-This paper studies analytically the dynamics of the opinion in multi-agent systems where two classes of agents coexist. The fact that the population of considered multi-agent systems is divided into two classes is meant to account for agents with different propensity to change their opinions. Skeptical agents, which are agents that are not inclined to change their opinions, can be modeled together with moderate agents, which are agents that are moderately open to take into account the opinions of the others. The studied analytic model of opinion dynamics involves only compromise, which describes how agents change their opinions to try to reach consensus. The adopted model of compromise is stochastic in order to give agents some level of autonomy in their decisions. Presented results show, analytically, that after a sufficient number of interactions consensus is reached, regardless of the initial distribution of the opinion. Analytic results are confirmed by simulations shown in the last part of the paper.
I. INTRODUCTION
This paper discusses an analytic framework that can be used to study collective and asymptotic properties of multiagent systems. The properties of studied multi-agent systems evolve because of binary interactions among agents, where the term (binary) interaction is used here to denote a message exchange among two agents. Each interaction identifies a single step of the evolution of the systems, regardless of how often interactions occur. Note that studied multi-agent systems are completely decentralized and that they involve no form of supervised coordination. In particular, in this paper we assume that each agent is associated with a scalar property, which changes because of interactions with other agents, and we assume that such a property represents the opinion of the agent on a given topic. Notably, we remark that the proposed approach can be used to study other collective and asymptotic properties of multi-agent systems, and that it is not limited to the study of the opinion, even if the development of specific analytic results is needed if different properties are considered.
We start by considering proper rules that describe the effects of interactions among agents on their opinions at a microscopic level. Then, the dynamics of macroscopic properties of the considered multi-agent system are analytically derived, taking inspiration from physical models. In detail, the proposed framework is related to the kinetic theory of gases, a branch of physics which studies the temporal evolution of macroscopic properties of gases starting from the description of microscopic collisions among molecules. The idea of generalizing the framework of kinetic theory of gases to study social phenomena relies on a proper parallelism between molecules and their collisions in gases, and agents and their interactions in multi-agent systems. Such an idea is not new and it recently gave birth to a discipline called sociophysics (see, e.g., [1]). Note that the details of collisions among molecules are different from the details of interactions among agents and, hence, analytic results obtained at the macroscopic level for the dynamics of the properties of multi-agent systems are significantly different from those of kinetic theory of gases. For the specific case of the study of opinion dynamics, the major advantage of the proposed approach consists in the derivation of analytic results, in contrast with simulation results that are typically studied in the literature on the subject.
While the validity of simulations depends on the specicfi tool adopted, and on the choice of the parameters of simulated models, analytic results are valid as long as the hypothesis used to derive them are valid.
We have already applied the proposed approach to the study of various models of opinion dynamics (see, [2]–[9]), and analytic results were always confirmed by independent simulations. In this paper, we enrich previous studies on the subject by considering multi-agent systems where two types of agents, grouped into two disjoint classes, coexist. Each class is associated with different parameters, such as a different number of agents, a different initial distribution of the opinion, and a different inclination of contained agents at changing their opinions. Results concerning a similar model, where only deterministic interactions among agents are considered, have already been presented in [10]. Here, we extend those results by adding stochastic parameters in order to describe the behavior of agents that exhibit some level of autonomy.
Moreover, we concentrate on the impact of the presence of skeptical agents in the multi-agent system. Note that, in order to account for the existence of two classes of agents, we take inspiration from kinetic theory of gas mixtures, which is used to study gases composed of different types of molecules.
This paper is organized as follows. Section II describes the details of the proposed kinetic framework. Section III derives relevant analytic results concerning the average opinion in the multi-agent system. Section IV shows simulation results that are used to confirm analytic results. Finally, Section V concludes the paper.
II. A KINETIC FRAMEWORK OF OPINION DYNAMICS
In kinetic theory of gases, each molecule of a gas is associated with specific physical quantities, such as its position and its velocity, and the macroscopic features of gases, concerning, for example, its temperature and pressure, are derived on the basis of a proper balance equation known as Boltzmann equation. Similarly, in the context of opinion dynamics, we assume that each agent is associated with a single scalar value which models its opinion, and macroscopic characteristics of the considered multi-agent system, such as the average opinion and the standard deviation of the opinion, are studied analytically using a proper balance equation, whose formulation is inspired by the Boltzmann equation.
Opinion is modeled as a continuous variable v, which is defined in a closed interval Iv. Without loss of generality [11], interval Iv is typically set to
Iv = [−1, 1] where values close to 0 represent moderate opinions while values close to −1 or to +1 correspond to extremal opinions.
The first step to derive analytic results consists in the definition of the microscopic rules that govern the effects of an interaction among two agents. As detailed in the introduction, we assume that two classes of agents characterize the considered multi-agent system. Let us consider an interaction between an agent of class s ∈ {1, 2} and an agent of class r ∈ {1, 2}. Denoting as v the pre-interaction opinion of the agent of class s, and as w the pre-interaction opinion of the agent of class r, we assume that the post-interaction opinions of the two interacting agents can be computed 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, and {Γsr}s2,r=1 are four mutually independent random variables.
Assuming that the support of random variables {Γsr}s2,r=1
is a subset of (
Iv ns(t) = fs(v, t)dv
We also denote the total number of agents at time t as n(t), which can be computed as
n(t) = n1(t) + n2(t).
(
This phenomenon can be reproduced in terms of the following
inequalities
Simple algebraic manipulations show that a sufficient
condition for (
The expected value of the sum of post-interaction opinions
of two interacting agents can be computed from (
E[v∗ + w∗] = v + w + (Γrs − Γsr)(v − w)
(
v∗ − w∗ = εrs(v − w).
where εrs = 1−(Γrs+Γsr) and, according to (
From (
We now describe the kinetic framework which allows
deriving analytic results starting from microscopic rules (
Observe that, using this notation, the number of agents of class s at time t, denoted as ns(t), can be computed as |v∗ − v| < |v∗ − w| |w∗ − w| < |w∗ − v|.
IΓ = 0,
.
(
Is =
2 X Qsr(fs, fr) r=1 s ∈ {1, 2}
s ∈ {1, 2},
and Qsr(fs, fr) is a proper operator that depends on
distribution functions fs(v, t) and fr(v, t). Equation (
Hence, the explicit expression of the collisional operator (
In order to obtain analytic results, the weak form of the
Boltzmann equation needs to be considered. Just like in kinetic
theory of gas mixtures, the weak form of the Boltzmann
equation is obtained by multiplying (
βΘsr(Γsr)Θrs(Γrs) · fs(v)fr(w)(φ(v∗(v, w)) − φ(v))dvdwdΓrsdΓsr X2 Z
Z Z
In this section, we show how the weak form of the
Boltzmann equation can be used to derive collective and
asymptotic properties of considered multi-agent systems. First,
we set φ(v) = 1, so that the weak form of the Boltzmann
equation with respect to such a test function becomes
d Z
dt Iv
fs(v, t)dv = 0
Recalling (
Let us now consider the test function φ(v) = v in (
X2 Z
Z Z
Z r=1 Iv Iv Bsr Brs
βΘsr(Γsr)Θrs(Γrs) · fs(v)fr(w)Γsr(w − v)dvdwdΓrsdΓsr.
Observe that (
s ∈ {1, 2}.
Note that the average opinion of the entire multi-agent system is computed as the following weighed sum of average opinions
1 u(t) =
n
Using (
(n1u1(t) + n2u2(t)) .
d 2 ns dt us(t) = βXΓsr r=1
Z
Iv2
fs(v)fr(w)(w − v)dvdw. (18)
(
From (18), it is evident that also its right-hand side can be expressed in terms of average opinions. Simple algebraic manipulations show that the following equalities hold d 2 dt us(t) = βXΓsrnr (ur(t) − us(t)) r=1 s ∈ {1, 2}. (19) 1) Θsr(·) and Θrs(·) are the distributions of random vari
ables Γsr and Γrs; The two equations (19) form a homogeneous system of
2) Bsr and Brs are the support of Γsr and Γrs, respec- first-order linear differential equations, which can be solved
tively; and analytically. In order to simplify notation, let us introduce the
3) β is the probability that two agents interact. following parameters
Note that using the fact that the integral with respect to v and
the derivative with respect to t commute, the left-hand side of a1 = βΓ12n2 a2 = βΓ21n1. (20)
(
(
Introducing the auxiliary function ( u˙ 1(t) = −a1(u1(t) − u2(t)) u˙ 2(t) = a2(u1(t) − u2(t)).
x(t) = u1(t) − u2(t)
(
a2 a1 + a2
a1 a1 + a2 where C is a constant. Recalling (22), the following relation between u1(t) and u2(t) can be found
u1(t) = u2(t) + Ce−(a1+a2)t.
Using this result in the second equation of system (21), the following differential equation for u2(t) is found
u˙ 2(t) = Ca2e−(a1+a2)t and, hence, u2(t) can be expressed as u2(t) = −C
e−(a1+a2)t + K, where K is a constant. Finally, inserting the expression of u2(t) in (25), we obtain u1(t) = C
e−(a1+a2)t + K where K is the same constant used in (27). The two constants C and K can be found by imposing that initial conditions are satisfied. Simple algebraic manipulations show that
K = u1(0) C = u1(0) − u2(0)
a2 a1 + a2 + u2(0)
a1 a1 + a2 where {us(0)}s2=1 are the initial average values of the opinions of the two classes of agents.
Therefore, it can be concluded that the solution of (21) can be expressed in closed form as u1(t) = C u2(t) = −C
a1 a1 + a2
a2 a1 + a2 e−(a1+a2)t + K
e−(a1+a2)t + K where C and K are computed in (29) using the initial distributions of the opinion of the two classes of agents. From (30), it can be observed that since a1 > 0 and a2 > 0, the following equalities hold
lim u1(t) = lim u2(t) = K.
t→+∞ t→+∞ Observe that, according to (20) and (29), K depends on the average initial opinions {us(0)}s2=1, on the number of agents {ns}s2=1 in each class, and on Γ12 and Γ21.
TABLE I THE CONSIDERED VALUES OF THE PARAMETERS FOR THE TWO CLASSES OF AGENTS IN SIMULATIONS: NUMBER OF AGENTS, n1 AND n2; INITIAL
DISTRIBUTIONS OF THE OPINION, f1(v, 0) AND f2(v, 0); AND
DISTRIBUTION OF RANDOM VARIABLES, Γ12 AND Γ21. n1 n2 f1(v, 0) f2(v, 0) Γ12
Γ21
In this section, we show analytic results obtained according
to the framework outlined in previous sections for proper
choices of parameters. In order to confirm the validity of
such results, we compare them against those obtained by
simulating a system composed of 103 agents, which interact
according to (
First, we consider the parameters shown in the first row of Table I. In this case, n1 = 900 and n2 = 100, meaning that 90% of the agents belong to class 1 and only 10% of the agents belong to class 2. Initial opinions of agents of class 1 are uniformly distributed in interval Iv, as shown in the third column. Therefore, the initial average opinion of agents of class 1 is 0. Initial opinions of agents of class 2, instead, are uniformly distributed in the smaller interval (3/4; 1). This choice implies that agents of class 2 have extremal opinions and that their initial average opinion is u2(0) = 7/8. Another feature that distinguishes the agents in the two classes concerns their inclination to change opinion.
The distributions of random variables Γ12 and Γ21 are related to such an inclination. As shown in Table I, we assume that Γ12 has uniform distribution in interval (0, 2/10), corresponding to an average value Γ¯12 of 1/10, and that Γ21 has uniform distribution in interval (0, 2/100), corresponding to an average value Γ¯21 of 1/100. According to such choices, Γ¯12 = 10Γ¯21, which means that the propensity of agents of class 2 to change their opinions in favor of those of agents of class 1 is much lower than the propensity of agents of class 1 to change their opinions in favor of those of agents of class 2. For this reason, agents of class 2 can be considered skeptical.
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 (31), u1(t) and u2(t) converge to the same value, which, according to this choice of parameters, corresponds to K ' 0.46. Fig. 1 also shows the values of {u˜s(t)}s2=1 obtained by simulation (dashed black lines), showing that analytic results are in agreement with those obtained by simulation.
We now consider the parameters shown in the second row of Table I, which differ from those in the first row only for {ns}s=1. The initial distributions of opinion {fs(v, t)}s2=1
2 and, hence, the initial values of the average opinion of the two classes of agents, are the same as in the first scenario.
This means that agents of class 2 still have extremal opinions.
Also the distributions of random variable Γ12 and Γ21 are the same as in the first scenario, so that agents of class 2 can still be considered skeptical. The only difference is that, in this case, skeptical agents are only 1%. In fact, 990 agents, corresponding to 99% of the total number of agents, belong to class 1, and only 10 agents, corresponding to 1% of the total number of agents, belong to class 2. Fig. 2 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 (31), the values of u1(t) and u2(t) converge to the same value, which now corresponds to K ' 0.08. The values of u˜1(t) and u˜2(t) (dotted black lines) obtained by simulation with the parameters in the second row of Table I are also shown. As in the first scenario, simulation results are in agreement with analytic ones.
In order to improve the analysis of considered scenarios, Fig. 3 shows the distributions f1(v, t) (solid blue lines) and f2(v, t) (dashed red lines) of the opinions of the two classes of agents obtained by simulating the multi-agent system with the parameters in the first row of Table I. In detail: 1) Fig. 3(a) shows the distributions fs(v, t) after 5 · 104 interactions, which corresponds to 100 interactions per agent on average; 2) Fig. 3(b) shows the distributions fs(v, t) after 105 interactions, which corresponds to 200 interactions per agent on average; 3) Fig. 3(c) shows the distributions fs(v, t) after 1.5 · 105 interactions, which corresponds to 300 interactions per agent on average; and 4) Fig. 3(d) shows the distributions fs(v, t) after 2 · 105 interactions, which corresponds to 400 interactions per agent on average.
From Fig. 3 it can be observed that not only the average opinions us(t) converge to the same value K ' 0.46, as already shown in Fig. 1, but also that consensus among agents is reached, since the opinions of all agents tend to the same value, as expected from the analytic model. Similar diagrams can be drawn using the parameters in the second row of Table I, obtaining similar results.
V. CONCLUSIONS
This paper presented an analytic model of opinion dynamics which assumes that the agents in the studied multi-agent system can be grouped into two classes on the basis of their characteristic parameters. Such classes can be used to model agents with different propensity to change their opinions after interactions, and they are used here to model the presence of skeptical agents in the multi-agent system. Among the sociological phenomena that can be used to describe the dynamics of the opinion, the presented model considers only compromise, which is the phenomenon that describes interactions that tend to consensus. The characteristic autonomy of agents is modeled in terms of random variables used at the microscopic level to perturb the classic model of compromise, which is deterministic. Analytic results ensure that the average opinion of the multi-agent system is conserved, and that consensus is always reached for a sufficiently large number of interactions. Simulations shown in the last part of the paper confirm such properties.
Ongoing research involves the extension of the presented model to account for major sociological phenomena (see, 10 8 0.5 0.5 0.5 Fig. 3. The distributions f1(v, t) (blue line) and f2(v, t) (red line) obtained by simulation using the parameters in the first row of Table I for: (a) t = 5 · 104; (b) t = 105; (c) t = 1.5 · 105; (d) t = 2 · 105. e.g., [8]). Among considered phenomena, ongoing research includes stochastic models for: • Diffusion, the phenomenon according to which the opin
ion of agents is influenced by the social context [13]; • Homophily, the process according to which agents interact
only with those with similar opinions [14]; • Negative influence, the idea according to which agents evaluate their peers, and they only interact with those with positive scores [15]; • Opinion noise, the process according to which a random additive variable may lead to arbitrary opinion changes with small probability [16]; and • Striving for uniqueness, the phenomenon according to
which agents want to distinguish from others [17].
Preliminary results on the deterministic study of such phenomena for multi-agent systems with multiple classes of agents are encouraging (see, e.g., [9]), and they show that major collective and asymptotic properties of multi-agent systems can be fruitfully studied analytically.