=Paper=
{{Paper
|id=Vol-1382/paper15
|storemode=property
|title=Simulations of Opinion Formation in Multi-Agent Systems Using Kinetic Theory
|pdfUrl=https://ceur-ws.org/Vol-1382/paper15.pdf
|volume=Vol-1382
|dblpUrl=https://dblp.org/rec/conf/woa/MonicaB15
}}
==Simulations of Opinion Formation in Multi-Agent Systems Using Kinetic Theory==
https://ceur-ws.org/Vol-1382/paper15.pdf
Proc. of the 16th Workshop “From Object to Agents” (WOA15) June 17-19, Naples, Italy
Simulations of Opinion Formation in
Multi-Agent Systems using Kinetic Theory
Stefania Monica, Federico Bergenti
Dipartimento di Matematica e Informatica
Parco Area delle Scienze 53/A, 43124 Parma, Italy
Email: {stefania.monica, federico.bergenti}@unipr.it
Abstract—In this paper we formulate the problem of opinion numerically the opinion evolution according to a model intro-
formation using a physical metaphore. We consider a multi- duced in [9], which mandates that each agent can change its
agent system where each agent is associated with an opinion own opinion because of two different reasons [10]. The first
and interacts with any other agent. Interpreting the agents as reason is related to compromise between interacting agents.
the molecules of a gas, we model the evolution of opinion in More precisely, the model assumes that both agents involved
the system according to a kinetic model based on the analysis
of interactions among agents. From a microscopic description
in an interaction can change their respective opinions in favor
of each interaction between two agents, we derive the stationary of that of the other agent. The second reason only involves
profiles of the opinion under given assumption. Results show that, each single agent and it is related to the fact that an agent
depending on the average opinion and on the parameters of the can change its opinion autonomously, giving rise to a process
model, different profiles can be found, but all stationary profiles known as diffusion.
are characterized by the presence just of one or two maxima.
Analytic results are confirmed by simulations shown in the last This paper is organized as follows. In Section II, we
part of the paper. 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
I. I NTRODUCTION results. Section V concludes the paper.
In this paper we describe a model for opinion formation II. K INETIC F ORMULATION OF O PINION F ORMATION
among agents. We assume that each agent is associated with
an opinion v ∈ I ⊆ R and that it can change its opinion Kinetic theory of gases describes, from a microscopic
each time it interacts with another agent. In the literature, point of view, the effects of interactions among molecules in
various approaches that study opinion evolution in a society gases. By reinterpreting the molecules of a gas as agents, one
of agents have been proposed, and many of them are based can use the kinetic framework to describe social interactions
on Cellular Automata (CA) because CA describe well global among agents. While molecules are typically associated with
effects of local phenomena (see, e.g., [1]). In order to overcome relevant physical properties, like their velocities, agents can be
the synchronism that CA assume, in recent years the use of associated with attributes that represent some of their charac-
microscopic models inspired from physics has been introduced teristics. In particular, since in this work we are interested in
to describe asynchronous social interactions among agents in a modeling the evolution of the opinion, we assume that each
society [2]. Such models are based on the idea that the laws of agent is associated with a scalar parameter v which represents
kinetic theory, which are typically used to describe the effects its opinion and which is defined in a given interval I. In
of interactions between two molecules of a gas, can also be the following, we consider I = [−1, 1], where ±1 represent
used to model interactions between two agents. extremal opinions.
Statistical mechanics and kinetic theory describe the details In order to use the kinetic approach, we need to define
of each interaction between two molecules in a gas but they a function, denoted as f (w, t), which represents the density
also allow finding classic laws which describe macroscopic of opinion w at time t, and which is defined for each opinion
properties of gases [3]. Analogously, from the microscopic w ∈ I and for each time t ≥ 0. According to such a definition,
Z
laws which describe the details of each interaction between two
agents, collective behaviour can be described from a macro- f (w, t)dw = 1. (1)
I
scopic point of view [4]. All the research challenges related to
the application of kinetic and statistical formalisms to describe In order to formulate the problem of opinion evolution in
multi-agent systems gave rise to new disciplines known as kinetic terms, we assume that the function f (w, t) evolves
econophysics and sociophysics [5]. Such new disciplines have on the basis of the Boltzmann equation, which, under our
been used to describe, e.g., wealth evolution [6] and market assumptions, can be written as
economy [7], and they have also been adopted to characterize ∂f
opinion evolution in a society [8]. = Q(f, f )(w, t) (2)
∂t
In this paper, we focus on a model for opinion formation where Q is denoted as collisional operator. According to (2),
based on kinetic theory of gases and we analyze the evolution the temporal evolution of the opinion density is governed by
of the opinion in a society of agents. In particular, we analyze the collisional operator Q, whose explicit formulation depends
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Proc. of the 16th Workshop “From Object to Agents” (WOA15) June 17-19, Naples, Italy
on the details of binary interactions between any pairs of expression of the collisional operator Q used in (2), which is
agents. Before deriving a formula for Q, let us describe the given by
effects of each binary interaction. Z Z
0 1 0
Q(f, f ) = W f ( w)f (0 v) − W f (w)f (v) dvdηdη∗
Denoting as (w, v) the opinions of two agents before their B2 I J
interaction, we assume that the following model holds
0 where B is the support of ϑ, 0 w and 0 v are the pre-interaction
w = w + γ(v − w) + ηD(|w|) variables which generate w and v, respectively, 0 W is the
(3) transition rate and J is the Jacobian of the transformation of
v 0 = v + γ(w − v) + η∗ D(|v|)
(0 w,0 v) in (w, v). The two addends in the previous equation
where: (w0 , v 0 ) are the post-interaction opinions of the two represent the gain and the loss of agents in dw, respectively [9].
agents; γ is a constant defined in (0, 12 ); η and η∗ are two
independent random variables with the same statistics; and In order to study the opinion evolution, we need to in-
D(·) is a function that describes the impact of diffusion in the troduce the weak form of the Boltzmann equation. Generally
considered interaction [9]. From (3) it can be observed that speaking, the weak form of a differential equation is obtained
the post-interaction opinions of the two agents are obtained by multiplying both sides by a test function, namely a smooth
by adding to their pre-interaction opinions two terms: the first function with compact support, and integrating. The weak form
one is related to compromise, while the second one is related of the Boltzmann equation can then be found by multiplying
to diffusion through function D(·). both sides of (2) by a test function φ(w) and integrating with
respect to w. Using a proper change of variable in the integral,
Observe that the contribution of the compromise is pro- the weak form of the Boltzmann equation can be written as:
portional to the difference between the two pre-interaction Z
opinions. Taking, for instance, the first equation in (3), we can d
f (w, t)φ(w)dw =
conclude that the second addend on the right hand side of (3) is dt I
Z Z (6)
positive if v > w, so that the opinion of the considered agent
W f (w)f (v)(φ(w0 ) − φ(w))dwdvdηdη∗ .
(whose pre-interaction opinion is w) increases if it interacts B2 I2
with an agent with greater opinion. At the opposite, if w > v
the second addend is negative, so that the contribution of Setting φ(w) = 1 in (6) leads to
compromise decreases the opinion of the considered agent Z
(towards that of the agent it interacts with). Observe that the d
f (w, t)dw = 0. (7)
contribution of compromise is negligible if γ ' 0, while it dt I
becomes relevant as γ increases. Such an equality corresponds to the fact that the number of
Concerning the diffusion term, we assume that function agents is time invariant. This property is also found in classic
D(·) depends on the absolute value of the opinion, meaning kinetic theory and it corresponds to mass conservation.
that the propensity of changing opinion is symmetrical with Considering φ(w) = w as a test function and using (3) in
respect to 0 (namely, with respect to the middle point of I). (6) gives
Moreover, we assume that D(·) is non increasing with respect Z
to the absolute value of the opinion, coherently with the fact d
f (w, t)wdw =
that, typically, extremal opinions are more difficult to change. dt I
Z Z
Finally, we assume that 0 ≤ D(|w|) ≤ 1 for all w ∈ I.
According to such assumptions, the contribution of diffusion γ W f (w)f (v)(v − w)dwdvdηdη∗ (8)
can be either positive or negative depending on the value of ZB2 ZI 2
η and η∗ . In the following, we denote the probability density + W f (w)f (v)ηD(|w|)dwdvdηdη∗ .
function of η and η∗ as ϑ(·) and we assume that B2 I2
Z Z Denoting as u(t) the average value of the opinion at time t,
ηϑ(η)dη = η∗ ϑ(η∗ )dη∗ = 0 namely Z
Z Z (4) u(t) = f (w, t)w dw (9)
η 2 ϑ(η)dη = η∗2 ϑ(η∗ )dη∗ = σ 2 . I
the left hand side of (8) corresponds to the derivative u̇(t)
Such a choice corresponds to considering η and η∗ as 0 mean of the average opinion. Moreover, observe that the right hand
random variable with standard deviation σ. side term of (8) is 0. As a matter of fact, the first integral
The effects of diffusion in the opinion evolution are taken is 0 for symmetry reasons, while the second integral is 0
into account through the transition rate, which is defined as because, according to (4), the average value of ϑ is 0. From
(8) it can then be concluded that u̇(t) = 0, and, therefore,
W (w, v, w0 , v 0 ) = ϑ(η)ϑ(η∗ )χI (w0 )χI (v 0 ) (5) the average opinion is conserved, namely u(t) = u(0) = u.
This property corresponds to the conservation of momentum
where χI is the indicator function of set I (equals to 1 if its in kinetic theory.
argument belongs to I, and to 0 otherwise). The indicator func-
tion in (5) is meant to impose that post-interaction opinions We are now interested in studying the asymptotic behaviour
still belong to interval I [9]. of the distribution function f (w, t). For this reason, in order
to simplify notation, let us define a new temporal variable
Now that we have completed the definition of the law
that describe each single interaction, we can write the explicit τ = γt (10)
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Proc. of the 16th Workshop “From Object to Agents” (WOA15) June 17-19, Naples, Italy
where γ is the coefficient which appears in (3) and it is related where C is a constant which is necessarily 0. As a matter of
to compromise. Assuming that γ → 0, namely that each fact, by integrating (20) one obtains
interaction causes small changes of opinions, the function Z Z v2
λ v2 ∂ 2
(1−|w|) g dw + (w −m)gdw = C(v2 +v1 )
g(w, τ ) = f (w, t) (11) 2 −v1 ∂w −v1
describes the asymptotic behaviour of f (w, t). In [9] it is from which, assuming that v1 → 1 and v2 → 1 one obtains
shown that by substituting f (w, t) with g(w, τ ) in (6) and
0 + m − m = 2C
using a Taylor series expansion of φ(w) around w in (6) the
following equation of g can be derived which corresponds to C = 0.
2
dg λ ∂ ∂ Let us start by considering w > 0, so that (20) can be
= (D(|w|)2 g) + ((w − u)g) (12)
dτ 2 ∂w2 ∂w written as
where λ ∂g
(1 − w)2 + [(w − m) + λ(w − 1)] g = 0. (21)
λ = σ 2 /γ. (13) 2 ∂w
Equation (12) is known in the literature as the weak form of Dividing both sides by g one obtains
the Fokker-Planck equation [11]. g0 2 2(m − w)
= + (22)
We are now interested in studying stationary solutions of g (1 − w) λ(1 − w)2
this equation, namely those which satisfy and observing that
dg
= 0. (14) 2(m − w) d 2 2(m − 1)
dτ 2
= − log(1 − w) + (23)
λ(1 − w) dw λ λ(1 − w)
In the following, we denote such solutions as g∞ . In next sec-
equation (22) can be written as
tion we analyze such solutions for different diffusion functions
0
D(|w|) and for different values of the parameter λ. 0 2
−2− λ 2(m − 1)
(log g(w)) = log(1 − w) + (24)
λ(1 − w)
III. R ESULTS
where we have used the facts that
In this section we derive relevant stationary profiles for the g0 0
opinion density g. Such profiles are defined as solutions of (14) = (log g(w))
and, therefore, they depend on the parameters u and λ, which g
(25)
represent the average opinion and the ratio σ 2 /γ, respectively, 2
= −2(log(1 − w))0 .
and on the choice of the diffusion function D. Recalling the (1 − w)
initial assumptions on D, we develop our results considering Integrating (24) and applying the exponential function, one
D(|w|) = 1 − |w| w∈I (15) finally obtains the following expression for the stationary
profile
which is symmetrical with respect to 0 and decreasing in |w|.
2 2(m − 1)
With this choice of the diffusion function, equations (3) g∞ (w) = c̃u,λ (1 − |w|)−2− λ exp (26)
λ(1 − |w|)
become 0
w = w − γ(w − v) + η(1 − |w|) where c̃u,λ is a normalization constant that depends on the
(16) average opinion u and on λ.
v 0 = v − γ(v − w) + η∗ (1 − |v|).
In order to guarantee that post-collisional opinions still belong Let us now consider w < 0 so that equation (20) becomes
to the considered interval I, we need to set the support of ϑ λ ∂g
to (1 + w)2 + [(w − m) + λ(w + 1)] g = 0. (27)
2 ∂w
B = (−(1 − γ), 1 − γ). (17)
Dividing both sides by g leads to
As a matter of fact, from (16) g0 2 2(m − w)
0 =− + (28)
|w | ≤ (1 − γ)|w| + γ|v| + |η|(1 − |w|) g (1 + w) λ(1 + w)2
(18)
≤ (1 − γ)|w| + γ + |η|(1 − |w|) and by applying analogous calculation to the case with w > 0
and if, according to (17), |η| ≤ (1 − γ) then one obtains
0
|w0 | ≤ (1 − γ)|w| + γ + (1 − γ)(1 − |w|) = 1 (19) 0 −2− λ2 2(m + 1)
(log g(w)) = log(1 + w) − . (29)
λ(1 + w)
Hence, we can conclude that if η ∈ B, then w0 ∈ I. Analogous
results can be derived for v 0 . Integrating (29) leads to the following formula for the station-
ary profile
Now, if we substitute the expression of D defined in (15)
in (12) the stationary solution g∞ can then be found, according 2 −2(m + 1)
g∞ (w) = ĉu,λ (1 − |w|)−2− λ exp (30)
to (14), by solving the following partial differential equation. λ(1 − |w|)
λ ∂ where w has been substituted by −|w| and ĉu,λ is a normal-
(1 − |w|)2 g + (w − m)g = C (20) ization constant.
2 ∂w
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Proc. of the 16th Workshop “From Object to Agents” (WOA15) June 17-19, Naples, Italy
Since g∞ is the solution of a differential equation it must
1.4
be continuous. From (26) and (30) it is evident that g∞ is λ=1/3
continuous for w > 0 and w < 0. Imposing that g∞ is also λ=1
continuous in w = 0, the following equality needs to hold 1.2 λ=3
2u −2u
c̃u,λ exp = ĉu,λ exp . (31) 1
λ λ
Finally, the solution of (20) is
g ∞ (w )
0.8
2
−2− λ 2(1 − uw)
g∞ (w) = cu,λ (1 − |w|) exp − (32) 0.6
λ(1 − |w|)
where cu,λ is the quantity in (31) and it needs to be determined 0.4
in order to ensure Z
0.2
g∞ (w) = 1. (33)
I
0
Observe that g∞ is piecewise C 1 and it is non-differentiable in −1 −0.8 −0.6 −0.4 −0.2
w
0 0.2 0.4 0.6 0.8 1
w = 0 (as the function D). Moreover, the solution is symmetric
if we change w and u with −w and −u, namely
Fig. 1. Stationary profiles g∞ for u = 0 and λ = 1/3 (blue line), λ = 1
g∞ (w; u, λ) = g∞ (−w; −u, λ). (34) (red line), and λ = 3 (green line)
If u = 0, from (34) we can conclude that g∞ is an even
function. Moreover, using a change of variable for negative Observe that this value is negative if and only if λ > u. Finally,
values of w, the integral of g∞ can be written as the following cases can be considered
Z 1 Z 1
2
−2− λ −2 • if u = 0 then g∞ 0
(w) = 0 in two points that are
g∞ (w)dw = 2c0,λ (1 − w) exp dw symmetric with respect to 0, namely w = ± λ+1
λ
−1 0 λ(1 − w)
−2 • if u > 0
and, using the change of variable t = λ(1−w) , the previous
integral can be expressed as - if 0 < λ < u then g∞ 0
(w) = 0 in a unique
point, namely w = λ+1
u+λ
! λ2 +1 Z
λ +∞
2
- if λ > u then g∞ 0
(w) = 0 in two points,
2c0,λ t λ e−t dt. (35) namely w = u±λ
λ+1
2 2
λ
• if u < 0
Finally, introducing the incomplete gamma function defined as - if 0 < λ < −u then g∞0
(w) = 0 in a unique
Z +∞ point, namely w = λ+1
u−λ
Γ(x, a) = tx−1 e−t dt, (36) - if λ > −u then g∞ 0
(w) = 0 in two points,
a
namely w = u±λ
λ+1
the value of c0,λ which satisfies (33) is then
2 −1 Observe that simple manipulations shows that
λ λ +1 2 2
c0,λ = 2 Γ + 1, . (37) 0
lim g∞ (w) > 0
2 λ λ w→0+
0 (38)
The case with u = 0 is the only one where the value of cu,λ can lim− g∞ (w) < 0
w→0
be found analytically. Other cases can be studied numerically.
so that w = 0, which is a non-differentiable point, can be
We are now interested in studying the derivative of g∞ in considered as a point of minimum.
order to find singular points which correspond to maximum or
minimum points. Deriving (26) it can be shown that if w > 0 IV. N UMERICAL S IMULATIONS
0
g∞ (w) = 0 ⇐⇒ 2λ(1 − w) + 2(1 − w) + 2(u − 1) = 0. In this section, relevant numerical results are shown for
Hence the (unique) singular point is stationary profiles for different values of u and λ. We focus
on values of u ≥ 0 as the stationary profiles relative to
u+λ negative values of u can be obtained by symmetry, according
w=
λ+1 to (34). The constant cu,λ , which appears in g∞ , is evaluated
and it is positive if and only if λ > −u. Deriving (30), instead, numerically, using Newton-Cotes formulas [12].
it can be shown that if w < 0 First, we assume that u = 0 so that the average opinion
0
g∞ (w) = 0 ⇐⇒ 2λ(1 + w) + 2(1 + w) + 2(u + 1) = 0 corresponds to the middle point of I. As already observed in
the previous section, if u = 0 then g∞ is symmetric with
leading to the following singular point respect to 0 and it has two maxima at
u−λ λ
w= . w=± .
λ+1 λ+1
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Proc. of the 16th Workshop “From Object to Agents” (WOA15) June 17-19, Naples, Italy
2.5 3.5
λ=1/3 λ=1/3
λ=1 λ=1
λ=3 3 λ=3
2
2.5
1.5
g ∞ (w )
g ∞ (w )
2
1.5
1
1
0.5
0.5
0 0
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
w w
Fig. 2. Stationary profiles g∞ for u = 1/4 and λ = 1/3 (blue line), λ = 1 Fig. 3. Stationary profiles g∞ for u = 1/2 and λ = 1/3 (blue line), λ = 1
(red line), and λ = 3 (green line) (red line), and λ = 3 (green line)
As λ increases, such points get nearer to the extremal values maxima correspond to w = ±5/8 and w = ∓3/8, as shown in
of I, namely to extremal opinions. Observe that increasing the Fig. 2 (red line). Finally, Fig. 2 also shows the results obtained
value of λ corresponds to incrementing the impact of diffusion with λ = 3 (green line). In this case the maximum points are
with respect to compromise. w = ±13/16 and w = ∓11/16. As in the previous case, the
Fig. 1 shows the stationary profiles g∞ (w) when u = 0 points of maximum get nearer to ±1, namely to the extremes
for different values of λ. As expected from (34), the function of I, as λ increases. Moreover, observe that largest values of
g∞ (w) is symmetric with respect to 0 and it has a minimum λ correspond to increasing the likelihood of negative opinions.
in correspondence of w = 0 and two maxima whose values Let us now increase the value of the average opinion to
depend on λ. The stationary profiles g∞ (w) in Fig. 1 corre- u = 1/2. The stationary profiles g∞ (w) are shown in Fig. 3
spond to λ = 1/3 (blue line), λ = 1 (red line), and λ = 3 for λ = 1/3 (blue line), λ = 1 (red line), and λ = 3 (green
(green line). If λ = 1/3 the two maxima are in correspondence line). Observe that if λ = 1/3 the function g∞ (w) has only
of w = ±1/4. Observe that in this case extremal distributions one maximum, namely the positive one. As a matter of fact,
are associated with a very low probability. If λ = 1, instead, according the results in the previous section, the negative one
the maxima correspond to w = ±1/2; while in λ = 3 they only exists if λ > u. The maximum point is w = 5/8. This
correspond to w = ±3/4. value is greater than the one obtained for the same λ in the
Therefore, it can be concluded that as λ increases the points case u = 1/4, accordingly with the fact that, in this case, we
of maximum move towards the extremes of the considered in- consider a higher average opinion u. When considering λ = 1,
terval I. Observe that, according to its definition, any increase the function g∞ (w) has two maxima since the condition λ > u
of λ corresponds to assuming that the contribution of diffusion is satisfied. Such points are w = 3/4 and w = −1/4. As in
is more relevant than that of compromise. Moreover, according Fig. 2, the value of the maximum corresponding to the negative
to the results in Fig. 1, any increase of λ leads to stationary value of w is less significant with respect to that relative to the
profiles with small values in correspondence of opinions in the positive value of w. If λ = 3 the two maxima correspond to
middle of the interval I. 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
In Fig. 2 the stationary profiles g∞ (w) are shown when of the results in Fig. 2 with those in Fig. 3 shows that in the
considering as average opinion the value u = 1/4. In this case, latter the values of the positive maxima are greater while those
the function g∞ (w) is not symmetric and, as expected, it has a of the negative maxima are smaller.
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 Finally, we consider a greater value of the average opinion,
maxima is two, since the condition λ > u is always satisfied. namely u = 3/4. This corresponds to considering an extremist
If λ = 1/3 the positive maximum point is w = 7/16 and the society. Fig. 4 shows the stationary profiles for λ = 1/3 (blue
negative one is w = −1/16. While the negative maximum is line), λ = 1 (red line), and λ = 3 (green line). As in the
near the middle of the interval I, the positive one is farther. previous case, since λ < u, the profile g∞ (w) has only one
In Fig. 2 the stationary profile g∞ (w) obtained with λ = 1/3 maximum if λ = 1/3. The point of maximum is w = 13/16
is shown (blue line) and it can be observed that the value of and it is closer to 1 than the other points of maximum obtained
the maximum in w = 7/16 is far more significant than that with the same λ for lower values of the average opinion u.
corresponding to w = −1/16, namely the positive opinions are From Fig. 4 it can be shown that, once again, the positive
far more likely than the negative ones. This is in agreement maximum point moves towards the extreme 1 as λ increases,
with the fact that the average opinion u is positive. If λ = 1 the since it corresponds to w = 7/8 if λ = 1 and to w = 15/16
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Proc. of the 16th Workshop “From Object to Agents” (WOA15) June 17-19, Naples, Italy
Further investigation on this subject is currently under
7
λ=1/3 development. In particular, we are interested in deriving the
λ=1 explicit expressions of the stationary profiles with a different
λ=3
6
choice of the diffusion function. We aim at studying the
properties of such stationary profiles for different parameters of
5
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
g ∞ (w )
4
analytic results with simulation experiments and in studying
3 the number of iterations necessary to approximate to a certain
degree an analytic stationary profile.
2
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1
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w September 2014.
[2] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic
Equations and Montecarlo Methods. Oxford: Oxford University Press,
Fig. 4. Stationary profiles g∞ for u = 3/4 and λ = 1/3 (blue line), λ = 1
2013.
(red line), and λ = 3 (green line)
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