Analytic Study of Opinion Dynamics in Multi-Agent Systems with Two Classes of Agents

Abstract

-This paper describes a model for opinion dynamics in multi-agent systems composed of two classes of agents. Each class is characterized by distinctive values of the parameters that govern opinion dynamics. The proposed model is inspired by kinetic theory of gases, according to which macroscopic properties of gases are described starting from microscopic interactions among molecules. By interpreting agents as molecules of gases, and their interactions as collisions among molecules, the equations that govern kinetic theory can be reinterpreted to model opinion dynamics in multi-agent systems. A key feature of the adopted kinetic-based approach is that it allows macroscopic properties of the system to be derived analytically. In order to take into account that the considered multi-agent system is composed of two classes of agents, kinetic theory of gas mixtures, which deals with gases composed of different kinds of molecules, is adopted. Presented results show that consensus is reached after a sufficiently large number of interactions, which depends on the parameters associated with the two classes of agents.

INTRODUCTION

Opinion dynamics and consensus formation are wellknown problems that deal with the identification of interaction rules which lead to proper distribution of opinion in multiagent systems [1]. Such problems are important topics of the research on multi-agent systems and distributed computing and they have applications in many areas, such as control theory, physics, biology, and sociology (e.g., [2]). Various approaches have been proposed in the literature to describe opinion dynamics and consensus formation, among which we can recall those based on thermodynamics (e.g., [3]), on Bayesian networks (e.g., [4]), and on gossip-based algorithms (e.g., [4]). The use of cellular automata to model consensus formation has also been investigated; in this case, opinion is modeled as a discrete variable and consensus is reached through proper transition rules (e.g., [5]). Another important framework which is useful to study opinion dynamics is related to graph theory. (e.g., [6]).

In this paper, we consider a model for opinion dynamics which is inspired by sociophysics, a discipline according to which social interactions and opinion dynamics in multiagent systems can be described using the formalism of the kinetic theory of gases [7]. Kinetic theory of gases aims at analyzing the effects of microscopic collisions among the molecules from a probabilistic point of view in order to derive macroscopic properties of gases by means of a proper balance equation, namely, the Boltzmann equation [8]. According to sociophysics, a parallelism can be done between the molecules of gases and agents in multi-agent systems: collisions among the molecules are reinterpreted as interactions among agents. A major advantage of the use of kinetic-based approaches to model opinion dynamics and consensus is that analytic results can be derived, while, at the opposite, opinion dynamics in multi-agent systems is typically investigated through simulations [9]. It is worth noting that common approaches to the analysis of interactions in multi-agent systems (e.g., [10]) are normally more interested in formalizing complex microscopic interactions rather than in studying the overall emergent behavior of the system.

Standard kinetic theory typically assumes that all the molecules are equal. However, gases are typically composed of molecules of different types and, therefore, a more accurate description of gases can be achieved using kinetic theory of gas mixtures, which takes into account that different species of molecules coexist in the same gas. Using the framework of kinetic theory of gas mixtures, it is then possible to describe multi-agent systems composed of different classes of agents, each of which is associated with different values of the parameters used to model opinion dynamics. The most important features that can be introduced to distinguish a specific type of agents are the propensity to change opinion when interacting with other agents, and the ability to change the opinions of interacting agents. In addition, different classes of agents can have different cardinalities and different initial distribution of opinions. According to this approach, phenomena such as extremism or skepticism can be studied [6]. In this paper, we consider multi-agent systems composed of two classes of agents; however, the proposed approach is general and, potentially, the number of classes of agents can be set equal to the number of agents, thus having one agent for each class.

This paper is organized as follows. In Section II the opinion dynamics problem is formulated using the kinetic framework. In Section III macroscopic properties of the considered multiagent system are derived. Section IV shows some illustrative results concerning the average opinions of specific multi-agent systems. Section V concludes the paper.

II.

KINETIC FORMULATION OF OPINION FORMATION The identification of agents with the molecules of a gas allows applying the framework of kinetic theory to different fields and, in particular, to distributed artificial intelligence and opinion dynamics in multi-agent systems. As in kinetic theory, we assume that each agent can interact with any other agent in the system and that each interaction involves two agents [11].

For this reason, we denote interactions as binary. While the molecules of gases are typically related to their velocities, in the context of opinion dynamics we assume that each agent is associated with a scalar attribute v that denotes its opinion. The opinion of each agent is updated at each interaction, according to proper rules. Various kinds of rules to update the opinion of agents after interactions have been studied to model different characteristics of agents [12], [13].

Let us denote as n(t) the total number of agents at time t and as n1(t) and n2(t) the number of agents of the type 1 and 2, respectively, so that n(t) = n1(t) + n2(t). With no loss of generality, we assume in the rest of this paper that the opinion of each agent is defined in the interval I = [−1, 1], where −1 and 1 represent extremal opinions. The considered model is aimed at describing the temporal evolution of the opinion by studying the effects of pairwise interactions.

A. Interaction Rules

In order to describe the microscopic effects of pairwise interactions, let us define the interaction rules. Assume that an agent of type s with opinion v interacts with another agent of type r with opinion w. The post-interaction opinions of the two interacting agents depend on their pre-interaction opinions according to the following rules v∗ = v − γsr (v − w) w∗ = w − γrs(w − v) where v∗ and w∗ are the opinions of the two agents after the interaction. Observe that the considered model involves 2 4 coefficients { γsr } s,r=1, where γsr measures the propensity of an agent of type s to change its opinion in favor of that of an agent of type r. As a matter of fact, considering, for instance, the first equation of ( 1 ) it is clear that an increment of γsr increases the propensity of agents of type s to change their opinions when interacting with agents of type r. In the following, we assume that the coefficients { γsr} s2,r=1 satisfy 1 0 < γsr < ∀ s, r ∈ { 1, 2} . ( 2 )

2 In agreement with the intended meaning of γsr explained above, according to ( 1 ), if γsr is nearly 0, the individuals of type s are not inclined to change their opinion towards that of agents of type r. For this reason, values of γsr close to 0 characterize skeptical agents. At the opposite, if in the first equation of ( 1 ) we set γsr ≃ 1/2, then v∗ ≃ 1/2(v + w), so that the first agent looses half of its opinion in favour of that of the second, which characterize easily influenced agents.

The sum of the opinions of two interacting agents after the interaction can be derived from ( 1 ) and it is given by v∗ + w∗ = v + w + (γrs − γsr )(v − w).

From ( 3 ), the opinion is not conserved and that it can change depending on the sign of (γrs − γsr )(v − w), namely on the values of the coefficients γrs and γsr and on the values of the pre-interaction opinions v and w. From ( 1 ) it can also be derived that the difference of the opinions of two interacting agents after the interaction is

v∗ − w∗ = εrs(v − w). where εrs = 1−(γrs +γsr ). Since, from ( 2 ), γrs +γsr ∈ ( 0, 1 ), it is easy to conclude that εrs ∈ ( 0, 1 ). Therefore, from ( 1 ) ( 3 ) ( 4 ) ( 4 ) we can conclude that the difference between the postinteraction opinions is smaller than the difference between the pre-interaction opinions of the two agents. Hence, it is reasonable to expect that, after a sufficiently large number of interactions, all agents end up with the same opinion, regardless of their class. Concerning differences of opinions, it can also be concluded that the post-interaction opinion of an agent is closer to its pre-interaction opinion than to the preinteraction opinion of the agent it interacts with. As a matter of fact, from ( 2 ) and ( 1 ), one can derive that | v∗ − v| = γsr| v − w| < (1 − γsr)| v − w| = | v∗ − w| | w∗ − w| = γrs| w − v| < (1 − γrs)| w − v| = | w∗ − v| . ( 5 )

We remark that, according to the model in ( 1 ), the postinteraction opinions v∗ and w∗ still belong to the interval I where the opinions are defined.

B. The Boltzmann Equation

Starting from the interaction rules in ( 1 ), it is possible to study opinion dynamics of multi-agent systems using simulations. Instead, we now show how to obtain analytical results by applying the framework of kinetic theory of gas mixtures to the considered opinion dynamics scenario. For this purpose, we introduce the Boltzmann equation, namely an integro-differential equation that allows deriving macroscopic properties of gases. In the considered scenario, which includes only two classes of agents, two equations need to be considered, whose unknowns are non-negative functions { fs(v, t)} s2=1 which represent the density of the opinion v ∈ I , relative to agents of class s, at time t ≥ 0. The temporal evolution of each distribution function can be described, in spatially homogeneous conditions, as ∂fs (v, t) = Is ∂t s ∈ { 1, 2} where Is is the collisional operator relative to the class s and it is written as

Is =

2 X Qsr(fs, fr) r=1 s ∈ { 1, 2} .

From ( 7 ) it is evident that the collisional operator relative to each class of agents depends on the distribution functions { fs} s2=1 of all species.

In order to obtain analytic results, the explicit expression of the collisional operator is needed. To simplify notation, in the derivation of the explicit expression of the collisional operator we neglect the dependence of the distribution functions { fs} s2=1 on time t, since all involved integrals are related to the opinion variable. Let us denote as

W (v, w, v∗, w∗)dv∗dw∗ the probability that after the binary interaction of two agents with opinion values v and w, the opinions of the two agents become v∗ and w∗, respectively. Hence, the loss of agents of class s in v and, simultaneously, of agents of class r in w can be denoted as

Qsr −(fs, fr) = W (v, w, v∗, w∗)fs(v)fr(w)dvdwdv∗dw∗. ( 6 ) ( 7 ) ( 8 ) Analogously, the gain of agents of class s in v and, simultaneously, of agents of class r in w, is given by Qsr+(fs, fr) = W (v∗, w∗, v, w)fs(v∗)fr(w∗)dv∗dw∗dvdw where v∗ and w∗ are the pre-interaction opinions of agent of class s and r, respectively, which lead to v and w as opinions of the two agents after the interaction [14].

According to kinetic theory of gas mixtures, the collisional operator Qsr relative to classes s and r can be written as [15] Qsr(v) =

W (v∗, w∗, v, w)fs(v∗)fr(w∗)dv∗dw∗dw W (v, w, v∗, w∗)fs(v)fr(w)dwdv∗dw∗ ( 10 ) ( 11 ) ( 12 ) ( 14 ) dt I 1 n d Z dt I =

X2 Z r=1 I4

fs(v, t)vdv

ANALYTIC STUDY OF MACROSCOPIC PROPERTIES From standard kinetic theory, we can describe the temporal evolution of the distribution function fs(v, t) according to the spatially homogeneous Boltzmann equation which, in case of a gas mixture, corresponds to ( 6 ), where the right hand side represents the collisional operator Is relative to the class s. We now show how the Boltzmann equation can be used to derive macroscopic properties of the considered multi-agent system. The number of agents of class s at time t can be expressed as Z

I fs(v, t)dv = ns(t)

s ∈ { 1, 2} . us(t) =

1 Z ns(t) I fs(v, t)vdv s ∈ { 1, 2} .

Observe that the global (i.e., referred to all the agents) average opinion is then defined as the sum of the average opinions of each class weighed by the number of agents of the corresponding class and divided by n, namely u(t) =

(n1(t)u1(t) + n2(t)u2(t)) .

Such definitions are related to two simple test functions φ(v) in ( 14 ). More precisely, setting φ(v) = 1 in ( 14 ) leads to fs(v, t)dv = 0 s ∈ { 1, 2} ( 9 )

Similarly, the average opinion of agents of class s at time t can be defined as ( 16 ) (17) (18) (19) (20) (21) (22) where the 0 on the right hand side is due to the fact that, since φ(v) is a constant function, the difference φ(v∗) − φ(v) inside the integral is 0. Since, from ( 16 ), the integral on the left hand side of (19) represents ns(t), equation (19) can be written as d dt ns(t) = 0 s ∈ { 1, 2} so that the number of individuals of each class is conserved. Observe that equation (20) also implies that d dt d dt n(t) =

(n1(t) + n2(t)) = 0 so that, as expected, that the total number of agents is constant. For these reasons, in the rest of this paper we omit the 2 dependence of n and { ns} s=1 on t. The conservation of the number of agents is a realistic property of the model.

Let us now consider the test function φ(v) = v in order to investigate the temporal evolution of the average opinion. Setting φ(v) = v in ( 14 ) and using (17) we obtain where the two integrals are obtained by integrating Qsr−(fs, fr) and Qsr+(fs, fr) with respect to all the variables except v, and they represents the gain and the loss of agents with opinion in (v, v + dv), respectively.

Let us now consider the weak form of the Boltzmann equation, which is obtained by multiplying ( 6 ) by a test function φ(v), namely a smooth function with compact support, and integrating the result with respect to v [16]. The weak form of the Boltzmann equation is then given by

Z ∂fs φ(v)dv =

I ∂t

X2Z r=1 I

Qsr(fs, fr)φ(v)dv where, according to ( 9 ), the integral in the sum on the right hand side can be written as

W (v∗, w∗, v, w)fs(v∗)fr(w∗)φ(v)dv∗dw∗dvdw

W (v, w, v∗, w∗)fs(v)fr(w)φ(v)dvdwdv∗dw∗

By applying the change of variables

(v∗, w∗, v, w) → (v, w, v∗, w∗) in the first integral in ( 11 ) one obtains that the weak form of the collisional operator Is in ( 11 ) can be written as

W (v, w, v∗, w∗)fs(v)fr(w)(φ(v∗) − φ(v))d4v ( 13 ) where, from now on, d4v denotes the products on the four differentials dvdwdv∗dw∗. By substituting ( 13 ) in ( 10 ) the weak form of the Boltzmann equation for each class s ∈ { 1, 2} can be finally written as

X2 Z r=1 I4 fs(v, t)φ(v)dv =

W (v, w, v∗, w∗)fs(v)· fr(w)(φ(v∗) − φ(v))d4v ∀ s ∈ { 1, 2} where on the left hand side we used the fact that for every test function (see [16])

Z ∂fs φ(v)dv = d Z

I ∂t dt I

W (v, w, v∗, w∗)fs(v)fr(w)(v∗ − v)d4v.

Since, from ( 1 ), the difference v∗ − v can be expressed as −γsr(v − w), the integral on the right hand side of equation (22) che be written as fs(v, t)φ(v)dv ∀ s ∈ { 1, 2} . ( 15 ) β(v, w)fs(v, t)fr(w, t)(w − v)dvdw (23) where represents the probability of interaction between an agent with opinion v and an agent with opinion w. Using this notation, the weak form of the collisional operator with φ(v) = v is We now assume that β does not depend on v and w, namely that the probability of interactions between two agents does not depend on their current opinion. Inserting ( 16 ) and (17) into (25) and dividing both sides by ns, the weak form of the Boltzmann equation relative to φ(v) = v can be written as 2 d dt us(t) = βXγsrnr (ur(t) − us(t))

r=1 The 2 equations in (26) represents a homogeneous system of linear differential equations of first order which can be solved analytically. As a matter of fact, let us introduce, for the sake of simplicity, the two parameters a1 = βγ12n2 a2 = βγ21n1.

The two equations in (26) can then be written explicitly as u˙ 1(t) = −a1(u1(t) − u2(t)) u˙ 2(t) = a2(u1(t) − u2(t)).

The solution of the system (28) can be found simply by subtracting the second equation from the first one and, defining x(t) = u1(t) − u2(t), we find that whose solution is x˙ (t) = −(a1 + a2)x(t)

x(t) = Ce−(a1+a2)t and C is an arbitrary constant. Equation (30) implies that u1(t) = u2(t) + Ce−(a1+a2)t.

By substituting (31) in the second equation of (28) one finds u˙ 2(t) = Ca2e−(a1+a2)t where the only unknown is u2(t) which turns out to be u2(t) = −C e−(a1+a2)t + K.

Substituting this result in (31) one finds that the explicit expression of u1(t) is u1(t) = C e−(a1+a2)t + K.

The two constants C and K can be found by imposing that the solutions satisfy the initial conditions, namely so that with a2 a1 + a2

a1 a1 + a2   u1(0) = C   u2(0) = −C

a1 a1 + a2

a2 a1 + a2 + K

+ K β(v, w)fs(v, t)fr(w, t)(w − v)dvdw.

(25) s ∈ { 1, 2} . (26) where C and K are defined in (36) and (37), respectively. From (38) it is clear that the following limits hold | u1(t) − u2(t)| = | C| e−(a1+a2)t | u1(t) − u2(t)| ≤ ε ⇐⇒ t ≥ tmin tmin =

1 a1 + a2 log | C| .

ε Observe that tmin is the minimum value of the time t which guarantees that the average opinions of the two classes of agents differ less than ε. The condition (43) is only relative to the average opinions and does not imply consensus. where { uj(0)} j2=1 are the initial average values of the opinions of the two classes of agents. By subtracting the second equation from the first one, it can be easily shown that

C = u1(0) − u2(0) and substituting this results in the first equation of (35) gives K = u1(0)

+ u2(0) a2 a1 + a2

a1 .

Finally, the solution of (28) obtained by taking into account the initial conditions are   u1(t) = C   u2(t) = −C

a1 Observe that, according to (37) and (27), the value of the limit K depends on the average intial opinions { us(0)} s2=1, on the number of agents { ns} s2=1 in each class, and on γ12 and γ21.

We are now interested in studying the convergence time. In particular, since | us(t)− K| represents the distance between the average opinion of the classes s at time t and its limit for t → +∞, we consider the following inequalities | u1(t) − K| ≤ ε | u2(t) − K| ≤ ε.

From (38) the first inequality in (40) is equivalent to e−(a1+a2)t ≤ ε a1 + a2 .

| C| a1

From (41) it can be concluded that

| u1(t) − K| ≤ ε ⇐⇒ t ≥ t1 = Analogous elaborations show that | u2(t) − K| ≤ ε ⇐⇒ t ≥ t2 = 1 1 a1 + a2 a1 + a2 log log

a1 | C| . a1 + a2 ε

a2 | C| . a1 + a2 ε Finally, one can evaluate the minimum time necessary to ensure that the solution u1(t) differs from u2(t) for no more than ε. From (38) one obtains (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42) (43) (44) 2 4 6

8 t

In this section, we show simulation results concerning the opinion dynamics according to the framework proposed in Section II. We remark that such results are obtained by implementing the microscopic equations in ( 1 ), thus neglecting the analytic framework relative to the Boltzmann equation. From now on we denote as { u˜s(t)} s2=1 the values of the average opinions of the class s found by simulation while { us(t)} s2=1 represent the analytic solutions in (38). We consider a system composed of n = 103 agents. Table I shows the values of the parameters relative to the two classes of agents which are considered to derive analytic and simulation results in this section. In particular, different values of the parameters are 2 considered for: (i) the number of agents { ns} s=1; (ii) the initial 2 distribution of opinion; and (iii) the values of { γsr} s,r=1.

First, we consider the parameters shown in the first row of Table I. In this case, n1 = n2 = 500, namely the two classes of agents have the same number of agents. The initial opinions of the agents of class 1 are uniformly distributed in the interval (−1; 1/3), so that the initial average opinion of the agents of class 1 is u1(0) = −1/3. The initial opinions of the agents of class 2, instead, are uniformly distributed in the interval (−1/3; 1), and, therefore, their initial average opinion is u2(0) = 1/3. The two classes of agents are not only distinguished by their initial opinion distribution but they are also characterized by different propensity at changing their opinions when interacting with other agents. More precisely, the value of γ12 is 5/100 while the value of γ21 is 10/100. Since γ21 = 2γ12, the agents of class 2 are more inclined to change their opinion than those of class 1. Fig. 1 shows the average opinion u1(t) of the agents of class 1 (blue line) and the average opinion u2(t) of the agents of class 2 (red line). As expected from (37), u1(t) and u2(t) converge to the same value, which, according to this choice of parameters, corresponds to K = −1/9. Fig. 1 also shows the values of { u˜s(t)} s2=1 obtained by simulation. More precisely, the dashed cyan line refers to u˜1(t) while the dashed magenta line refers to u˜2(t). It can be observed that analytic results are in agreement with those obtained by simulating pairwise interactions according to ( 1 ). In Fig. 1, the value of the average opinion u(t) defined in (18) is also shown (dash-dotted black line). As expected from Section III, u(t) also converges to K .

Fig. 2 shows the distribution f1(v, t) (blue lines) and f2(v, t) (red lines) of the opinions of the two classes of agents obtained by simulating the multi-agent system with the parameters shown in the first line of Table I. More precisely: Fig. 2 (a) shows the distributions fs(v, t) after 104 interactions; Fig. 2 (b) shows the distributions fs(v, t) after 2 · 104 interactions; Fig. 2 (c) shows the distributions fs(v, t) 1 0.8 0.6 sn0.4 o i in0.2 p eo 0 g ra−0.2 e v a−0.4 −0.6 −0.8 −10 1.5 1 0.5 0 −1 −0.5 6 5 4 3 2 1 0 −1 −0.5 v0 (a) v0 (c) 0.5 1 0.5

1 after 3 · 104 interactions; and Fig. 2 (d) shows the distributions fs(v, t) after 105 interactions. From Fig. 2 it can be observed that not only the average opinions us(t) converge to the same value K , but also that, as discussed in previous sections, consensus among agents is reached, since the opinions of each agents tend to the same value.

We now consider the parameters shown in the second row of Table I. In this case, the two classes of agents differ not only because of their initial distribution of opinions and their values of γsr (which are equal to those previously considered), but also because of the number of agents. More precisely, agents of class 1 represent 75% of the population. Fig. 3 (a) shows the average opinion u1(t) of the agents of class 1 (blue line) and the average opinion u2(t) of the agents of class 2 (red line). As expected from (37), the values of u1(t) and u2(t) converge 2 to the same value, which, with these new values of { ns} s=1, corresponds to K ≃ −0.24. The values of u˜1(t) (dashed cyan line) and u˜2(t) (dashed magenta line) obtained by simulation are also shown in Fig. 3 and they are in agreement with those obtained analytically. Fig. 3 (a) also shows the value of the average opinion u(t) (dash-dotted black line) defined in (18), which converges to the same value K .

Finally, we consider the parameters shown in the third row of Table I. In this case, n1 = 900 and n2 = 100, i.e., agents of class 2 represent only 10% of the entire population. Under this assumption we consider that the initial opinions of the −0.6 −0.8 0.8 0.6 −0.6 −0.8 (a) 1.5 agents of class 1 are uniformly distributed in the interval I (so that u1(0) = 0) and the initial opinions of the agents of class 2 are uniformly distributed in the interval (3/4; 1) (so that u2(0) = 7/8). This choice corresponds to considering agents of class 2 as extremists, since their opinions are very close to one of the extremes of the interval I . In agreement with the idea that extremal opinions are typically more difficult to be changed, we assume that the value of γ21 is smaller than γ12. More precisely, we consider γ12 = 1/10 and γ21 = 1/100, so that γ21 is equal to a tenth of γ12. According to the choice γ21 = 1/100, agents of class 2 are skeptical.

Fig. 3 (b) shows the average opinion u1(t) of the agents of class 1 (blue line) and the average opinion u2(t) of the agents of class 2 (red line) as functions of time t. As in the previous cases, u1(t) and u2(t) converge to the same value, which, according to this choice of parameters and (37), corresponds to K ≃ 0.46. Fig. 3 (b) also shows the values of u˜1(t) (dashed cyan line) and u˜2(t) (dashed magenta line) obtained by simulation. Once again, analytic results obtained according to the kinetic approach are in agreement with those obtained by simulations. For the sake of completeness, Fig. 3 (b) also shows the value of the average opinion u(t) (dash-dotted green line) defined in (18). As expected, u(t) also converges to K . A wide variety of choices for the parameters of the model could be taken and the results shown here are only illustrative of some particular configurations. The agreement between analytic and simulation results indicates that the framework based on kinetic theory is consistent and, therefore, it can be properly used to analytically study opinion dynamics.

V.

CONCLUSIONS

In this paper, we study analytically a model for opinion dynamics based on kinetic theory. We start from the description of the effects of microscopic interactions among agents, which are assumed to be binary, and we describe macroscopic properties related to opinion dynamics in the considered multiagent system, using proper balance equations. More precisely, we take inspiration from kinetic theory of gas mixtures, which allows describing the behavior of gases composed of different kinds of molecules. Similarly, we aim at describing a multiagent system composed of different classes of agents. The considered different classes of agents have different characteristics, namely: (i) cardinality, (ii) initial average opinions, and (iii) propensity to change opinions.

References

[1] H. V. Parunak , T. C. Belding , R. Hilscher , and S. Brueckner , “ Modeling an managing collective cognitive convergence ,” in Proceedings of 7th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2008 ), Estoril, Portugal, May 2008 .

[2] M. Ma ¨s, A . Flache, and D. Helbing , “ Individualisazion as driving force of clustering phenomena in humans , ” PLOS One , vol. 6 , no. 10 , 2010 .

[3] F. Schweitzer and J. Holyst , “ Modelling collective opinion formation by means of active brownian particles , ” European Physical Journal B , vol. 15 , pp. 723 - 732 , 2000 .

[4] D. Kempre , A. Dobra , and J. Gehrke , “ Gossip-based computation of aggregate information ,” in Proceedings of the 44th Annual IEEE Symposium Foundations of Computer Science (FOCS 03) , Hannover, Germany, March 2008 , pp. 71 - 78 .

[5] S. Monica and F. Bergenti , “ A stochastic model of self-stabilizing cellular automata for consensus formation ,” in Proceedings of 15th Workshop “Dagli Oggetti agli Agenti” (WOA 2014 ), Catania, Italy, September 2014 .

[6] A. Tsang and K. Larson , “ Opinion dynamics of skeptical agents ,” in Proceedings of 13th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2014 ), Paris, France, May 2014 .

[7] L. Pareschi and G. Toscani , Interacting Multiagent Systems: Kinetic Equations and Montecarlo Methods . Oxford University Press, 2013 .

[8] W. Weidlich , Sociodynamics: a systematic approach to mathematical modelling in the social sciences . Harwood Academic Publisher , 2000 .

[9] S. Monica and F. Bergenti , “ Kinetic description of opinion evolution in multi-agent systems: Analytic model and simulations,” in Proceedings of the 18th International Conference on Principles and Practice of Multi-Agent Systems (PRIMA) , Bertinoro, Italy, 2015 , pp. 483 - 491 .

[10] M. Baldoni , C. Baroglio , F. Bergenti , A. Boccalatte , E. Marengo, M. Martelli , V. Mascardi , L. Padovani , V. Patti , A. Ricci , G. Rossi, and A. Santi , “ Mercurio: An interaction-oriented framework for designing, verifying and programming multi-agent systems,” in Multi-Agent Logics, Languages, and Organisations Federated Workshops, (MALLOW), Lyon , France, 2010 , pp. 134 - 149 .

[11] S. Monica and F. Bergenti , “ A study of consensus formation using kinetic theory,” in Proceedings of the 13th International Conference on Distributed Computing and Artificial Intelligence (DCAI 2016 ), Sevilla, Spain, June 2016 , pp. 213 - 221 .

[12] S. Monica and F. Bergenti , “ Simulations of opinion formation in multiagent systems using kinetic theory,” in Proceedings of 16th Workshop “Dagli Oggetti agli Agenti” (WOA '15) , Napoli, Italy, June 2015 .

[13] S. Monica and F. Bergenti , “ A kinetic study of opinion dynamics in multi-agent systems,” in Atti del Convegno (AI*IA '15) , Ferrara, Italy, September 2015 .

[14] G. Toscani , “ Kinetic models of opinion formation , ” Communications in Mathematical Sciences , vol. 4 , pp. 481 - 496 , 2006 .

[15] A. V. Bobylev and I. M. Gamba , “ Boltzmann equation for mixtures of maxwell gases: exact solutions and power-like tails , ” Journal of Statistical Physics , vol. 124 , pp. 497 - 516 , 2006 .

[16] W. Rudin , Functional Analysis .

McGraw-Hill , 1973 .