Analytic Study of Opinion Dynamics in Multi-Agent Systems with Two Classes of Agents Stefania Monica and Federico Bergenti Dipartimento di Matematica e Informatica Università degli Studi di Parma Parco Area delle Scienze 53/A, 43124 Parma, Italy Email: {stefania.monica, federico.bergenti}@unipr.it Abstract—This paper describes a model for opinion dynamics the molecules are reinterpreted as interactions among agents. in multi-agent systems composed of two classes of agents. Each A major advantage of the use of kinetic-based approaches class is characterized by distinctive values of the parameters that to model opinion dynamics and consensus is that analytic govern opinion dynamics. The proposed model is inspired by results can be derived, while, at the opposite, opinion dynam- kinetic theory of gases, according to which macroscopic properties ics in multi-agent systems is typically investigated through of gases are described starting from microscopic interactions simulations [9]. It is worth noting that common approaches among molecules. By interpreting agents as molecules of gases, and their interactions as collisions among molecules, the equations to the analysis of interactions in multi-agent systems (e.g., that govern kinetic theory can be reinterpreted to model opinion [10]) are normally more interested in formalizing complex dynamics in multi-agent systems. A key feature of the adopted microscopic interactions rather than in studying the overall kinetic-based approach is that it allows macroscopic properties emergent behavior of the system. of the system to be derived analytically. In order to take into account that the considered multi-agent system is composed of Standard kinetic theory typically assumes that all the two classes of agents, kinetic theory of gas mixtures, which molecules are equal. However, gases are typically composed deals with gases composed of different kinds of molecules, is of molecules of different types and, therefore, a more accurate adopted. Presented results show that consensus is reached after description of gases can be achieved using kinetic theory of a sufficiently large number of interactions, which depends on the gas mixtures, which takes into account that different species parameters associated with the two classes of agents. of molecules coexist in the same gas. Using the framework of kinetic theory of gas mixtures, it is then possible to describe I. I NTRODUCTION multi-agent systems composed of different classes of agents, each of which is associated with different values of the pa- Opinion dynamics and consensus formation are well- rameters used to model opinion dynamics. The most important known problems that deal with the identification of interaction features that can be introduced to distinguish a specific type of rules which lead to proper distribution of opinion in multi- agents are the propensity to change opinion when interacting agent systems [1]. Such problems are important topics of the with other agents, and the ability to change the opinions of research on multi-agent systems and distributed computing interacting agents. In addition, different classes of agents can and they have applications in many areas, such as control have different cardinalities and different initial distribution of theory, physics, biology, and sociology (e.g., [2]). Various opinions. According to this approach, phenomena such as approaches have been proposed in the literature to describe extremism or skepticism can be studied [6]. In this paper, opinion dynamics and consensus formation, among which we consider multi-agent systems composed of two classes we can recall those based on thermodynamics (e.g., [3]), on of agents; however, the proposed approach is general and, Bayesian networks (e.g., [4]), and on gossip-based algorithms potentially, the number of classes of agents can be set equal (e.g., [4]). The use of cellular automata to model consensus to the number of agents, thus having one agent for each class. formation has also been investigated; in this case, opinion is modeled as a discrete variable and consensus is reached This paper is organized as follows. In Section II the opinion through proper transition rules (e.g., [5]). Another important dynamics problem is formulated using the kinetic framework. framework which is useful to study opinion dynamics is related In Section III macroscopic properties of the considered multi- to graph theory. (e.g., [6]). agent system are derived. Section IV shows some illustrative results concerning the average opinions of specific multi-agent In this paper, we consider a model for opinion dynamics systems. Section V concludes the paper. which is inspired by sociophysics, a discipline according to which social interactions and opinion dynamics in multi- II. K INETIC F ORMULATION OF O PINION F ORMATION agent systems can be described using the formalism of the kinetic theory of gases [7]. Kinetic theory of gases aims at The identification of agents with the molecules of a gas analyzing the effects of microscopic collisions among the allows applying the framework of kinetic theory to different molecules from a probabilistic point of view in order to derive fields and, in particular, to distributed artificial intelligence and macroscopic properties of gases by means of a proper balance opinion dynamics in multi-agent systems. As in kinetic theory, equation, namely, the Boltzmann equation [8]. According to we assume that each agent can interact with any other agent in sociophysics, a parallelism can be done between the molecules the system and that each interaction involves two agents [11]. of gases and agents in multi-agent systems: collisions among For this reason, we denote interactions as binary. While the 17 molecules of gases are typically related to their velocities, in (4) we can conclude that the difference between the post- the context of opinion dynamics we assume that each agent is interaction opinions is smaller than the difference between associated with a scalar attribute v that denotes its opinion. The the pre-interaction opinions of the two agents. Hence, it is opinion of each agent is updated at each interaction, according reasonable to expect that, after a sufficiently large number to proper rules. Various kinds of rules to update the opinion of of interactions, all agents end up with the same opinion, agents after interactions have been studied to model different regardless of their class. Concerning differences of opinions, characteristics of agents [12], [13]. it can also be concluded that the post-interaction opinion of an agent is closer to its pre-interaction opinion than to the pre- Let us denote as n(t) the total number of agents at time t interaction opinion of the agent it interacts with. As a matter and as n1 (t) and n2 (t) the number of agents of the type 1 and of fact, from (2) and (1), one can derive that 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 |v ∗ − v| = γsr |v − w| < (1 − γsr )|v − w| = |v ∗ − w| of each agent is defined in the interval I = [−1, 1], where −1 (5) |w∗ − w| = γrs |w − v| < (1 − γrs )|w − v| = |w∗ − v|. 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. We remark that, according to the model in (1), the post- interaction opinions v ∗ and w∗ still belong to the interval I where the opinions are defined. A. Interaction Rules In order to describe the microscopic effects of pairwise B. The Boltzmann Equation interactions, let us define the interaction rules. Assume that an agent of type s with opinion v interacts with another agent Starting from the interaction rules in (1), it is possi- of type r with opinion w. The post-interaction opinions of the ble to study opinion dynamics of multi-agent systems using two interacting agents depend on their pre-interaction opinions simulations. Instead, we now show how to obtain analytical according to the following rules results by applying the framework of kinetic theory of gas  ∗ mixtures to the considered opinion dynamics scenario. For v = v − γsr (v − w) (1) this purpose, we introduce the Boltzmann equation, namely w∗ = w − γrs (w − v) an integro-differential equation that allows deriving macro- where v ∗ and w∗ are the opinions of the two agents after scopic properties of gases. In the considered scenario, which the interaction. Observe that the considered model involves includes only two classes of agents, two equations need to 4 coefficients {γsr }2s,r=1 , where γsr measures the propensity be considered, whose unknowns are non-negative functions of an agent of type s to change its opinion in favor of that {fs (v, t)}2s=1 which represent the density of the opinion v ∈ I, of an agent of type r. As a matter of fact, considering, for relative to agents of class s, at time t ≥ 0. The temporal instance, the first equation of (1) it is clear that an increment evolution of each distribution function can be described, in of γsr increases the propensity of agents of type s to change spatially homogeneous conditions, as their opinions when interacting with agents of type r. In the ∂fs following, we assume that the coefficients {γsr }2s,r=1 satisfy (v, t) = Is s ∈ {1, 2} (6) ∂t 1 0 < γsr < ∀ s, r ∈ {1, 2}. (2) where Is is the collisional operator relative to the class s and 2 it is written as In agreement with the intended meaning of γsr explained 2 above, according to (1), if γsr is nearly 0, the individuals of X Is = Qsr (fs , fr ) s ∈ {1, 2}. (7) type s are not inclined to change their opinion towards that r=1 of agents of type r. For this reason, values of γsr close to 0 characterize skeptical agents. At the opposite, if in the first From (7) it is evident that the collisional operator relative equation of (1) we set γsr ≃ 1/2, then v ∗ ≃ 1/2(v + w), so to each class of agents depends on the distribution functions that the first agent looses half of its opinion in favour of that {fs }2s=1 of all species. of the second, which characterize easily influenced agents. In order to obtain analytic results, the explicit expression The sum of the opinions of two interacting agents after the of the collisional operator is needed. To simplify notation, in interaction can be derived from (1) and it is given by the derivation of the explicit expression of the collisional op- erator we neglect the dependence of the distribution functions v ∗ + w∗ = v + w + (γrs − γsr )(v − w). (3) {fs }2s=1 on time t, since all involved integrals are related to From (3), the opinion is not conserved and that it can change the opinion variable. Let us denote as depending on the sign of (γrs − γsr )(v − w), namely on the values of the coefficients γrs and γsr and on the values of W (v, w, v ∗ , w∗ )dv ∗ dw∗ (8) the pre-interaction opinions v and w. From (1) it can also be the probability that after the binary interaction of two agents derived that the difference of the opinions of two interacting with opinion values v and w, the opinions of the two agents agents after the interaction is become v ∗ and w∗ , respectively. Hence, the loss of agents of v ∗ − w∗ = εrs (v − w). (4) class s in v and, simultaneously, of agents of class r in w can be denoted as where εrs = 1−(γrs +γsr ). Since, from (2), γrs +γsr ∈ (0, 1), it is easy to conclude that εrs ∈ (0, 1). Therefore, from Qsr − (fs , fr ) = W (v, w, v ∗ , w∗ )fs (v)fr (w)dvdwdv ∗ dw∗ . 18 Analogously, the gain of agents of class s in v and, simulta- III. A NALYTIC S TUDY OF M ACROSCOPIC P ROPERTIES neously, of agents of class r in w, is given by From standard kinetic theory, we can describe the temporal Qsr + (fs , fr ) = W (v∗ , w∗ , v, w)fs (v∗ )fr (w∗ )dv∗ dw∗ dvdw evolution of the distribution function fs (v, t) according to the spatially homogeneous Boltzmann equation which, in case of where v∗ and w∗ are the pre-interaction opinions of agent of a gas mixture, corresponds to (6), where the right hand side class s and r, respectively, which lead to v and w as opinions represents the collisional operator Is relative to the class s. We of the two agents after the interaction [14]. now show how the Boltzmann equation can be used to derive macroscopic properties of the considered multi-agent system. According to kinetic theory of gas mixtures, the collisional The number of agents of class s at time t can be expressed as operator Qsr relative to classes s and r can be written as [15] Z Z fs (v, t)dv = ns (t) s ∈ {1, 2}. (16) Qsr (v) = W (v∗ , w∗ , v, w)fs (v∗ )fr (w∗ )dv∗ dw∗ dw I ZI 3 (9) Similarly, the average opinion of agents of class s at time t ∗ ∗ ∗ ∗ can be defined as − W (v, w, v , w )fs (v)fr (w)dwdv dw I3 Z 1 us (t) = fs (v, t)vdv s ∈ {1, 2}. (17) where the two integrals are obtained by integrating ns (t) I Qsr − (fs , fr ) and Qsr + (fs , fr ) with respect to all the variables Observe that the global (i.e., referred to all the agents) average except v, and they represents the gain and the loss of agents opinion is then defined as the sum of the average opinions of with opinion in (v, v + dv), respectively. each class weighed by the number of agents of the correspond- Let us now consider the weak form of the Boltzmann equa- ing class and divided by n, namely tion, which is obtained by multiplying (6) by a test function 1 φ(v), namely a smooth function with compact support, and u(t) = (n1 (t)u1 (t) + n2 (t)u2 (t)) . (18) n integrating the result with respect to v [16]. The weak form of the Boltzmann equation is then given by Such definitions are related to two simple test functions Z 2 Z φ(v) in (14). More precisely, setting φ(v) = 1 in (14) leads to ∂fs X Z φ(v)dv = Qsr (fs , fr )φ(v)dv (10) d I ∂t r=1 I fs (v, t)dv = 0 s ∈ {1, 2} (19) dt I where, according to (9), the integral in the sum on the right where the 0 on the right hand side is due to the fact that, since hand side can be written as φ(v) is a constant function, the difference φ(v ∗ ) − φ(v) inside Z the integral is 0. Since, from (16), the integral on the left hand W (v∗ , w∗ , v, w)fs (v∗ )fr (w∗ )φ(v)dv∗ dw∗ dvdw side of (19) represents ns (t), equation (19) can be written as I 4Z (11) d ∗ ∗ ∗ ∗ ns (t) = 0 s ∈ {1, 2} − W (v, w, v , w )fs (v)fr (w)φ(v)dvdwdv dw (20) I4 dt so that the number of individuals of each class is conserved. By applying the change of variables Observe that equation (20) also implies that (v∗ , w∗ , v, w) → (v, w, v ∗ , w∗ ) (12) d d n(t) = (n1 (t) + n2 (t)) = 0 (21) in the first integral in (11) one obtains that the weak form of dt dt the collisional operator Is in (11) can be written as 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 Z X dependence of n and {ns }2s=1 on t. The conservation of the W (v, w, v ∗ , w∗ )fs (v)fr (w)(φ(v ∗ ) − φ(v))d4 v (13) number of agents is a realistic property of the model. r=1 I4 Let us now consider the test function φ(v) = v in order where, from now on, d4 v denotes the products on the four to investigate the temporal evolution of the average opinion. differentials dvdwdv ∗ dw∗ . By substituting (13) in (10) the Setting φ(v) = v in (14) and using (17) we obtain weak form of the Boltzmann equation for each class s ∈ {1, 2} Z can be finally written as d fs (v, t)vdv dt I Z X2 Z 2 Z (22) d X fs (v, t)φ(v)dv = W (v, w, v ∗ , w∗ )fs (v)· = W (v, w, v ∗ , w∗ )fs (v)fr (w)(v ∗ − v)d4 v. dt I r=1 I 4 (14) r=1 I4 ∗ fr (w)(φ(v ) − φ(v))d4 v ∀ s ∈ {1, 2} Since, from (1), the difference v ∗ − v can be expressed as where on the left hand side we used the fact that for every test −γsr (v − w), the integral on the right hand side of equation function (see [16]) (22) che be written as Z Z X2 Z ∂fs d φ(v)dv = fs (v, t)φ(v)dv ∀ s ∈ {1, 2}. (15) γsr β(v, w)fs (v, t)fr (w, t)(w − v)dvdw (23) I ∂t dt I r=1 I2 19 where where {uj (0)}2j=1 are the initial average values of the opinions ZZ of the two classes of agents. By subtracting the second β(v, w) = W (v, w, v ∗ , w∗ )dv ∗ dw∗ (24) equation from the first one, it can be easily shown that I2 represents the probability of interaction between an agent with C = u1 (0) − u2 (0) (36) opinion v and an agent with opinion w. Using this notation, and substituting this results in the first equation of (35) gives the weak form of the collisional operator with φ(v) = v is a2 a1 X2 Z K = u1 (0) + u2 (0) . (37) a1 + a2 a1 + a2 γsr β(v, w)fs (v, t)fr (w, t)(w − v)dvdw. (25) r=1 I2 Finally, the solution of (28) obtained by taking into account We now assume that β does not depend on v and w, namely the initial conditions are that the probability of interactions between two agents does  a1 not depend on their current opinion. Inserting (16) and (17)   u1 (t) = C e−(a1 +a2 )t + K into (25) and dividing both sides by ns , the weak form of the a1 + a2 a2 (38) Boltzmann equation relative to φ(v) = v can be written as   u2 (t) = −C e−(a1 +a2 )t + K a1 + a2 X2 d where C and K are defined in (36) and (37), respectively. us (t) = β γsr nr (ur (t) − us (t)) s ∈ {1, 2}. (26) dt r=1 From (38) it is clear that the following limits hold The 2 equations in (26) represents a homogeneous system of lim u1 (t) = lim u2 (t) = K. (39) t→+∞ t→+∞ linear differential equations of first order which can be solved analytically. As a matter of fact, let us introduce, for the sake Observe that, according to (37) and (27), the value of the limit of simplicity, the two parameters K depends on the average intial opinions {us (0)}2s=1 , on the number of agents {ns }2s=1 in each class, and on γ12 and γ21 . a1 = βγ12 n2 a2 = βγ21 n1 . (27) We are now interested in studying the convergence time. The two equations in (26) can then be written explicitly as In particular, since |us (t)−K| represents the distance between  the average opinion of the classes s at time t and its limit for u̇1 (t) = −a1 (u1 (t) − u2 (t)) (28) t → +∞, we consider the following inequalities u̇2 (t) = a2 (u1 (t) − u2 (t)). The solution of the system (28) can be found simply by |u1 (t) − K| ≤ ε |u2 (t) − K| ≤ ε. (40) subtracting the second equation from the first one and, defining From (38) the first inequality in (40) is equivalent to x(t) = u1 (t) − u2 (t), we find that a1 + a2 ẋ(t) = −(a1 + a2 )x(t) (29) e−(a1 +a2 )t ≤ ε . (41) |C|a1 whose solution is From (41) it can be concluded that x(t) = Ce −(a1 +a2 )t (30)   1 a1 |C| |u1 (t) − K| ≤ ε ⇐⇒ t ≥ t1 = log . and C is an arbitrary constant. Equation (30) implies that a1 + a2 a1 + a2 ε u1 (t) = u2 (t) + Ce−(a1 +a2 )t . (31) Analogous elaborations show that   By substituting (31) in the second equation of (28) one finds 1 a2 |C| |u2 (t) − K| ≤ ε ⇐⇒ t ≥ t2 = log . −(a1 +a2 )t a1 + a2 a1 + a2 ε u̇2 (t) = Ca2 e (32) Finally, one can evaluate the minimum time necessary to where the only unknown is u2 (t) which turns out to be ensure that the solution u1 (t) differs from u2 (t) for no more a2 than ε. From (38) one obtains u2 (t) = −C e−(a1 +a2 )t + K. (33) a1 + a2 |u1 (t) − u2 (t)| = |C|e−(a1 +a2 )t (42) Substituting this result in (31) one finds that the explicit expression of u1 (t) is so that |u1 (t) − u2 (t)| ≤ ε ⇐⇒ t ≥ tmin (43) a1 u1 (t) = C e−(a1 +a2 )t + K. (34) a1 + a2 with   1 |C| The two constants C and K can be found by imposing that tmin = log . (44) the solutions satisfy the initial conditions, namely a1 + a2 ε  a1 Observe that tmin is the minimum value of the time t which   u1 (0) = C +K a1 + a2 guarantees that the average opinions of the two classes of a2 (35) agents differ less than ε. The condition (43) is only relative to   u2 (0) = −C +K a1 + a2 the average opinions and does not imply consensus. 20 TABLE I. T HE CONSIDERED VALUES OF THE PARAMETERS FOR THE 1 TWO CLASSES OF AGENTS : NUMBER OF AGENTS ( FIRST AND SECOND 0.8 COLUMN ); INITIAL DISTRIBUTIONS OF THE OPINION ( THIRD AND FOURTH 0.6 COLUMN ); PARAMETERS γsr ( FIFTH AND SIXTH COLUMN ). average opinions 0.4 0.2 n1 n2 f1 (v, 0) f2 (v, 0) γ12 γ21 0 500 500 U ((−1; 1/3)) U ((−1/3; 1)) 5/100 10/100 −0.2 −0.4 750 250 U ((−1; 1/3)) U ((−1/3; 1)) 5/100 10/100 −0.6 900 100 U ((−1; 1)) U ((3/4; 1)) 10/100 1/100 −0.8 −1 0 2 4 6 8 10 t 4 x 10 IV. V ERIFICATION OF R ESULTS BY S IMULATION Fig. 1. The average opinions u1 (t) (blue line) and u2 (t) (red line) derived In this section, we show simulation results concerning the analytically with the parameters in the first row of Table I are shown. The opinion dynamics according to the framework proposed in corresponding average opinion u(t) is also shown (dash-dotted black line). Section II. We remark that such results are obtained by imple- The values of ũ1 (t) (dashed cyan line) and ũ2 (t) (dashed magenta line) obtained by simulation are shown. menting the microscopic equations in (1), thus neglecting the analytic framework relative to the Boltzmann equation. From 1.5 3.5 now on we denote as {ũs (t)}2s=1 the values of the average 2.5 3 opinions of the class s found by simulation while {us (t)}2s=1 1 2 represent the analytic solutions in (38). We consider a system 0.5 1.5 composed of n = 103 agents. Table I shows the values of 1 0.5 the parameters relative to the two classes of agents which are 0 −1 −0.5 0 0.5 1 0 −1 −0.5 0 0.5 1 v v considered to derive analytic and simulation results in this (a) (b) section. In particular, different values of the parameters are 6 25 considered for: (i) the number of agents {ns }2s=1 ; (ii) the initial 5 20 distribution of opinion; and (iii) the values of {γsr }2s,r=1 . 4 15 3 10 2 First, we consider the parameters shown in the first row 1 5 of Table I. In this case, n1 = n2 = 500, namely the two 0 −1 −0.5 0 0.5 1 0 −1 −0.5 0 0.5 1 v v classes of agents have the same number of agents. The initial (c) (d) opinions of the agents of class 1 are uniformly distributed in Fig. 2. The opinion distributions f1 (v, t) (solid blue line) and f2 (v, t) the interval (−1; 1/3), so that the initial average opinion of (dashed red line) relative to the parameters in the first row of Table I are the agents of class 1 is u1 (0) = −1/3. The initial opinions shown: (a) after 104 interactions; (b) after 2 · 104 interactions; (c) after 3 · 104 of the agents of class 2, instead, are uniformly distributed interactions; (d) after 105 interactions. in the interval (−1/3; 1), and, therefore, their initial average opinion is u2 (0) = 1/3. The two classes of agents are not after 3 · 104 interactions; and Fig. 2 (d) shows the distributions only distinguished by their initial opinion distribution but they fs (v, t) after 105 interactions. From Fig. 2 it can be observed are also characterized by different propensity at changing their that not only the average opinions us (t) converge to the same opinions when interacting with other agents. More precisely, value K, but also that, as discussed in previous sections, the value of γ12 is 5/100 while the value of γ21 is 10/100. consensus among agents is reached, since the opinions of each Since γ21 = 2γ12 , the agents of class 2 are more inclined agents tend to the same value. to change their opinion than those of class 1. Fig. 1 shows We now consider the parameters shown in the second row the average opinion u1 (t) of the agents of class 1 (blue line) of Table I. In this case, the two classes of agents differ not only and the average opinion u2 (t) of the agents of class 2 (red because of their initial distribution of opinions and their values line). As expected from (37), u1 (t) and u2 (t) converge to the of γsr (which are equal to those previously considered), but same value, which, according to this choice of parameters, also because of the number of agents. More precisely, agents corresponds to K = −1/9. Fig. 1 also shows the values of class 1 represent 75% of the population. Fig. 3 (a) shows the of {ũs (t)}2s=1 obtained by simulation. More precisely, the average opinion u1 (t) of the agents of class 1 (blue line) and dashed cyan line refers to ũ1 (t) while the dashed magenta the average opinion u2 (t) of the agents of class 2 (red line). line refers to ũ2 (t). It can be observed that analytic results As expected from (37), the values of u1 (t) and u2 (t) converge are in agreement with those obtained by simulating pairwise to the same value, which, with these new values of {ns }2s=1 , interactions according to (1). In Fig. 1, the value of the average corresponds to K ≃ −0.24. The values of ũ1 (t) (dashed cyan opinion u(t) defined in (18) is also shown (dash-dotted black line) and ũ2 (t) (dashed magenta line) obtained by simulation line). As expected from Section III, u(t) also converges to K. are also shown in Fig. 3 and they are in agreement with those obtained analytically. Fig. 3 (a) also shows the value of the Fig. 2 shows the distribution f1 (v, t) (blue lines) and average opinion u(t) (dash-dotted black line) defined in (18), f2 (v, t) (red lines) of the opinions of the two classes of which converges to the same value K. agents obtained by simulating the multi-agent system with the parameters shown in the first line of Table I. More Finally, we consider the parameters shown in the third row precisely: Fig. 2 (a) shows the distributions fs (v, t) after 104 of Table I. In this case, n1 = 900 and n2 = 100, i.e., agents interactions; Fig. 2 (b) shows the distributions fs (v, t) after of class 2 represent only 10% of the entire population. Under 2 · 104 interactions; Fig. 2 (c) shows the distributions fs (v, t) this assumption we consider that the initial opinions of the 21 1 1 0.8 0.8 0.6 0.6 average opinions average opinions 0.4 0.4 0.2 0.2 0 0 −0.2 −0.2 −0.4 −0.4 −0.6 −0.6 −0.8 −0.8 −1 −1 0 2 4 6 8 10 0 0.5 1 1.5 2 t 4 x 10 t 5 x 10 (a) (b) Fig. 3. The average opinions u1 (t) (blue line) and u2 (t) (red line) derived analytically are compared to ũ1 (t) (dashed cyan line) and ũ2 (t) (dashed magenta line) obtained by simulation when considering the parameters: (a) in the second row of Table I and (b) in the third row of Table I. The corresponding average opinion u(t) is also shown (dash-dotted black line). agents of class 1 are uniformly distributed in the interval I (so R EFERENCES that u1 (0) = 0) and the initial opinions of the agents of class [1] H. V. Parunak, T. C. Belding, R. Hilscher, and S. Brueckner, “Modeling 2 are uniformly distributed in the interval (3/4; 1) (so that an managing collective cognitive convergence,” in Proceedings of u2 (0) = 7/8). This choice corresponds to considering agents 7th International Conference on Autonomous Agents and Multiagent of class 2 as extremists, since their opinions are very close to Systems (AAMAS 2008), Estoril, Portugal, May 2008. one of the extremes of the interval I. 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Monica and F. Bergenti, “A stochastic model of self-stabilizing of class 2 (red line) as functions of time t. As in the previous cellular automata for consensus formation,” in Proceedings of 15th cases, u1 (t) and u2 (t) converge to the same value, which, Workshop “Dagli Oggetti agli Agenti” (WOA 2014), Catania, Italy, according to this choice of parameters and (37), corresponds September 2014. to K ≃ 0.46. Fig. 3 (b) also shows the values of ũ1 (t) [6] A. Tsang and K. Larson, “Opinion dynamics of skeptical agents,” in Proceedings of 13th International Conference on Autonomous Agents (dashed cyan line) and ũ2 (t) (dashed magenta line) obtained and Multiagent Systems (AAMAS 2014), Paris, France, May 2014. by simulation. Once again, analytic results obtained according [7] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic to the kinetic approach are in agreement with those obtained Equations and Montecarlo Methods. Oxford University Press, 2013. by simulations. For the sake of completeness, Fig. 3 (b) also [8] W. Weidlich, Sociodynamics: a systematic approach to mathematical shows the value of the average opinion u(t) (dash-dotted green modelling in the social sciences. Harwood Academic Publisher, 2000. line) defined in (18). As expected, u(t) also converges to K. [9] S. Monica and F. Bergenti, “Kinetic description of opinion evolution in A wide variety of choices for the parameters of the model multi-agent systems: Analytic model and simulations,” in Proceedings could be taken and the results shown here are only illustrative of the 18th International Conference on Principles and Practice of Multi-Agent Systems (PRIMA), Bertinoro, Italy, 2015, pp. 483–491. of some particular configurations. The agreement between [10] M. Baldoni, C. Baroglio, F. Bergenti, A. Boccalatte, E. Marengo, analytic and simulation results indicates that the framework M. Martelli, V. Mascardi, L. Padovani, V. Patti, A. Ricci, G. Rossi, and based on kinetic theory is consistent and, therefore, it can be A. Santi, “Mercurio: An interaction-oriented framework for designing, properly used to analytically study opinion dynamics. verifying and programming multi-agent systems,” in Multi-Agent Log- ics, Languages, and Organisations Federated Workshops, (MALLOW), Lyon, France, 2010, pp. 134–149. V. C ONCLUSIONS [11] S. Monica and F. Bergenti, “A study of consensus formation using kinetic theory,” in Proceedings of the 13th International Conference on In this paper, we study analytically a model for opinion Distributed Computing and Artificial Intelligence (DCAI 2016), Sevilla, dynamics based on kinetic theory. We start from the descrip- Spain, June 2016, pp. 213–221. tion of the effects of microscopic interactions among agents, [12] S. Monica and F. Bergenti, “Simulations of opinion formation in multi- which are assumed to be binary, and we describe macroscopic agent systems using kinetic theory,” in Proceedings of 16th Workshop “Dagli Oggetti agli Agenti” (WOA ’15), Napoli, Italy, June 2015. properties related to opinion dynamics in the considered multi- agent system, using proper balance equations. More precisely, [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, we take inspiration from kinetic theory of gas mixtures, which September 2015. allows describing the behavior of gases composed of different [14] G. Toscani, “Kinetic models of opinion formation,” Communications in kinds of molecules. Similarly, we aim at describing a multi- Mathematical Sciences, vol. 4, pp. 481–496, 2006. agent system composed of different classes of agents. The [15] A. V. Bobylev and I. M. Gamba, “Boltzmann equation for mixtures considered different classes of agents have different charac- of maxwell gases: exact solutions and power-like tails,” Journal of teristics, namely: (i) cardinality, (ii) initial average opinions, Statistical Physics, vol. 124, pp. 497–516, 2006. and (iii) propensity to change opinions. [16] W. Rudin, Functional Analysis. McGraw-Hill, 1973. 22