What is a leader of opinion formation in bounded confidence models?

Transcription

1 What is a eader of opinion formation in bounded confidence modes? E. Kurmyshev a, b * and H. A. Juárez a a Departamento de Ciencias Exactas y Tecnoogía, Centro Universitario de os Lagos, Universidad de Guadaajara. Lagos de Moreno, México b Centro de Investigaciones en Óptica A. C. León, México. *Correponding author E-mai addresses: ekurmyshev@cuagos.udg.mx, kev@cio.mx (E. Kurmyshev), hjuarez@cuagos.udg.mx (H.A. Juárez) Abstract Taking a decision in democratic socia groups (societies) is based on the opinion of the majority or on the consensus, if it is the case. Personaities and opinions of individuas constituting a socia group in different stages of its deveopment are used to be heterogeneous. The difference in opinions is indeed a motive to changes in a socia group. So, the study of opinion dynamics is of great interest in anayzing socia phenomena. Among the different modes of opinion dynamics, bounded confidence modes have been studied in different contexts and shown interesting dynamics, particuary in custering, poarization and fragmentation of opinions, and infuence of extremists [1-3]. In [4] we proposed a new bounded confidence mode and studied the sef-formation of opinion in heterogeneous societies composed by agents of two psychoogica types, concord (C-) and partia antagonism (PA-) agents. In this work we study the infuence of eaders on the custering of opinions in sma word (SW) networks, starting the opinion dynamics from the uniform initia distribution of opinions in the network. Mixed C/PA-societies aong with the pure C- and PA-society are studied. The infuence of the eader's connectivity in the network, his toughness or toerance and his opinion on the opinion dynamics is studied as a function of the initia opinion uncertainty (toerance) of the popuation. Numerica resuts obtained with eaders at ow, high and average toerance show compex bifurcation patterns of the group opinion; a decrease or even the tota ost of contro of the eader over the society is observed in different intervas of toerance of agents in the case of C/PA-societies. We found that in the C-society a eader showing high opinion toerance has more contro over the popuation. In the PA-society a eader changes the bifurcation pattern of group opinion in a drastic and unexpected way, contrary to the common sense, and generates stronger poarization in the opposite opinion groups; the connectivity of the eader is an important factor that usuay improves the adhesion of agents to the eader's opinion. A ow toerance (authoritarian) eader has greater contro over a PA-society than that of a high toerance (democratic) one; the opposite resut is obtained in the C- society. Keywords: Leadership, opinion formation, agent-based simuation, socia power, bounded confidence.

2 1. Introduction One of the major efforts in the fied of compex networks is directed to the studying of socia phenomena on networks, where the individuas (agents) are the nodes of a network and the reations between them are the inks. Most of the socia phenomena can be understood or interpreted through the dynamics of opinion formation. Among them the search for agreements and the reaching of consensus are important aspects of socia group dynamics that finay enhance its impact on the society. A socia group is used to have a eader (or eaders) which infuence the socia ife, in particuar through the opinion formation. In this work we mode and study a eader s infuence over the opinion formation in heterogeneous socia groups. The eadership can be defined in a variety of manners, but most of the definitions consider the eadership as the abiity of a person to convince the others to reach certain objectives [5, 6]. In order to convince persons a eader executes his power using the foowing toos basicay [6-9]. Compensation is done when a eader disposes and provides psychoogica and/or economica stimuus in exchange for the oyaty to the idea. Coercive power is based on fear, and is executed through bruta force, threats and punishments. Lega power is associated with the roe of a chief by his forma position. Natura or charismatic power is based on the admiration of a eader by the foowers. Finay, a eader can be an expert in a particuar area or topic, which has a mayor capacity than others to anayze, contro and execute the tasks. Each eader has its persona stye of guiding a group. In genera, the stye depends on the personaity of the eader, even though this can be modified by personaities of the subordinates or foowers, the nature of the task and/or the structure of a company. A eader possesses a persona socia-psychoogica stye of executing a power; it can be authoritarian, paternaistic, advisory or democratic [6-9]. In addition, socioogists used to distinguish among eaders of socia groups or organizations, eaders of socia networks and mass media, even though there is much in common among those. Any mode of socia behavior and/or opinion formation is imited, so it is hardy possibe to bring together a features mentioned above and study the eaders infuence in a mathematica mode. How and which of the above mentioned characteristics of a eader can be incorporated in a mode of opinion dynamics? Under what conditions can an agent, possessing some eader s quaities in terms of the variabes and parameters of the mode, act as a eader? Among the works on the socia groups and networks there is a reativey sma number of works on the mathematica modeing of a eader infuence on the opinion formation. The socia-psychoogica aspect of a eadership is used to be studied in socia sciences and those are descriptive mainy. They propose or define the attributes of a eader and describe his/her behavior in the society [6-12]. The authors of [10] outine a method to acceerate the dissemination of innovations using opinion eaders, present optima procedure and report on computer simuations that show how much faster diffusion occurs when initiated by opinion eaders. The topic of diffusion of opinions in networks is aso discussed in [11]. Severa attributes reated to opinion eaders are reported in [12] after the revision of the concept of opinion eadership. On the other hand, the mathematica modeing of opinion formation is based mainy on the socia physics modes that take into account dynamica nature of socia interactions and different roes of individuas [2, 13-15]. Those are physica modes adapted for the socia phenomena, in particuar, the opinion formation. Being imited for the description of opinion formation, they have reduced capacity for incorporating eaders of different types in the mode [16-22]. Another type of mathematica modes are needed, they

3 shoud arise from the socia and psychoogica behavior of individuas. Mathematica variabes and parameters of the mode shoud describe the concepts and notions of socia science with high fideity from the very beginning of the modeing. Combining socio-psychoogica approaches with the quantitative mathematica toos and statistica physics can hep to achieve reiabe insights. Both approaches are important to improve the predictive capabiity of the opinion formation modes. To our opinion, the bounded confidence, reative agreement modes (BC/RA) of opinion formation [1-4] are cose to the ast requirement. Sef-organization of opinion not infuenced by eaders was studied extensivey in the frame of C/PA mode [4], giving interesting bifurcation patterns in opinion space as a function of opinion toerance of agents. This agent-based mode aows simuating the opinion formation in a mixed society constituted of agents of the two psychoogica types, concord and partia antagonism agents. Modes of opinion dynamics are usuay composed of the foowing basic eements: a) Opinion Space, either discrete or continuous; b) Updating Rue describes the change of opinion of a given agent in dependence on opinions of agents reated to him; c) Updating Dynamics characterizes the interaction between agents, that can be either pair or group interaction and d) Socia Network describes reationships between agents (socia organization). Each and every of these eements contribute different way to the opinion formation in a society. Since the eadership is an intrinsic characteristic of a socia group, the infuence of a eader on the opinion evoution and formation is an important thing to study. In [4] was found that the opinion formation to be simiar in the networks of the foowing three types: Erdös Rényi random graphs, Watts-Strogatz sma-word networks and compete graphs. In this work we imited our study to the sma word networks ony. The eadership is the abiity of an individua to convince the others to reach certain objectives. Leader of opinion has to infuence the opinion of subordinates-foowers in a way that they adopt the vision of the eader; he/she has to be firm. Because a eader maintains a course in order to approach the goa, we consider as a eader an agent which maintains an invariabe opinion in the process of socia opinion formation. Another distinguishing feature that heps a eader to execute the power is its reativey high connectivity into the socia network. In addition, we contro the eader s toerance to simuate authoritarian, paternaistic, advisory or democratic eader. The socia power of a eader is modeed partiay via the connectivity of the eader in the network and through the reative agreement of the updating rue. In order to see separatey different effects of the eader s infuence on the opinion formation, in this work we consider a unique eader of the socia group. It shoud be noted that a mass media can be treated somehow as a eader of opinion by itsef, or as a too of a eader. This paper is organized as foows. In Section 1 we give a brief quaitative description of a eader and trends in modeing of opinion dynamics in socia groups with eaders. A mode of opinion dynamics in a mixed C/PA society is presented in Section 2. In Section 3 we incorporate a eader into the C/PA mode of opinion dynamics, using variabes and parameters of the mode. Section 4 presents resuts of computer simuation. Finay, concusions and discussion is given in Section C/PA mode of opinion dynamics Socia groups are constituted of persons of different psychoogica types [14, 15]. In order to capture this important feature of socia organization, in [4] was proposed a mode of opinion in a mixed C/PA society which consists of agents of two psychoogica types, partia antagonism (PA-) and concord (C-) agents. The mode uses

4 interva S = [ 1, 1] as a continuous opinion space. In addition, each agent i is characterized by his opinion uncertainty or toerance u i (0, 1]. An opinion segment s i = [x i u i, x i + u i ] is assigned to each agent i. For two agents, i and j, their opinion segments overap if and ony if h ij = min(x i + u i, x j + u j ) max(x i u i, x j u j ) > 0, where h ij is caed an opinion overap. In the C/PA-mode the opinion and uncertainty of passive (receptive) agent j are changed when 0 h ij. Agents of the two types differs each other in the way they update their opinion in pair interaction. The updating rue for the opinion and uncertainty of a passive C-agent j is as foows: x u h ( x x ) hij + µ ( u i u j ) u 2 i (2) ij j : = x j + µ 1 i j 2u (1) i j : = u j 2 where the active agent can be either a C- or PA-agent. Note that the interaction of C- agents is aways attractive simiar to that of [3], that gives the name to concord agents. Agents with partia antagonism represent the other psychoogica type. Since rea ife interaction between persons is usuay repusive-attractive, we define the updating factor of this type of agents in such a way that opinions of two interacting agents can diverge. The new updating rue for the opinion and uncertainty of a passive PA-agent j is as foows: x u h h µ ( x i x j ) (3) hij hij = u j + µ ( u i u j ) u 1 i u 2 i (4) ij ij j : = x j + 1 u 1 2 i ui j : 2 ij u i where active agent i can be C- or PA-type. Note that the reative agreement (h ij / 2u i )(h ij / u i 1) [ 0.125, 1], thus the opinions of PA-agents can diverge from that of active agents when ( h / 1) 0. The C/PA society of N agents consists of two subpopuations, M C and M PA of pn and (1 p)n size respectivey, where a fraction of C- agents, 0 p 1 is a parameter of the mode. When an active agent i interacts with a passive one j, agent j changes its opinion and uncertainty foowing the updating rues for the C-agents, Eqs. (1) and (2), if j M C or those for the PA-agents, Eqs. (3) and (4), if j M PA. Varying the vaue of p from 0 to 1, we mode mixed C/PA-societies with popuations ranging through the pure C- to the PA-society. The behaviora patterns of the two types (C- and PA-) proposed and expoited in our work are typica for socia groups but each of them has no unique socio-psychoogica mechanism; other patterns can vary around in detais and differ in mathematica representation depending on the variabes and parameters of the mode. The every day experience suggests that the simpest way to see psychoogica reaction of an individua is to refuse or accept the opinion of the opponent. When the opinion of the opponent is accepted, the most probabe resut is that the opinions getting coser. Nevertheess, in order to accept an opinion of the other, an individua have to have an opinion cose enough to that of the opponent and, in addition, to be confident in the opponent psychoogicay. The atter is a psychoogica factor that, to certain extent, shoud to be measured independenty of the opinion. If an individua has no sufficient

5 confidence in the opponent, he wi most probaby refuse the opponent s opinion; if the unidirectiona confidence is sufficienty high, the opinions most probaby get coser. This is a tiny moment not simpe to express mathematicay. Being imited in the variabes, we were forced to express this mechanism mathematicay in terms of opinion and uncertainty (opinion toerance) ony, as it is described in Eqs. (3) and (4). It is difficut to encounter the situation when individuas having very cose opinions and confidence each in other wi reject the opinion of opponent. Because the refuse of opponent s opinion can hardy be tota, we consider the situation of partia antagonism. In our mode we treat competey confident individuas as a specia group of concord agents. No socio-psychoogica study has been done to justify the specific mathematica reation (PA) used in these works. These are mosty heuristic reations. 3. Mode eader in the mixed C/PA-society We consider eadership as the abiity of a person to guide individuas (agents) to a goa. So, a eader needs to be abe to communicate ideas to agents and convince them [5, 6]. To communicate impies that the eader can interact with a reativey arge portion of popuation, directy or impicity. To convince impies having some kind of power of opinion manipuation. This power can be either natura (charismatic eaders), given by some degree of reputation or experience, or impemented by some means such as a reward or even coercion. Another important characteristic of the eader is his abiity to take into consideration other s opinion. From this point of view, a eader can be democratic or autocratic (dictator). Since any opinion mode is imited, we represent these features in the foowing way. Our eader is an agent connected with a significant part of popuation; his degree of connectivity with other agents of the society, denoted by d runs through 0.2 to 1. Because a eader has a goa (an objective) and strong conviction to reach it, we assign an invariabe opinion to the eader, denoted hereafter by x. This can be viewed as a straightforward strategy of the eader. The democracy eve of the eader is represented by his uncertainty or toerance, u. We suppose that a dictator (authoritarian eader) has invariabe ow toerance u = 0.2, whereas a democratic eader has invariabe, reativey high toerance u = 0.5. We aso expect that the eader s toerance can be equa to that of the rest of popuation, u = U, where U is the average of initia toerance of popuation; in that case the eader is caed popuist. Socia power is rather compicated socio-psychoogica notion. In this mode it is reguated by the two magnitudes. First, it is the degree of connectivity of a eader in the network that is a refection of his communicabiity and his socia power. Second, the reative agreement between agents i and j reguates the magnitude of changing in the opinion of passive agent j. Therefore, it represents the socia power of an active agent i, and, in particuar, of a eader that is considered as an active agent aways. For the passive agent j of the C-type, the reative agreement is equa to a = h / u, Eq. (1). When the passive agent j is of the PA-type, the reative agreement is equa to a = h / u 1) h / 2u = ( a 1) a / 2, Eq. (3). PA ( j j C C In practice, every eader usuay uses a combination of powers, but not the ony one. Different combinations of connectivity and toerance can be considered as a combined, integra representation of eader s powers. For exampe, in a professiona group, probaby the most important of the eader s powers is the professiona reputation, but charisma and reward can not be excuded and certainy are present; they enhance the reputation power. In this mode, the reputation can be considered as a combination of a C ij i

6 reativey high toerance and connectivity. Charisma is a psychoogica quaity, and in the frame of this mode it can be partiay attributed to a reativey high toerance; the atter is a feature of a popuist aso. As for the connectivity, it can be considered as a part of a strategy that can be used by a eader to reach a desired resut. When a eader is a dictator (coercive power, perhaps enhanced with a reputation and rewards), he used to have a ow toerance (no charisma) and fixed opinion; and, most probaby, he wi need high connectivity and/or hierarchica structure of the society to contro the opinion or, at east, the opinion expression. Note that the way we conceive and impement the eader can be used to mode different situations where a kind of eadership is necessary, for exampe, eaders of socia groups or organizations or eaders of socia networks. In particuar, adjusting parameters attributed to the eader we can simuate the effect of mass media in the opinion dynamics of popuation. 4. Resuts of computer simuation In this work we are mainy interested in the asymptotic, stationary distribution of opinion in the society. This impies that a eader has quite enough time to work with the agents and convince them if it is the case. Since we are interested in the fina state of the opinion, we do not study opinion convergence in time, eaving this important detai for future work. In order to appreciate changes in the opinion formation caused by the presence of a eader in a socia network, we studied numericay the patterns of opinion distribution on the Sma Word (SW) networks of PA- and C-agents, in function of the parameter U ranging through 0.3 to 1.2, where U is the average of initia toerance of popuation. Resuts for U 1 has theoretica but not practica sense, because it is hardy expect that amost a popuation has so big opinion uncertainty in the opinion space x i [ 1,1]. Leader s attributes, his connectivity d to the network, toerance (uncertainty of eader s opinion) u and opinion x were varied in a wide range of vaues. Note that the opinion x is kept invariabe during the evoution of the society opinion. We wonder to know, how does a mode eader change the bifurcation pattern of opinion compared to that in the society without eader? In order to simuate the infuence of a eader on the opinion formation in the mixed C/PA-mode we use the foowing methodoogy. The society of individuas is a sma word network with N = 1000 agents. The state variabes of the society are the opinion x i and the uncertainty u i of individuas (agents), i = 1,2, N. Uniform probabiity distribution is used to choose the initia opinion of agents from the interva [-1, 1] and uncertainty from the interva [U 0.2, U + 0.2], where U is the average initia opinion uncertainty of the popuation. We add to the network a eader with the opinion x and uncertainty u by connecting him randomy to d N agents, where d is the connectivity of the eader in the network. At each time step, we choose randomy N edges (pairs of couped agents) of the network. If one of these agents is the eader, then his partner is chosen to be the passive agent aways; this guarantees that the eader does not change his opinion and uncertainty. If neither of the pair of agents is a eader, then one of them is seected at random to be the active agent i, whie the other is considered to be the passive one, j. In our study we suppose that, in the process of opinion evoution in time, the dictator has invariabe ow toerance u = 0.2, whereas a democratic eader has invariabe, reativey high toerance u = 0.5 through a the range of the average initia opinion uncertainty U. As for the

7 popuist eader we expect that his/her toerance is simiar to the rest of popuation, u = U. The vaue of the opinion overap h ij is computed, and we update variabes of the passive agent if and ony if h ij 0. As mentioned above, the composition of heterogeneous society is reguated with the parameter 0 p 1. When an active agent i interacts with a passive one j, agent j changes its attitude according to the C-type rues if j M C, or the PA-type rues if j M PA. The vaue of the parameter U runs through 0.3 to 1.2 at the step of At each vaue of U, we execute 350 iterations (time steps). Because we are basicay interested in the steady state of opinion dynamics, we pot the density of asymptotic opinion distribution as a function of parameter U. The resuts shown in the foowing figures represent an average over 50 simuations for each vaue of U. Figure 1 shows bifurcation patterns for the societies without eaders as the reference; those are cacuated just under the same conditions as for the societies with eaders, presented in this work. This comparison is instructive for the studying of natura competition between the sef-organization of the society and the infuence of a eader on the opinion formation. a) b) c) Figure 1. Bifurcation patterns of opinion in the C/PA-societies without eaders on the SW networks: a) C- society (p = 1); b) Mixed C/PA-society (p = 0.5) and c) PA-society (p = 0.0) Our experiments pretend to answer the question: Under what conditions does an agent act as a eader of opinion? That is, when he attracts opinions of the majority of individuas? To answer the question, in computer simuations we varied the foowing parameters of the mode. For each C/PA composition of the society running through 0/100, 20/80, 40/60, 60/40, 80/20 and 100/0 (p = 0, 0.2, 0.4, 0.6, 0.8 and 1), the eader s parameters were given the foowing vaues: opinion x = 0, 0.2, 0.4, 0.6 and 0.8; his toerance u = 0.2, 0.5 and U; his degree d = 0.2, 0.4, 0.6, 0.8 and 1. Bifurcation patterns and pots of tempora evoution of opinion demonstrated a variety of interesting detais and tendencies; those are hardy possibe to comment in detai in one artice. So, in order to resume most of the resuts in this artice we seected ony few of them, those being most representative cases in our opinion. Behavior of concord agents is rather simpe and predictabe. In Figure 1 we see that the consensus centered at the opinion x = 0 prevais in the C-society without eader, for the uniform initia distribution of opinion at vaues U 0. 4, athough the sma fragmentation of opinion is observed at vaues U ess than 0.4. As expected, we found that the centrist eader, x = 0, of any type ony enhances the convergence to the unique opinion x = 0. The consensus is aways observed in the C-society, for the democratic (u = 0.5) and the popuist (u = U) eader; two sma groups of separatists with opinions x =

8 0.5 and x = -05 are observed at U cose to 0.3 with a dictator (u = 0.2) at a vaues of his degree. So, the dominant opinion distribution in the C-society is the consensus, x = 0 in the presence of a centrist eader of any type, having the degree d = 0.2, 0.4, 0.6, 0.8 and 1; for that reason we omit the corresponding pots. The patterns of asymptotic opinion distribution in the PA-society strongy differ to that of the C-society, with and without eaders (for exampe, compare the bifurcation patterns of Figure 1a and 1c). Computer simuations on the opinion formation in the PA-society with the centrist eader show that the asymptotic opinion distribution over the network is simiar, except sma quantitative detais, in the society guided by the democratic eader, u = 0.5 and by the popuist, u = U. For that reason we present opinion distribution ony for the democratic eader, see Figure 2, ine A. In both cases we observe strong generation of opposed extremists and incapacity of a eader to dominate the opinion of sef-organized reguar agents, in the interva 0.5 < U < 1.0 (see Figure 1c aso). These effects are modified quantitativey (but not quaitativey) with the degree of a eader (see Figure 2, ine A). Nevertheess, a ow toerance eader (dictator) used to dominate the society opinion at a reativey high connectivity, d > 0.6; that is not the case at d = 0.2 when the dictator has itte contro in the interva 0.3 < U < 0.6 and ooses it in the interva of 0.6 < U < 1.0 (see Figure 2, ine B). A) B) (a) (b) (c) Figure 2. Bifurcation patterns of opinion in the PA-society, on the SW networks, guided by a centrist democratic eader, u = 0.5, ine A and a dictator, u = 0.2, ine B: coumn (a) d = 0.2; (b) d = 0.6 and (c) d = 1.0 In rea ife it is hardy possibe to meet homogeneous socia groups composed of agents of the same psychoogica type. So, having in mind specific features of the opinion formation in pure C- and PA-society we wonder to see the infuence of a eader on the opinion formation in mixed societies. Even though we have resuts of computer simuation for the fractions of C/PA agents through 0/100, 20/80, 40/60, 60/40, 80/20 and 100/0, we consider appropriate to show and anayze here the pots in the case of C/PA-40/60 that reveas genera tendencies, Figure 3.

9 A) B) C) (a) (b) (c) Figure 3. Bifurcation patterns of opinion in the C/PA-40/60 society, on the SW networks, guided by a centrist democratic eader, u = 0.5, ine A; popuist eader u = U, ine B and a dictator, u = 0.2, ine C: coumn (a) d = 0.2; (b) d = 0.6 and (c) d = 1.0 Asymptotic opinion distribution over the network is simiar, except some quantitative detais, in the society guided by a democratic eader, u = 0.5 and a popuist u = U; see Figure 3, ines A and B. Nevertheess, the ow toerance eader ( dictator ) shows surprisingy different resuts in opinion formation; those are given in Figure 3, ine C. Dictator demonstrates much better performance; he dominates the society opinion for U > 0.4 and does not oose the governabiity in any interva of U in contrast to that of the popuist or democratic eader. Another interesting feature of this opinion dynamics is that centrist eaders of any toerance used to generate extremists. Bifurcation patterns are modified in function of the eader s degree (connectivity). Those genera tendencies are aso observed in the mixed societies at C/PA ratio equa to 20/80, 40/60, 60/40 and 80/20, with notabe variations in quantitative detais through the society composition. A)

10 B) C) (a) (b) (c) Figure 4. Bifurcation patterns of opinion in the C/PA-80/20 society guided by a centrist democratic eader, u = 0.5, ine A; popuist eader u = U, ine B and a dictator, u = 0.2, ine C: coumn (a) d = 0.2; (b) d = 0.6 and (c) d = 1.0 From Figure 4 we see that the presence of even 20% of PA-agents changes significanty both the dynamics and the fina distribution of opinion in the mixed C/PA- 80/20 society compared to that of the pure C-society; in particuar, we observe intervas of U where the centrist eader notaby ooses contro over the opinion of reguar agents. It is a resut of competition between the sef-organization of opinion of agents and the open oop contro of the eader over their opinions. There is aso an initia, symmetric repusion of agents from the eader and then their sef-organization in the opposite groups. Note that Figure 4 has notabe correation with the case of C/PA 40/60, Figure 3. A) B)

11 C) (a) (b) (c) Figure 5. Bifurcation patterns of opinion in the SW network as a function of U, for the eader s opinion x = 0.6 and the degree d = 0.2. Row A, pure C-society (p = 1); row B, mixed C/PA-society (p = 0.4) and row C, pure PA-society (p = 0). Coumns: (a) dictator, u = 0.2; (b) popuist, u = U and (c) democratic eader, u = 0.5. Figure 5 shows, in series of pots, the infuence of a eader, having we defined right orientation, x = 0. 6 and a reativey ow degree of connectivity, d = 0. 2 on the opinion formation in the C-, PA- and mixed C/PA-40/60-society; eaders of the three types are considered. Comparing figures in coumns (a) and (c) in Figure 5, we see that in genera a democratic eader and a dictator, with reativey ow degree-connectivity (0.2), get simiar resuts; those differ from that of the popuist (compare coumns (a) and (c) versus (b)). In the C-society, the popuist, u = U and democratic eader, u = 0. 5 with reativey ow degree (of connectivity), d = 0.2 have amost the same contro over the fina opinion of reguar agents; they ceary dominate agents opinions if the initia toerance (uncertainty) of the popuation U 0. 4, the distribution of opinion having the bifurcation and fragmentation of opinion at vaues U < The dictator, u = 0. 2, aso gets contro over the opinion of agents, but in the societies with the higher toerance, U 0.5. For the ess vaues of U, the group opinion suffers notabe fragmentation and the dictator does not get contro over it. In the C/PA-40/60-society (Figure 5, ine B), a dictator and a democratic eader get simiar contro over the agents opinions, and they are quite efficient in that issue for the societies with agents whose toerance U But the popuist is ess efficient and he gets ony partia contro over the opinion of popuation in some intervas of the initia toerance of popuation, U. Leader of any type induces generation of the eft extremists, x = 1, in the interva 0.6 U 1. In case of the PA-society, the patterns of opinion bifurcation ook more compicated, Figure 5, ine C. The contro of agents opinion by a dictator and a democratic eader is simiar, except quantitative detais; in genera, they contro efficienty the opinion of popuation at vaues U As for the popuist, his contro of opinion is itte efficient. In the interva 0.6 U 1. 2, he gets a number of foowers, but at the same time he provokes a generation of extremists, both eft and right orientation (see Figure 5 (C, b)). Extremists opposing eader s opinion even exceed his foowers in number. Bifurcation of opinion is observed at U = As a resut of this bifurcation a arge group of eft opponents, then becoming eft extremists, emerges. In addition, a group of right oriented opponents, becoming right extremists, spits of the group of the eader s foowers. This bifurcation pattern can be interpreted as the competition of the sef-organization of opinions of agents and the open oop contro of a eader over their opinions; a eader with opinion x = 0.6 at U > 0.55 attracts a supporting group of agents, which repuse the opinion of the opposite groups to the eft

12 and right extremes accompanied by the effects of sef-organization of the extreme groups. The pots of coumn (b) in Figure 5 show that the popuist eader succeeds in dominating of pubic opinion in the C-society ony; in mixed C/PA-societies there are intervas of U where the popuist ooses the contro over the pubic opinion, and this effect is more notorious in societies with a big PA-component of popuation. It shoud be noted that this tendency is observed for different vaues of p and connectivity of the eader. (a) (b) (c) Figure 6. Bifurcation patterns of opinion distribution as a function of U in the mixed C/PA-40/60 society for x = 0.6 and high degree of a eader d = 0.8: (a) dictator, u = 0.2; (b) popuist, u = U and (c) democratic eader, u = 0.5. In order to show the infuence a eader with a reativey high connectivity, in Figure 6 we present the bifurcation patterns of opinion in the mixed C/PA-40/60 society for x = 0.6 and d = 0.8 (high connectivity). Even though the dictator used to attract the opinion of a numerous group of agents for the society with the toerance in the interva 0.3 U 1.2, his contro becomes efficient for 0.8 U ony (see Figure 6a). The popuist is an efficient eader for the societies with a modest toerance, 0.45 U 0. 85, but then he ooses a contro (see Figure 6b). From Figure 6c we see that the democratic eader joins the power of both, the popuist and dictator, getting an efficient contro over the society in the interva 0.3 U Democratic eader (u = 0.5) becomes a kind of a dictator for the agents with U > 0.5. For that reason his behavior is simiar to that of the dictator for U > 0.5 (see Fig. 6a). In this situation the democratic eader contros opinions of agents; there is no gap, as it is observed in the case of the popuist. Groups of eft extremists and dissidents are observed in Figure 6. These tendencies where observed in different mixed societies with highy connected eaders. Comparing of Figure 6 with Figure 5, row B shows the importance of eader s degree for its performance. Note that eaders with high degree of connectivity in the networks can be interpreted as mass media to some extent. (a) (b) (c)

13 Figure 7. Bifurcation patterns of opinion distribution as a function of U in a pure PA-society for x = 0.2 and d = 0.2: (a) dictator, u = 0.2; (b) popuist, u = U and (c) democratic eader, u = 0.5. In order to see more detais and anayze the infuence of a eader with a modest right or eft orientation, that is the intermediate case between the centrist and extreme oriented eader, we present in Figure 7 the series of graphics for the pure PA-society and eader s opinion x = 0.2 and d = 0.2; the atter means that the eader is amost centrist and has reativey ow degree of connectivity. In this situation a dictator succeeds to get a contro over the society, even though sma groups of dissidents are observed for U > 0.4, Figure 7a. However, both the popuist and democratic eader ooses contro over the society in the interva 0.3 U 1. 1, Figure 7b, c. Moreover, there are two groups of dissidents positioned in opposite sides of opinion space. The opinion distribution is not symmetric and the group of eft dissidents and extremists is aways arger than that of the right oriented agents (note that the eader is right oriented modesty). Bifurcation patterns of Figure 7 show that the characteristics attributed to the mode eader are not aways sufficient to guarantee a contro over the society. 5. Concusions and discussion In this work we have proposed the mode and studied the competition between the seforganization of opinion of agents in the network and the infuence of a eader over the pubic opinion. We have introduced mode eaders of the authoritarian (dictator), democratic and popuist types in the heterogeneous C/PA mode of opinion dynamics. A eader of the network opinion is an agent to which are assigned an invariabe opinion and uncertainty (toerance), and a reativey high degree of connectivity in the network. In this work we have shown, those are necessary but not sufficient conditions for an agent to be a eader. To be an efficient eader one needs to posses some more quaities than high degree of connectivity d and invariabe opinion x and toerance u ; mode eader, at east, has to assess the psychoogica type of participants (C/PA-agents) of the group and their toerance U. In some respects the hubs coud be thought as mode eaders since they are high degree nodes in scae-free networks. Nevertheess, it is not intrinsic feature of an agent, but it is a topoogica property of the network that refects a socia position of the hub to some extent. In addition, hubs are numerous formations in the scae-free networks. The question is who is a eader? In this sense, any pair of hubs considered as eaders can have identica, simiar or opposite opinions; these situations wi produce different resuts. A of these quaities of hubs compicate, if not confuse the understanding of the eader s infuence upon the opinion formation in a socia group (network). In order to better the understanding of the mechanism of a eadership, in this work we have considered a unique eader in homogeneous networks. Sma word networks were used to mode the opinion dynamics of a society. We have found that the opinion formation in a society is sensitive to both the psychoogica type of agents (C- and PA-agents) and the type of a eader. In our experiments we observed the foowing tendencies. Individua behavior of C-agents is rather simpe, Eq. (1) and (2). Nevertheess, even in the pure C-society a contro of pubic opinion by a eader is quite compicated. The opinion distribution in the C-society, in the presence of the dictator with the opinion in the interva x [ 0.4, 04], is the consensus centered at the eader s opinion x, for the

14 eader s degrees of connectivity d = 0.2, 0.4, 0.6, 0.8, 1 and U However, when a dictator has more extremist position, x = ±0.6, ± 0. 8, he amost ooses contro over the pubic opinion for 0.3 U 0. 6, getting a sma number of foowers at high degrees d = 0.8, 1 ony. For the societies with a high uncertainty (toerance), U > 0. 6, even a dictator with a radica opinion used to estabish contro and dominate pubic opinion. In the C-society the popuist and democratic eader in genera have better performance than that of a dictator in a range of parameters, the democratic eader being the best. In genera, mode eader of any type has more probems with the contro of opinion in the C/PA-societies with a notabe portion of PA-agents. The popuation with at most 20% of PA-agents used to suffer mutipe opinion fragmentations. There are intervas of opinion toerance U, especiay for societies with a notabe PA-component, when the mode eader ooses contro over the society (see Figures 2, 3, 4, 5, 7). In the case with the popuist or democratic eader, we observe strong generation of opposed extremists and incapacity of a eader to convince reguar agents, in the interva 0.5 < U < 1.0 (see Figures 2A, 3A and 3B, 4A and 4B, 5B and 5C). In the societies with a notabe portion of PA-agents the overa performance of the dictator in guiding the pubic opinion is better than that of the popuist or democratic eader; that is in contrast with the C-society. In genera, the attraction of a eader increases in function of the eader s degree of connectivity with the network but it is not a inear effect. With the goba study of asymptotic trends in opinion evoution it is possibe to detect interesting or unusua situations in opinion formation under the eaders infuence. When an interesting goba situation is observed, then it is just the case to study the scenery in more detais, incuding time evoution of opinion of individuas and their positioning or connectivity in the network. Resuts of our modeing and simuation suggest that, besides the features mentioned above, eaders need more sophisticated strategies to maintain the contro over societies characterized by a high degree of mistrust and antagonism. We think that some of the resuts can be appied aso to the effect of mass media onto the dynamics of pubic opinion formation. Competition of eaders, eader s strategies, different kinds of the socia power of eaders, and the network topoogy are expected to affect the opinion formation; those are natura extensions of this work for the future. We expect that the C/PA-mode with eaders heps to expain some tendencies in opinion formation in heterogeneous societies composed of agents simiar to those of the C- and PA-type. Acknowedgments Authors are gratefu to Dr. Héctor Caderón Gama for hepfu discussions and suggestions. Appendix Time evoution of opinions is occurred in a variety of ways, depending on the C/PAsociety composition, eader s opinion x and toerance u, eaders connectivity d and the average opinion uncertainty of agents U. So many parameters suggest the topic of the time convergence of opinions to be studied mathematicay rigorousy or by means of practica computer runs for particuar sceneries described by a specific set of vaues of mode parameters. Since the aim of this work is to see the goba statistica trends in opinion formation in societies composed of agents of different psychoogica type, we

15 did not use any forma mathematica criterion to contro the time convergence of opinions. Based on the previous experience and some probe runs, we chose for this work an intermediate way to contro the convergence. In each experiment we have fixed the vaues of the foowing parameters: C/PA-society composition, eader s opinion and toerance x and u, eaders connectivity d and the average opinion uncertainty of agents U; besides, we generate new network (of the same type and same number of agents), new uniform distribution of opinions and uncertainties of agents. We et the system evove for 500 time steps. Each time step invoves N pair interactions. Then fina opinion distributions of 50 experiments are averaged to give the trend of opinion formation (one point in a pot of Figures 1 to 7). We verified visuay the convergence of the opinion formation in many samped (not in a) runs, and observed that 500 time steps is reasonabe evoution time to see the asymptotic opinion distribution. Moreover, if the experiments were not stabe asymptoticay or if there was no convergence in opinion formation, then the opinion distribution averaged over 50 experiments woud give amost uniform distribution, and that is not the case. Three figures iustrate some ways of convergence, but we observed many more interesting detais in other sceneries that worth to be studied and commented rigorousy in a separate work. Figure A. Time evoution of opinions in a pure C-society: PA000/C100, x = 0.6, u = 0.2, d = 0.6 and U = 0.6. Figure B. Time evoution of opinions in a pure PA-society: PA100/C000, x = 0, u = 0.5, d = 0.6 and U = 0.4.

16 Figure C. Time evoution of opinions in a pure mixed C/PA-society: PA60/C40, x = 0, u = 0.2, d = 0.2 and U = 0.6. Group opinion convergence in time was observed in a cases of samped evoution pots, even though interesting detais are observed at eary stages of opinion evoution, especiay. Time convergence of opinions in presence of a popuist (u = U) and democratic eader (u =.50) is simiar. In any case, 500 time steps show the trends of opinion formation ceary. Some of the observed features of opinion evoution are as foows. C-society (Figure A): Two stages of group opinion evoution are ceary observed. First stage is a reativey rapid sef-organization of opinions in a compact custer (20 to 50 time steps, depends on d and U), and then the custer of opinions tends to the opinion of the eader, if it is the case. PA-society (Figure B): Opinion groups are usuay formed at 50 to 120 time steps. Mixed PA60/C40-society (Figure C): Sef-organization of opinions of agents and adhesion of opinions of other agents to the opinion of eader are observed simutaneousy. The arger U, the ess is the convergence time. The arger d, the sower is the convergence in sef-organized groups of opinion that do not share the opinion of a eader. Opinion convergence in a mixed C/PA-society is usuay sower than in the case of pure C- or PA-societies at the same vaues of parameters. 6. References [1] J. Lorenz, Continuous Opinion Dynamics under bounded confidence: A Survey, Int. J. Mod. Phys. C, 18 (12) (2007) [2] C. Casteano, S. Fortunato, V. Loreto, Statistica physics of socia dynamics, Rev. Mod. Phys. 81 (2) (2009) [3] G. Deffuant, Comparing extremism propagation patterns in continuous opinion modes, J. Artif. Soc. Soc. Simuat. 9 (3) (2006) URL: [4] E. Kurmyshev, H.A. Juárez, R.A. Gonzáez-Siva, Dynamics of bounded confidence opinion in heterogeneous socia networks: Concord against partia antagonism, Physica A, 390 (16) (2011)

17 [5] J.A.F. Stoner, C. Wanke, Management, 3rd edition, Prentice-Ha (Engewood Ciffs, N.J.), [6] B.M. Bass and R. Bass, The Bass handbook of eadership: theory, research, and manageria appications, 4th ed, Free Press, [7] J.R.P. French, The forma theory of socia power, Psychoogica Review, 63 (1956) [8] S. Atman, E. Vaenzi, R.M. Hodgetts, Organizationa behavior: Theory and practice. Academic Press. (1985) [9] N.E. Friedkin, A forma theory of socia power, J. Math. Socio. 12 (1986) [10] T.W. Vaente, R.L. Davis, Acceerating the diffusion of innovations using opinion eaders, Annas AAPSS, 566 (1999) [11] D.J. Watts, P.S. Dodds, Infuentias, networks, and pubic opinion formation, J. Consum. Res. 34 (2007) [12] G. Weimann, D.H. Tustin, D. van Vuuren, J.P.R. Joubert, Looking for opinion eaders: traditiona vs. modern measures in traditiona societies, Int. J. Pubic. Opin. Res. 19 (2) (2007) [13] P. Sobkowicz, M. Kaschesky, G. Bouchard, Opinion mining in socia media: Modeing, simuating, and forecasting poitica opinions in the web, Government Information Quartery 29 (2012) [14] H.-X. Yang, Z.-X. Wu, C. Zhou, T. Zhou, B.-H. Wang, Effects of socia diversity on the emergence of goba consensus in opinion dynamics, Phys. Rev. E, 80 (2009) [15] H.-X. Yang, B.-H. Wang, Effects of socia diversity on the evoutionary game and opinion dynamics, Physics Procedia, 3 (2010) [16] K. Kacperski, J.A. Hoyst, Opinion formation mode with strong eader and externa impact: a mean fied approach, Physica A 269 (1999) [17] P. Sobkowicz, Effect of eader's strategy on opinion formation in networked societies with oca interactions, Int. J. Mod. Phys. C, 21 (839) (2010) [18] M. He, H. Xu, Q. Sun, Opinion evoution on a scae-free network with eaders, Int. J. Mod. Phys. C 15 (7) (2004) [19] D. Ramirez-Cano, J. Pitt, Foow the Leader: Profiing Agents in an Opinion Formation Mode of Dynamic Confidence and Individua Mind-sets, Proceedings of the IEEE/WIC/ACM Internationa Conference on Inteigent Agent Technoogy (IAT'06)

18 [20] B. During, P. Markowich, J.-F. Pietschmann, M.-T. Wofram, Botzmann and Fokker Panck equations modeing opinion formation in the presence of strong eaders, Proc. R. Soc. A, 465 (2009) [21] Mahdi Jaii, Effects of eaders and socia power on opinion formation in compex networks, Simuation: Transaction of the Society for the Modeing and Simuation Internationa, In press, doi: / [22] A.H. Rodriguez, Y. Moreno, Effects of Mass Media action on the Axerod Mode with Socia Infuence, Phys. Rev. E 82 (2010)