneou attitude called cluter [15, 16]. Thee odel are uually called bounded confidence odel. Thee odel of bounded confidence are able to exhibit polariation that they aue, accordingly to the literature, to be an attraction for a inority attitude [17] or a inority group [18, 19]. Thi i alo in relation with ego-involveent ince the ocial judgeent theory [14] aue the ego-involveent level ake the enitivity of the agent to peruaion: the influence increae and then decreae with the dicrepancy between the attitude of the ource and the attitude of the participant. The level of ego-involveent defined the inflexion point of the influence: the ore ego-involved, the thicker the dicrepancy before the inflexion point and the le enitive to opinion change i the participant. Typically, the iulation tart with a population including two type of agent: extreit who have extree and very certain attitude (they are not eaily influenced by other) and oderate whoe attitude are uniforly ditributed and who can eaily be influenced. Depending on the initial nuber of extreit and the initial uncertainty of the oderate, three convergence type appear: central cluter, double extree cluter in which alot everyone have adopted one or the other extreit attitude, ingle extree cluter iplying everyone ha the ae extreit attitude [20-22]. Recently [23] how that a new tationary tate appear when the uncertaintie do not change over tie and for very large uncertaintie of the oderate : the attitude of oderate agent keep fluctuating without clutering but the ditribution of the attitude over the tie reain table. Thee odel provide pattern of attitude evolution that can be

4 related to obervation. However, they uppoe that extreit are initially preent; they do not explain how thee initial extreit appear. Thu, ore recently, everal reearcher invetigated if polariation can take place fro a tarting population with only oderate by adding a rejection echani in oe condition. Soe of thee odel conider binary opinion [24] [25] [26]. In ot of the odel, the rejection take place when the attitude are very far apart ([27, 28]), a uggeted by oe reearche in ocial pychology [29]. Another hypothei for the condition in which rejection take place i to conider it coe fro a dionance between oeone value and a inor dicued iue. Thi hypothei i grounded on tudie and theorie uggeting that polariation i connected with identity iue [4-6]. In particular, the et of experient preented in [3] ugget that the level of elf-relevance of the group-ource for the agent i very iportant in the rejection echani. Indeed, the experient how that in cae of agreeent on a inor iue and diagreeent on a highly elf-relevant iue (i.e. the group eberhip in the experientation), a ubject tend to hift away fro alter poition on the inor iue. Moreover, in cae of agreeent on a highly elf-relevant iue, the ubject tend to be attracted on inor iue whatever the level of preliinary diagreeent. A firt invetigation by iulation of oe identity iue in attitude dynaic ha been done by [30] which ha propoed a odelling of the elf-categorization theory [31]. However agent have to olve an optiization proble to define their eberhip. Depite the great interet of thi work, thi doe not ee u ufficiently iple to reflect real coon huan dynaic. Alo [28] [32] have propoed an ipleentation of the ocial judgent theory with a bound for attraction or rejection defined by the level of ego-involveent of the agent. However, their deign, in one or two dienion, doe not avoid a poible exploion of opinion. Another approach ha been propoed by [33] inpired by the [3] experient. They conider pair interaction in which two agent who can hare or not attitude on a fundaental iue influence each other on le iportant (econdary) attitude and on thi fundaental (ain) attitude. On each of the two dienion (i.e. two attitude), they uppoe that the attitude can take continuou value between -1 and +1. When the agent are cloe to each other on the ain dienion, they tend to attract each other on the econdary dienion, even if their diagreeent i trong. When they diagree on the ain dienion, and are cloe on the econdary, then they tend to reject each other' econdary attitude. Agent have threhold to decide if the ditance between their attitude and oeone ele attitude i cloe. Then, to odel a diagreeent, they copare the ditance between attitude with a threhold (called attraction threhold u on the ain dienion, and rejection threhold u on the econdary dienion). They how thi dynaic reduce the nuber of cluter ince agent are driven ainly by the ain dienion and the nuber of cluter can be directly deterined fro the value of u ; the nuber of econdary dienion doe not play a role in the definition of the nuber of cluter. Moreover, they oberve polariation on the econdary dienion due to the rejection. However, the dynaic they have conidered i only valuable for agent conidering the ain dienion a highly elf-relevant a outlined by [3].

5 Thee author (pp. 1190, 1191) alo indicate that if the participant i not highly negatively elf-involved in the group of the ource, he doe not change opinion when he agree with the ource on econdary iue but not on the fundaental iue. Rather he decreae the elf-relevance of the ource (i.e. the level of iportance given to the ource group) in a way not to differentiate fro it in the future. They alo precie that participant relatively indifferent to the group ource deontrated little hift in their attitude judgent (p. 1186). Then in thi paper, we will aue that oe agent are indifferent to their group eberhip and do not change their opinion if they diagree on at leat one dienion. Thi choice i done for iplicity purpoe, a well a the one conidering a tatic elf-relevance or level of iportance given to the ain attitude. In the following we conider a ixed population copoed of a tatic part of highly involved agent in one ain attitude and the copleentary part of noore-iportant-attitude agent. The dynaic of the firt type of agent i baed on [33] while the dynaic of the econd type i baed onto a claical 2D bounded confidence odel [15]. They are preented with detail in the next Section. 2 The odel and the experiental deign We conider a population of N individual. A part of the are conidered a highly elf-involved in the ain dienion while the other part i not. The odel include four paraeter: phi, the part of highly elf-involved people having a dynaic cloed to the people conidering the group eberhip a highly elf-relevant in [29], u and u, repectively threhold for opinion change on the ain and the econdary dienion, and μ ruling the intenity of influence at each eeting (copried between 0 and 0.5). An individual ha two attitude x (on the ain dienion) and x (on the econdary dienion) taking real value between -1 and +1. During an iteration, a couple of individual X and Y i randoly choen and can influence each other. The algorith i the following: Chooe randoly a couple (X,Y) of individual in the population; X and Y change their attitude at the ae tie, according to the influence function choen accordingly to their tatu (i.e. the potentially influenced i a non-highly elf-involved agent or a highly elf-involved agent). 2.1 Influence on a non-highly elf-involved agent (non-hsi) The calculation of the influence of Y on X if X i a non-hsi agent (of coure the influence of X on Y if Y i a non-hsi agent i found by inverting X and Y) i the following. Let (x, x ) and (y, y ) be the attitude of X and Y repectively. The influence of Y on X i not null if x y u and x y u indicating agent X agree with Y. Both attitude of X are going to get cloer to thoe of Y, proportionally to the attitudinal ditance on each dienion: x ( t 1) y ( t) x ( t 1) y ( t)

6 For every other cae of partial agreeent (only cloe on one dienion) or total diagreeent (far on two dienion), X reain indifferent to Y and doe not change their opinion. 2.2 Influence on a highly elf-involved agent (HSI) The calculation of the influence of Y on X if X i a HSI agent (of coure the influence of X on Y if Y i a HSI i found by inverting X and Y) i baed on [33] and can be decribed a follow. Let (x, x ) and (y, y ) be the attitude of X and Y repectively. We firt conider the ain attitude dienion. If x y u, individual X agree with Y on the ain dienion. Both attitude of X are going to get cloer to thoe of Y, proportionally to the attitudinal ditance on each dienion: x ( t 1) y ( t) x ( t 1) y ( t) Indeed, whatever the agreeent level on the econdary dienion i, X i going to be globally cloer to Y. If it wa already cloe, it get cloer. If x y u, agent X diagree with Y on the ain dienion and if x y u : Agent X feel it i too cloe to Y on the econdary dienion, becaue of their diagreeent on the ain dienion. To olve the conflicting ituation, X ove away fro Y on thi dienion. The attitude change i proportional to the ditance to reach the rejection threhold: if y 0 x ( t 1) u y ( t) x then ele x ( t 1) u y ( t) In the other cae X i not odified by Y. Moreover, we confine the attitude in the interval [-1, +1]: if then where ign() i a function which return -1 if it arguent i trictly negative, +1 otherwie. The attitude of Y i calculated i the ae way conidering the ituation of the eeting with X and if Y i HSI. 2.3 The experiental deign and the eaured indicator Our population of agent have attitude initially drawn at rando. We iulate their evolution, for different value of the paraeter in order to copare the behaviour of a ixed population to the known behaviour of pure population of highly involved agent or not involved agent. We vary the following paraeter conidering 10 replicate for each et of value: N and phi: 5000 (known a ufficient to oberve table tate of the already known odel of pure population, ie phi equal 0 or 1 (correponding to [33]), agent for phi equal to 0.1, (and agent to have ore viible graph); u and u : fro 0.05 to 1 by tep of We top the exploration of u and u at 1 arbitrarily for thi firt tudy. For a ae et of paraeter value, we conider two cae for coparion purpoe: on the ec-

7 ondary dienion, the opinion i bounded between [-1;+1]; it i not bounded. The econd cae ha been tudied to prove the bound are not reponible for the polariation. μ reain contant on the two dienion for every experiental condition. It i valued 0.5. Two indicator are eaured during the iulation and then averaged over the replicate: (1) the nuber of cluter (which i claically counted conidering one agent i eber of a cluter if it i located at a ditance aller than a threhold fro at leat one eber of the conidered group); (2) the average abolute opinion on each dienion (it i valued 0.5 at the beginning for a unifor rando initialiation of opinion). In thi conference paper, ince the ot unexpected type of polariation i a polariation of the whole population (or alot), we focu on the paraeter ubpace where agent are ajorly gathered into one cluter to tudy if it polarie. Practically, it ean our nuber of large cluter i one and the average abolute opinion on each dienion i cloe to 0 (ince everyone goe to the centered poition in pure population cae. Thu a value ufficiently higher than 0 indicate a polariation while a value cloe to 0 indicate a centraliation of the opinion. 3 The clutering proce Thi ection i intereting in the nuber of cluter generated by the dynaic and reind what we already know fro the behaviour of pure population. For a pure population of non-hsi, the nuber of ufficiently large cluter can be approxiated by 1/u *1/u. If the population i only copoed fro HSI agent, we know the nuber of cluter i approxiated by 1/u ince the ain dienion drive the clutering [33]. Thi ake the nuber of cluter independent fro the nuber of dicued econdary dienion and iplify the ocial pace a it can be een by an agent (i.e. becaue of the aller nuber of cluter). Fig. 1 on the right how what occur for a ixed population of non-hsi and HIS agent (phi=0.1): a behaviour quite iilar to the one oberved for phi =1 aking the nuber of cluter equal to 1/u. A few part of highly elf-involved people i enough to iniie the nuber of cluter and to ake the proble of the nuber of cluter alway equivalent to a one-dienion proble (a for a pure population of HSI agent). Fig. 1 on the left reind about the exitence of inor cluter containing a very all inority of agent when μ i ufficienly large (a it i the cae in our current experiental deign). They have been clearly identified in the BC odel and are located between the variou cluter and onto the extrea of the opinion pace, epecially for large value of u norally leading to one ajor cluter containing at leat 95% of the population [34] [35]. We will ee thee inor cluter becoe iportant when the population contain HIS agent. The two following next Section preent the reult regarding polariation.

8 Fig. 1. On the left: Nuber of cluter containing at leat 4% of the population (for a bounded opinion 2D pace) and phi = 0.1 (10000 agent). Abcia u, ordinate : u. While light grey draw the area of total conenu (only one cluter), other colour repreent where agent gather into everal cluter. On the right: Minor cluter appear for larger value of µ in an opinion pace varying fro 0 to 1, fro Laguna et al [34] (ee alo [35] who how ize of inor cluter varie with u [for a denity odel]) 4 The polariation trajectorie In the one-cluter paraeter ubpace, we dicovered eerging extree inor cluter leading to a polariation on inor iue, but alo on fundaental one. A very long tranitory tate of ingroup polariation on econdary iue ha been exhibited alo. The following preent the repective trajectorie after a firt ubection explaining how they are repreented. 4.1 How to read the trajectorie. The following figure preent the initial tate a well a the type of graph ued to how the trajectorie of the odel (on the left). All along the iulation, dot on graph repreent an agent opinion: brown dot repreent the non-his agent, and green dot repreent the HSI agent. Since the cluter are ufficiently fored, they are alo repreented in denity over the paraeter pace (on the right). The cale for the denity i on the right of the figure.

9 Fig. 2. On the left, initial tate of unifor rando opinion for 1000 agent, and in the center for agent (25000 agent have been choen for illutration becaue dynaic i ore viible onto the graph): each dot repreent one couple of opinion (x,x ) of an agent. On Abcia, opinion 1 i the ain opinion. On ordinate, opinion 2 i the econdary dienion. Brown dot are non-his agent, and green dot and non-his agent. The opinion denity ditribution of each dienion are viible on the right and the top of the graph. 4.2 Polariation on the ain dienion: a ajor ingle extree due to extree inoritie A firt type of polariation we oberved occur on the ain dienion for variou couple of value (u, u ) with u < u. Fig. 3 how an exaple of trajectory leading to uch a ingle extree cluter on the ain dienion ince oe very extree inor cluter have eerged. It i ainly ruled by attraction between agent. Due to the high value of u, agent go quickly to the center and oe agent are forgotten on the extree value of the ain dienion by the dynaic (previouly hown by [36] and [37], ee alo Section 3, fig. 1 on the right). For u < u the initial depolariation onto the econdary dienion i lower than the depolariation on the ain dienion. Then, there are le forgotten agent on the extree of the econdary dienion. Moreover, HSI agent go quicker to the center copared to non-hsi. Since everyone i gathered into a centered unique large cluter, thi cluter lowly ocillate in the opinion pace due to HIS agent which want to be cloer to forgotten agent on the border. Thi ove lead the larger cluter to an extreiation, only topped when all forgotten people cloe on the ain dienion are gathered into it. Thi i obviou HSI agent in green are attracted on both dienion by the extree corner of non-hsi agent. To u-up, thi i the tability of the eerging corner all cluter which i reponible for the polariation of the larger cluter. Thi tability coe fro non-hsi agent which require cloene onto the two dienion to be influenced. Then, conidering the relation between the non-his of the larger and the corner all cluter, they ignore each other becaue they are different onto the econdary dienion. Thi type of polariation doe not depend on rejection (it appear alo for a iilar odel with no rejection).

10 Fig. 3. Polariation on the ain dienion: Single extree. Stable tate for agent, u = 0.8, u = 0.3, phi = 0.1 (dot in green are HSI and thoe in brown are non-hsi agent).

11 4.3 Polariation on the econdary dienion: another extree inoritie effect Fro the tudy of a pure population of highly elf-involved agent in [33], we know the unique large cluter can polarie for very large value of u due to the preence of inor cluter in the opinion pace. For large u, agent attract each other very fat to the center of the opinion pace on the ain dienion. Then oe agent are forgotten onto the extrea of the ain dienion. Since alot everyone ha gathered into a large central cluter, thee reaining inor cluter, far on the ain dienion, puh agent fro the central cluter to polarie onto the econdary dienion. Fig. 4. Polariation on the econdary dienion. Stable tate for agent, u = 0.7, u = 1.0, phi = 0.11 (dot in green are HSI and thoe in brown are non-hsi agent). Blue circle indicate extree cluter of non-his agent and green one indicate HSI agent. Fig. 4 how thi alo occur in ixed population. For our particular cae, coparing to inor effect leading to polarization on ain dienion, agent gather into the central cluter very fat on the econdary dienion. Since u i very large, there i no forgotten (or very few) agent on the econdary dienion. Mot of the forgotten agent on the ain dienion gather altogether on quite central econdary dienion poition on the ain extrea. Only when there are forgotten people onto the two dienion, inor cluter can eerge in the corner. Thee inor cluter are

12 otly copoed fro non-hsi agent. However, fro the point of view of the HSI agent, eber of the larget initially central cluter, they are conidered a cloe onto the econdary dienion while far on the ain dienion. Then HSI agent of the larget cluter reject the while iultaneouly attract the non-hsi agent of the larget cluter. On the other hand extree HSI agent on ain dienion ake the ae (ee green circle on Fig. 4). Thi ake the large cluter polarized onto the econdary dienion until the inor cluter are not anyore conidered a cloe. Then, they reain at the end, on the other extree than the one occupied by the inor cluter on the ain dienion. Thi latter indeed, are aborbed by the larger cluter becaue of the HSI-agent conidering the a attractive due to the proxiity onto the ain dienion. Thi type of polariation depend on rejection (it doe not appear for a iilar odel with no rejection). 4.4 Polariation on the econdary dienion: a long tranitory in-group polariation We have alo noticed a tranitory in-group polariation occurring when u i all (and u >0.5). It i due to the tie required to reach the convergence which increae when u decreae. For aller value of u, the unique central cluter cover alot all the econdary dienion for a long tie, copoed fro variou ubcluter of non-hsi agent which are joined altogether by oving HSI agent (ee Fig. 5). At the end, everyone i gathered in the center due to their ipact of the HSI agent. Thi perfectly decribe how HSI increae the level of coheion of the population copared to a population of non-hsi. Fig. 5. Polariation on the econdary dienion u = 0.7, u = 0.1, phi = 0.1, 5000 agent: the clutering proce in the depolariation area onto the econdary dienion i very long when u i all. Then, the agent reain polaried for long inide a unique cluter but really clutering onto the two dienion. 5 The polariation ap for large u A already aid, we focu on polariation on the ot expected depolariation area: the area leading to one cluter that we define a u >0.5 according to Fig. 1 and the literature on the BC odel. We draw ap of polariation for variou u and variou

13 u on the ain and on the econdary dienion. The blue color indicate a perfect centraliation of the opinion (no polariation) while other color how variou level or polariation (ee 2.3 for explanation). Fig. 6. Average abolute opinion on ain dienion for, fro the left to the right, phi equal to 0 (5000 agent), 0.1 (10000 agent) and 1 (5000 agent). Abcia u, ordinate: u. We ee on the fig. 6 the polariation on the ain dienion. Thi polariation coe only fro the co-preence of highly elf-involved people and non-highly elf-involved people (in the centered graph) ince it doe not occur for phi = 0 (graph on the left) a well a for phi=1 (graph on the right). For thee latter cae, only depolariation occur. Thi polariation leading to a ingle extree cluter occur only for a particular arrow-hape of couple (u, u ) value for phi=0.1 which reain to explain. We notice epecially it doe not occur for u >u. Fig. 7. Average abolute opinion on econdary dienion for, fro the left to the right, phi equal to 0 (5000 agent), 0.1 (10000 agent) and 1 (5000 agent). Abcia u, ordinate: u. The polariation on the econdary dienion doe not occur for a pure population of non-hsi agent (on the left of the Fig. 7) and HIS agent (on the right of the Fig. 7), at leat for u 1. It doe for a very all part of HSI agent in the population (Fig. 7 in the central for phi=0.1). Fro the centered graph of Fig. 7, we notice two variou area of polariation for: (a) large value of u > u and u large enough (ie cloe or higher than 1 not to oberve forgotten agent onto the econdary dienion; (b) all value of u. They correpond to two trajectory type preented in Section 4.

14 The large value of u correpond to the ingle extree cluter. The all value of u correpond to the tranitory in-group polariation that, a expected, doe not depend on the value of u ince u i higher than 0.5. One can think the polariation can be explained by the fact the opinion pace i bounded. But it i not the cae a it can be een in the following Fig. 8 howing reult for an unbounded attitude pace. We ee on both figure the ae polariation area. u 0,55 0,6 0,65 0,7 0,75 0,8 0,85 0, ,95 0,9 0,85 0,8 0,75 0,7 0,65 0,6 0,55 0,5 0,45 0,4 0,35 0,3 0,25 0,2 0,15 0,1 0,05 u 0,65-0,75 0,55-0,65 0,45-0,55 0,35-0,45 0,25-0,35 0,15-0,25 0,05-0,15 Fig. 8. Average abolute opinion on the ain (on the left) and on the econdary (on the right) dienion for agent evaluating in an unbounded opinion pace, phi =0.1 Abcia u, ordinate: u. 6 Concluion We tudy how involveent can explain polariation in an agent-baed odel. We conider a population copoed of two different type of agent: the highly elfinvolved (HSI) agent having a dynaic baed onto [33] inpired fro a particular et of experient [3]; the non-highly elf-involved agent (non-hsi) with a dynaic baed onto a two-dienional bounded confidence odel ([15]). HIS dynaic correpond to the particular behaviour in ter of attitude change of people who are highly elf-involved in one dienion and conider it a defining theelve. Fro the odelling point of view, it aue that one dienion i ore table and ore iportant than the other. Thi ore table dienion, called ain dienion, repreent the highly elf-relevant iue for the agent and rule the attitude change on a econdary dienion. If two agent are cloe on the ain dienion (eparated by a ditance aller than u ), then they attract each other on the ain and on the econdary dienion, whatever their diagreeent on the econdary dienion. If they are far fro each other on the ain dienion, then proxiity on the econdary dienion (at a ditance le than u ) i uncofortable, and generate rejection on thi dienion. The proxiity on a dienion i defined by coparing the attitude ditance with the agent threhold on thi dienion. For non-hsi agent, neither the ain nor the econdary dienion i ore iportant than other. Then, to be influenced, the non-his agent hould conider the peer ource a cloe iultaneouly onto the two dienion. One can think thi latter condition for influence i very deanding. But [3] precie in their paper that participant who do not conider the ource a highly

15 elf-relevant do not change their opinion, and [36-39] how in their experient the level of attraction for peer depend on the rate of iilarity with the. A expected, iulation how agent gathering into attitudinal group. The nuber of group i defined by how high they feel cloe onto the ain dienion of the ret of the population. Even few highly elf-involved people are enough to iniie the nuber of cluter. In thi paper, for the tudy of polariation, we focu on u >0.5 eaning the table tate i only one ajor cluter (containing ot of the population). Polariation on ain iue appear for oe particular couple of value (u,u ) and u u. They lead to polariation becaue on the one hand to the rejection proce, on the other hand to the ayetry of influence between highly elf-involved and non-highly elf-involved agent. In both cae, they allow the eergence of corner inor cluter, extree on the two dienion and copoed of non-highly elfinvolved agent. Since extree inor cluter and central ajor cluter have eerged, the attraction of highly elf-involved agent of the central cluter for non-highly involved agent of the inor extree cluter lead the ajor cluter to polarie on the ain attitude. The polariation on fundaental iue i a pecificity of ixed population. It i aociated to the blindne of non-highly elf-involved agent which ake the relationhip with highly elf-involved agent ayetric and forced the to adopt exactly their poition or to reject the. Thi type of polariation doe not depend on rejection. It occur at leat, but not alway, when openne on the ain iue i higher or equal than the one on the econdary iue. Thi i not really intuitive ince fundaental iue ake u ore table and le influenced norally. However, thi i totally coherent with the idea a high uncertainty i reponible for extreiation of people. A econd type of polariation of the central cluter i onto the econdary dienion. It occur when the paraeter value do not allow the appearance of corner inor cluter but oe agent reain forgotten onto the extree of the ain dienion (large value of u > u and u large enough (ie cloe or higher than 1 not to oberve forgotten agent onto the econdary dienion). It i due to a rejection of highly elf-involved agent between the iultaneouly with an attraction for the agent of their own group (ajor on the one hand, extree on ain attitude on the other hand). Thi ound in accordance with theorie related to group and identity, epecially the elf-categorization theory [26-28] arguing agent look for a coproie between difference and iilarity to other at the group level. Thi type of polariation depend on rejection. It i ore intuitive ince people are ore certain about their fundaental iue copared to the econdary. The appearance of a long tranitory in-group polariation aociation to a large certainty (ie a all uncertainty) on econdary iue i alo of interet. It alo perfectly decribe how HSI increae the level of coheion of the population copared to a population of non-hsi. Overall, thee firt reult ee to reconcile variou point of view about the poible ource of polariation: attraction for extree inority on ain iue, or rejection of the on econdary iue. In the two cae, the key of the polariation lay in the eergence of thee extree inoritie which require a too fat convergence of opin-

16 ion (ie a ufficiently high value of µ). Our reult alo offer oe new inight about thi eergence copleenting the work of [34, 35] who have hown inor cluter appear between the big cluter and on the border of the attitude pace in the bounded confidence odel. The in-group polariation, even tranitory, exhibited for very all value of u i alo very intereting and new. Thi i only a firt tudy which deerve ore invetigation about the foraliation of the behaviour and what occur for aller value of u. The explanation of why the polariation on the ain iue occur only for oe particular couple of value (u,u ) and u u i alo a challenge to deal with in the futre. 7 Reference 1. Petty, R., J. Cacioppo, and C. Haugtvedt, Ego-involveent and Peruaion: An appreciative look at the Sherif' contribution to the tudy of elfrelevance and attitude change, in Social judgent and intergroup relation: Eay in honor of Muzafer Sherif, D. Granberg and G. Sarup, Editor. 1992, Springer/Verlag: New York. p Sherif, C.W., and Muzafer Sherif, ed.. Sherif, Carolyn W., and Muzafer Sherif, ed. Attitude, Ego-involveent, and Change. Wiley, in Sypoiu of the Pennylvania State Univerity New York. 3. Wood, W., et al., Self-definition, defenive proceing, and influence: the norative ipact of ajority and inority group. Journal of Peronality and Social Pychology, 1996(71): p Brewer, M.B., The Social Self: On Being the Sae and Different at the Sae Tie. Per. Soc. Pychol. Bulletin, (5): p Tajfel, H. and J. Turner, An integrative theory of intergroup conflict. The ocial pychology of intergroup relation, (47): p Turner, J.C., Social identification and pychological group foration, in The ocial dienion, H. Tajfel, Editor. 1984, Cabridge Univerity Pre: Cabridge. p Allport, G.W., Social Pychology. 1924: Boton: Houghton Mifflin. 8. Sherif, M., A tudy of oe ocial factor in perception. Archive of Pychology, (187). 9. Ach, E.S., Foring Ipreion of Peronality. Journal of Abnoral and Social Pychology, 1946(41): p Axelrod, R., The dieination of culture a odel with local convergence and global polarization. Journal of Conflict Reolution, (2): p Latané, B. and M.J. Bourgeoi, Dynaic ocial ipact and the conolidation, clutering, correlation and continuing diverity of culture (chapter 10), in Group Procee: Blackwell handbook of ocial pychology, M.A.H.R.S. Tindale, Editor. 2001b, Oxford: Blackwell. p

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