-In this paper we formulate the problem of opinion formation using a physical metaphore. We consider a multiagent system where each agent is associated with an opinion and interacts with any other agent. Interpreting the agents as the molecules of a gas, we model the evolution of opinion in the system according to a kinetic model based on the analysis of interactions among agents. From a microscopic description of each interaction between two agents, we derive the stationary profiles of the opinion under given assumption. Results show that, depending on the average opinion and on the parameters of the model, different profiles can be found, but all stationary profiles are characterized by the presence just of one or two maxima. Analytic results are confirmed by simulations shown in the last part of the paper.
In this paper we describe a model for opinion formation among agents. We assume that each agent is associated with an opinion v ∈ I ⊆ R and that it can change its opinion each time it interacts with another agent. In the literature, various approaches that study opinion evolution in a society of agents have been proposed, and many of them are based on Cellular Automata (CA) because CA describe well global effects of local phenomena (see, e.g., [1]). In order to overcome the synchronism that CA assume, in recent years the use of microscopic models inspired from physics has been introduced to describe asynchronous social interactions among agents in a society [2]. Such models are based on the idea that the laws of kinetic theory, which are typically used to describe the effects of interactions between two molecules of a gas, can also be used to model interactions between two agents.
Statistical mechanics and kinetic theory describe the details of each interaction between two molecules in a gas but they also allow finding classic laws which describe macroscopic properties of gases [3]. Analogously, from the microscopic laws which describe the details of each interaction between two agents, collective behaviour can be described from a macroscopic point of view [4]. All the research challenges related to the application of kinetic and statistical formalisms to describe multi-agent systems gave rise to new disciplines known as econophysics and sociophysics [5]. Such new disciplines have been used to describe, e.g., wealth evolution [6] and market economy [7], and they have also been adopted to characterize opinion evolution in a society [8].
In this paper, we focus on a model for opinion formation based on kinetic theory of gases and we analyze the evolution of the opinion in a society of agents. In particular, we analyze numerically the opinion evolution according to a model introduced in [9], which mandates that each agent can change its own opinion because of two different reasons [10]. The first reason is related to compromise between interacting agents. More precisely, the model assumes that both agents involved in an interaction can change their respective opinions in favor of that of the other agent. The second reason only involves each single agent and it is related to the fact that an agent can change its opinion autonomously, giving rise to a process known as diffusion.
This paper is organized as follows. In Section II, we describe the considered kinetic model from an analytic point of view. In Section III, we derive a stationary profile of opinion in a specific case. In Section IV, we show relevant simulation results. Section V concludes the paper.
II.
KINETIC FORMULATION OF OPINION FORMATION Kinetic theory of gases describes, from a microscopic point of view, the effects of interactions among molecules in gases. By reinterpreting the molecules of a gas as agents, one can use the kinetic framework to describe social interactions among agents. While molecules are typically associated with relevant physical properties, like their velocities, agents can be associated with attributes that represent some of their characteristics. In particular, since in this work we are interested in modeling the evolution of the opinion, we assume that each agent is associated with a scalar parameter v which represents its opinion and which is defined in a given interval I. In the following, we consider I = [−1, 1], where ±1 represent extremal opinions.
In order to use the kinetic approach, we need to define a function, denoted as f (w, t), which represents the density of opinion w at time t, and which is defined for each opinion w ∈ I and for each time t ≥ 0. According to such a definition, f (w, t)dw = 1.
In order to formulate the problem of opinion evolution in
kinetic terms, we assume that the function f (w, t) evolves
on the basis of the Boltzmann equation, which, under our
assumptions, can be written as
= Q(f, f )(w, t)
(
I
∂f
∂t
where Q is denoted as collisional operator. According to (
Denoting as (w, v) the opinions of two agents before their
interaction, we assume that the following model holds
w0 = w + γ(v − w) + ηD(|w|)
v0 = v + γ(w − v) + η∗D(|v|)
(
Observe that the contribution of the compromise is
proportional to the difference between the two pre-interaction
opinions. Taking, for instance, the first equation in (
Concerning the diffusion term, we assume that function D(·) depends on the absolute value of the opinion, meaning that the propensity of changing opinion is symmetrical with respect to 0 (namely, with respect to the middle point of I). Moreover, we assume that D(·) is non increasing with respect to the absolute value of the opinion, coherently with the fact that, typically, extremal opinions are more difficult to change. Finally, we assume that 0 ≤ D(|w|) ≤ 1 for all w ∈ I. According to such assumptions, the contribution of diffusion can be either positive or negative depending on the value of η and η∗. In the following, we denote the probability density function of η and η∗ as ϑ(·) and we assume that Such a choice corresponds to considering η and η∗ as 0 mean random variable with standard deviation σ.
The effects of diffusion in the opinion evolution are taken into account through the transition rate, which is defined as
W (w, v, w0, v0) = ϑ(η)ϑ(η∗)χI (w0)χI (v0)
where χI is the indicator function of set I (equals to 1 if its
argument belongs to I, and to 0 otherwise). The indicator
function in (
Now that we have completed the definition of the law
that describe each single interaction, we can write the explicit
(
Z Z
B2 I 0W 1 J
f (0w)f (0v) − W f (w)f (v) dvdηdη∗ where B is the support of ϑ, 0w and 0v are the pre-interaction variables which generate w and v, respectively, 0W is the transition rate and J is the Jacobian of the transformation of (0w,0 v) in (w, v). The two addends in the previous equation represent the gain and the loss of agents in dw, respectively [9].
In order to study the opinion evolution, we need to
introduce the weak form of the Boltzmann equation. Generally
speaking, the weak form of a differential equation is obtained
by multiplying both sides by a test function, namely a smooth
function with compact support, and integrating. The weak form
of the Boltzmann equation can then be found by multiplying
both sides of (
B2 I2 f (w, t)φ(w)dw =
W f (w)f (v)(φ(w0) − φ(w))dwdvdηdη∗.
Setting φ(w) = 1 in (
f (w, t)dw = 0.
Such an equality corresponds to the fact that the number of agents is time invariant. This property is also found in classic kinetic theory and it corresponds to mass conservation.
Considering φ(w) = w as a test function and using (
Z Z γ +
B2 I2 Z Z
B2 I2 f (w, t)wdw =
Z
I Denoting as u(t) the average value of the opinion at time t, namely u(t) =
f (w, t)w dw
the left hand side of (
We are now interested in studying the asymptotic behaviour
of the distribution function f (w, t). For this reason, in order
to simplify notation, let us define a new temporal variable
τ = γt
(
((w − u)g) ∂ ∂w where
λ = σ2/γ.
Equation (
We are now interested in studying stationary solutions of this equation, namely those which satisfy dg
= 0. (
III.
RESULTS
In this section we derive relevant stationary profiles for the
opinion density g. Such profiles are defined as solutions of (
With this choice of the diffusion function, equations (
In order to guarantee that post-collisional opinions still belong to the considered interval I, we need to set the support of ϑ to
B = (−(1 − γ), 1 − γ).
|w0| ≤ (1 − γ)|w| + γ|v| + |η|(1 − |w|)
≤ (1 − γ)|w| + γ + |η|(1 − |w|)
and if, according to (
|w0| ≤ (1 − γ)|w| + γ + (1 − γ)(1 − |w|) = 1 Hence, we can conclude that if η ∈ B, then w0 ∈ I. Analogous results can be derived for v0.
Now, if we substitute the expression of D defined in (
Z v2
−v1 (1 − |w|)2g dw + (w − m)gdw = C(v2 + v1) from which, assuming that v1 → 1 and v2 → 1 one obtains 0 + m − m = 2C which corresponds to C = 0.
Let us start by considering w > 0, so that (20) can be written as λ (1 − w)2 ∂g + [(w − m) + λ(w − 1)] g = 0. (21) 2 ∂w Dividing both sides by g one obtains g0 g =
2 (1 − w) + 2(m − w) λ(1 − w)2 and observing that 2(m − w) d λ(1 − w)2 = dw
2 − λ log(1 − w) + 2(m − 1) λ(1 − w) equation (22) can be written as (log g(w))0 =
log(1 − w)−2− λ2 + where we have used the facts that 2(m − 1) 0 λ(1 − w) g0 g 2 (1 − w) = (log g(w))0 = −2(log(1 − w))0.
Integrating (24) and applying the exponential function, one finally obtains the following expression for the stationary profile g∞(w) = c˜u,λ(1 − |w|)−2− λ2 exp 2(m − 1) λ(1 − |w|) where c˜u,λ is a normalization constant that depends on the average opinion u and on λ.
Let us now consider w < 0 so that equation (20) becomes λ (1 + w)2 ∂g + [(w − m) + λ(w + 1)] g = 0.
2 ∂w Dividing both sides by g leads to g0 2 g = − (1 + w) + 2(m − w) λ(1 + w)2 and by applying analogous calculation to the case with w > 0 one obtains (log g(w))0 = 2(m + 1) 0 log(1 + w)−2− λ2 − λ(1 + w)
Since g∞ is the solution of a differential equation it must be continuous. From (26) and (30) it is evident that g∞ is continuous for w > 0 and w < 0. Imposing that g∞ is also continuous in w = 0, the following equality needs to hold c˜u,λ exp = cˆu,λ exp −2u λ 2(1 − uw) g∞(w) = cu,λ(1 − |w|)−2− λ2 exp − λ(1 − |w|) where cu,λ is the quantity in (31) and it needs to be determined in order to ensure
g∞(w) = 1.
Observe that g∞ is piecewise C1 and it is non-differentiable in w = 0 (as the function D). Moreover, the solution is symmetric if we change w and u with −w and −u, namely
g∞(w; u, λ) = g∞(−w; −u, λ).
If u = 0, from (34) we can conclude that g∞ is an even function. Moreover, using a change of variable for negative values of w, the integral of g∞ can be written as 2u λ Z
I and, using the change of variable t = λ(1−−2w) , the previous integral can be expressed as Finally, introducing the incomplete gamma function defined as the value of c0,λ which satisfies (33) is then c0,λ = 2 λ λ2 +1 2 Γ 2 λ + 1, 2 λ −1 .
The case with u = 0 is the only one where the value of cu,λ can be found analytically. Other cases can be studied numerically.
We are now interested in studying the derivative of g∞ in order to find singular points which correspond to maximum or minimum points. Deriving (26) it can be shown that if w > 0 g∞0(w) = 0 ⇐⇒ 2λ(1 − w) + 2(1 − w) + 2(u − 1) = 0. Hence the (unique) singular point is and it is positive if and only if λ > −u. Deriving (30), instead, it can be shown that if w < 0
g∞0(w) = 0 ⇐⇒ 2λ(1 + w) + 2(1 + w) + 2(u + 1) = 0 leading to the following singular point w = u + λ λ + 1 u − λ w = .
λ + 1 (31) (32) (33) (34) (35) (36) (37) 1.4 1.2 1 Observe that this value is negative if and only if λ > u. Finally, the following cases can be considered if u = 0 then g∞0(w) = 0 in two points tλhat are symmetric with respect to 0, namely w = ± λ+1 if u > 0 - if 0 < λ < u then g0 (w) = 0 in a unique point, namely w = uλ++∞λ1 - if λ > u then g0 (w) = 0 in two points, namely w = uλ±+λ1 ∞ if u < 0 - if 0 < λ < −u then g0 (w) = 0 in a unique point, namely w = uλ−+λ1∞ - if λ > −u then g∞0(w) = 0 in two points, namely w = uλ±+λ1 Observe that simple manipulations shows that wl→im0+ g∞0(w) > 0 lim g0 (w) < 0 w→0− ∞ so that w = 0, which is a non-differentiable point, can be considered as a point of minimum.
IV.
In this section, relevant numerical results are shown for stationary profiles for different values of u and λ. We focus on values of u ≥ 0 as the stationary profiles relative to negative values of u can be obtained by symmetry, according to (34). The constant cu,λ, which appears in g∞, is evaluated numerically, using Newton-Cotes formulas [12].
First, we assume that u = 0 so that the average opinion corresponds to the middle point of I. As already observed in the previous section, if u = 0 then g∞ is symmetric with respect to 0 and it has two maxima at λ w = ± λ + 1 . (38) 2.5 2 As λ increases, such points get nearer to the extremal values of I, namely to extremal opinions. Observe that increasing the value of λ corresponds to incrementing the impact of diffusion with respect to compromise.
Fig. 1 shows the stationary profiles g∞(w) when u = 0 for different values of λ. As expected from (34), the function g∞(w) is symmetric with respect to 0 and it has a minimum in correspondence of w = 0 and two maxima whose values depend on λ. The stationary profiles g∞(w) in Fig. 1 correspond to λ = 1/3 (blue line), λ = 1 (red line), and λ = 3 (green line). If λ = 1/3 the two maxima are in correspondence of w = ±1/4. Observe that in this case extremal distributions are associated with a very low probability. If λ = 1, instead, the maxima correspond to w = ±1/2; while in λ = 3 they correspond to w = ±3/4.
Therefore, it can be concluded that as λ increases the points of maximum move towards the extremes of the considered interval I. Observe that, according to its definition, any increase of λ corresponds to assuming that the contribution of diffusion is more relevant than that of compromise. Moreover, according to the results in Fig. 1, any increase of λ leads to stationary profiles with small values in correspondence of opinions in the middle of the interval I.
In Fig. 2 the stationary profiles g∞(w) are shown when considering as average opinion the value u = 1/4. In this case, the function g∞(w) is not symmetric and, as expected, it has a local minimum in w = 0. We consider the same values of λ as in the previous case. For each of these values, the number of maxima is two, since the condition λ > u is always satisfied. If λ = 1/3 the positive maximum point is w = 7/16 and the negative one is w = −1/16. While the negative maximum is near the middle of the interval I, the positive one is farther. In Fig. 2 the stationary profile g∞(w) obtained with λ = 1/3 is shown (blue line) and it can be observed that the value of the maximum in w = 7/16 is far more significant than that corresponding to w = −1/16, namely the positive opinions are far more likely than the negative ones. This is in agreement with the fact that the average opinion u is positive. If λ = 1 the maxima correspond to w = ±5/8 and w = ∓3/8, as shown in Fig. 2 (red line). Finally, Fig. 2 also shows the results obtained with λ = 3 (green line). In this case the maximum points are w = ±13/16 and w = ∓11/16. As in the previous case, the points of maximum get nearer to ±1, namely to the extremes of I, as λ increases. Moreover, observe that largest values of λ correspond to increasing the likelihood of negative opinions.
Let us now increase the value of the average opinion to u = 1/2. The stationary profiles g∞(w) are shown in Fig. 3 for λ = 1/3 (blue line), λ = 1 (red line), and λ = 3 (green line). Observe that if λ = 1/3 the function g∞(w) has only one maximum, namely the positive one. As a matter of fact, according the results in the previous section, the negative one only exists if λ > u. The maximum point is w = 5/8. This value is greater than the one obtained for the same λ in the case u = 1/4, accordingly with the fact that, in this case, we consider a higher average opinion u. When considering λ = 1, the function g∞(w) has two maxima since the condition λ > u is satisfied. Such points are w = 3/4 and w = −1/4. As in Fig. 2, the value of the maximum corresponding to the negative value of w is less significant with respect to that relative to the positive value of w. If λ = 3 the two maxima correspond to w = 5/8 and w = −7/8 and they are nearer to the extremes of I with respect to those obtained with lower λ. A comparison of the results in Fig. 2 with those in Fig. 3 shows that in the latter the values of the positive maxima are greater while those of the negative maxima are smaller.
Finally, we consider a greater value of the average opinion, namely u = 3/4. This corresponds to considering an extremist society. Fig. 4 shows the stationary profiles for λ = 1/3 (blue line), λ = 1 (red line), and λ = 3 (green line). As in the previous case, since λ < u, the profile g∞(w) has only one maximum if λ = 1/3. The point of maximum is w = 13/16 and it is closer to 1 than the other points of maximum obtained with the same λ for lower values of the average opinion u. From Fig. 4 it can be shown that, once again, the positive maximum point moves towards the extreme 1 as λ increases, since it corresponds to w = 7/8 if λ = 1 and to w = 15/16 7 6 5 )4 w ( ∞ g3 2 1 0 −1 Fig. 4. Stationary profiles g ∞ for u = 3/4 and λ = 1/3 (blue line), λ = 1 (red line), and λ = 3 (green line) if λ = 3. Moreover, Fig. 4 shows that an increase of the value of the average opinion significantly reduces the value of the negative maximum, which in this case is negligible. This result is not surprising since if the average opinion is near the positive extreme, then the number of agents with negative opinion has to be small.
V.
CONCLUSIONS
In this paper we discussed the opinion dynamics in a multi-agent system through a kinetic approach. We considered an opinion evolution model inspired from the interactions of molecules in a gas and we studied, from an analytic point of view, the asymptotic behaviour of the opinion distribution. Assuming that the opinion of an agent can change each time it interacts with any other agent because of compromise and diffusion, we showed that the average opinion of the system is conserved. The stationary profiles can have different characteristics, depending on the parameters of the model and on the explicit expressions of the function which represents the diffusion. For a particular diffusion function, we showed that the asymptotic distribution is characterized by one or two maxima, depending on the parameters of the model. The stationary profiles are shown for different values of the average opinion and for different parameters of the model.
We recognize in kinetic models the possibility of describing complex and decentralized systems that can exhibit interesting emergent behaviours. In particular, we are mainly interested in using kinetic models as a conceptual framework that captures essential characteristics of opinion formation in multi-agent systems, and to adopt it in the design of mobile scenarios that would eventually use general-purpose industrial strength technology (see, e.g., [13], [14]). Moreover, we recognize that kinetic models can be effectively used to model agent-based cooperation (like the ones discussed in, e.g., [15]), and that they can be used to study large scale systems (like the ones discussed in, e.g., [16]). Finally, we are interested in modeling the emergent behaviors of wireless sensor networks used to support accurate localization (see, e.g., [17], [18]).
Further investigation on this subject is currently under development. In particular, we are interested in deriving the explicit expressions of the stationary profiles with a different choice of the diffusion function. We aim at studying the properties of such stationary profiles for different parameters of the model. At the same time, we are studying the application of kinetic models to multi-agents systems also from a simulative point of view. More precisely, we are interested in comparing analytic results with simulation experiments and in studying the number of iterations necessary to approximate to a certain degree an analytic stationary profile.