A Stochastic Model of Diffusion in Opinion Dynamics

Stefania Monica and Federico Bergenti

Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Universit

a degli Studi di Parma, 43124 Parma, Italy

Keywords:

Opinion Dynamics, Multi-Agent Systems, Complex Systems.

Abstract:

This paper studies analytically the dynamics of the opinion in multi-agent systems when only the sociological

phenomenon known as diffusion is considered. First, the paper recalls a framework for the analytic study of

opinion dynamics which has been already applied to describe the effects of a number of sociological phe-

nomena. Then, the framework is specialized to the study of diffusion, according to which the opinion of an

agent can be inﬂuenced by the social context. Diffusion is introduced in the framework by stating stochastic

rules meant to describe at the microscopic level how diffusion contributes to change the opinion of an agent.

The obtained model is used to derive collective and asymptotic properties of multi-agent systems when only

diffusion is considered, which are veriﬁed against speciﬁc simulations shown in the last part of the paper. The

paper is concluded with a recapitulation of presented results and an outline of future work.

1 INTRODUCTION

This paper describes a stochastic model that is used

to derive analytic results on collective and long-time

asymptotic properties of the opinion in multi-agent

systems. Considered multi-agent systems are com-

pletely decentralized and without supervised coordi-

nation, and they are studied using a generic frame-

work that depends only on the description of the ef-

fects of microscopic interactions among agents. Mi-

croscopic rules that govern how the opinion of two in-

teracting agents change are used to derive analytically

observable macroscopic properties of the opinion in

the multi-agent system, under proper assumptions.

In this context, the term interaction is used to

denote a symmetric message exchange among two

agents, and it is assumed that message exchanges are

asynchronous. Time is modeled as a sequence of

discrete steps, which may not have the same dura-

tion, and each step corresponds to a single interaction

among two agents. Each agent is free to interact with

any other agent, and no restriction is imposed on the

topological properties of the multi-agent system. The

adopted framework assumes that each agent is asso-

ciated with a scalar attribute, and since the target ap-

plication of the framework in this paper regards the

study of the dynamics of the opinion, we assume that

such an attribute represents the opinion of an agent on

a fact, which may vary within a ﬁnite interval.

Most of existing agent-based models used to study

opinion dynamics are based on simulations (Deffuant

et al., 2000; Hegselmann and Krause, 2002; Monica

and Bergenti, 2014). Hence, their validity depends on

the speciﬁc type of multi-agent system that is simula-

ted, and on the actual values assigned to the parame-

ters of simulations. Actually, their validity depends

on how representative are simulated scenarios with re-

spect to studied phenomena (Flache et al., 2017). At

the opposite, the framework discussed in this paper is

analytic, and it provides parametric results which are

valid as long as its hypotheses remain valid.

Even if analytic models are typically simpler than

models based on simulations, they are preferred when

the interest is on how collective and long-time asymp-

totic behaviours are affected by the parameters of the

models, or when no representative cases to be simu-

lated can be found. Simulations are still very impor-

tant even if analytic models are available because they

become a means to independently verify analytic mo-

dels in interesting cases. Simulations reported in the

last part of the paper are performed independently of

discussed analytic results, and they are uniquely in-

tended to verify analytic results.

Note that the proposed framework is not limited

to the study of opinion dynamics and its approach

can be adopted to describe other collective behaviours

of multi-agent systems, provided that studied pheno-

mena emerge from decentralized interactions.

Monica, S. and Bergenti, F.

A Stochastic Model of Diffusion in Opinion Dynamics.

DOI: 10.5220/0006659601150124

In Proceedings of the 10th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2018) - Volume 1, pages 115-124

ISBN: 978-989-758-275-2

115

The analytic framework adopted in this paper and

described in next section is inspired by a branch of

physics known as kinetic theory of gases and its ge-

neralizations, as described, e.g., in (Bellouquid and

Delitala, 2006). According to this approach, analy-

tic models of microscopic interactions among mole-

cules can be considered to derive macroscopic cha-

racteristics of gases concerning, for instance, tempe-

rature and pressure. Analytic frameworks that follow

the approach of the kinetic theory of gases has been

applied successfully in different ﬁelds. For instance,

the kinetic approach has been successfully applied in

economics to describe the evolution of market eco-

nomy (Cordier et al., 2005) and the distribution of

wealth (Slanina, 2004), giving birth to a discipline

known as econophysics (Chakrabarti et al., 2006). In

this paper, we focus on an application of the kinetic

approach to describe sociological processes and the

dynamics of the opinion in multi-agent systems. The

idea is not new, and it is part of an emerging discipline

known as sociophysics (Galam et al., 1982). Many

analogies between the kinetic theory of gases and the

study of opinion dynamics in multi-agent systems can

be found. First, it is evident that a parallelism bet-

ween molecules in gases and agents in multi-agent sy-

stems can be drawn. Similarly, collisions among mo-

lecules in gases can be re-interpreted as interactions

among agents in multi-agent systems. Obviously, the

detailed mathematical modeling of collisions among

molecules is based on the laws of physics and it is dif-

ferent from that of interactions among agents, which

is typically suggested by sociological studies. Due to

such differences, only few results of kinetic theory of

gases hold in the study of the dynamics of the opinion.

Many sociological phenomena can be studied ana-

lytically using the kinetic approach (Monica and Ber-

genti, 2017a), like:

• Compromise, which is the tendency of agents to

move their opinions towards those of agents they

interact with (De Groot, 1974);

• Diffusion, according to which the opinion of each

agent can be inﬂuenced by the social context (Bo-

nabeau, 2002);

• Homophily, according to which agents interact

only with those with similar opinions (Nowak

et al., 1990; Mark, 2003);

• Negative Inﬂuence, according to which agents

evaluate their peers, and they only interact with

some peers (M

as and Flache, 2013);

• Opinion Noise, according to which a random ad-

ditive process may lead to arbitrary opinion chan-

ges with small probability (Pineda et al., 1982);

and

• Striving for Uniqueness, which is based on the

idea that agents want to distinguish from others

and, hence, they decide to change their opinions

if too many agents share the same opinion (M

et al., 2010).

Kinetic models that study compromise alone have

been already proposed in (Monica and Bergenti,

2016a; Monica and Bergenti, 2016b; Monica and Ber-

genti, 2017b; Monica and Bergenti, 2017c). In ad-

dition, in (Monica and Bergenti, 2015a; Monica and

Bergenti, 2015b; Monica and Bergenti, 2015c), both

compromise and diffusion were analytically modeled

and investigated under proper assumptions. In this

paper, instead, we focus uniquely on diffusion and

we analytically derive results concerning the dyna-

mics of the opinion when only this phenomenon is

considered. The major contribution of this paper is

to study analytically collective and asymptotic pro-

perties of the opinion when only diffusion is relevant,

so that the properties of diffusion can be isolated and

studied independently of other phenomena. Note that

other models of opinion formation based on the ki-

netic theory of gases were also proposed to account

for the possibility of having different types of agents

in a multi-agent system (Bergenti and Monica, 2016;

Bergenti and Monica, 2017).

This paper is organized as follows. Section 2 sum-

marizes the main ideas of the kinetic approach and

it outlines the adopted kinetic framework. Section 3

presents a stochastic model to describe the sociologi-

cal phenomenon of diffusion. Section 4 shows ma-

jor analytic results concerning collective and asymp-

totic properties of a multi-agent system when only

diffusion is considered. Section 5 describes simu-

lations that were ran to verify analytic results. Fi-

nally, Section 6 concludes the paper and outlines fu-

ture work.

2 A KINETIC FRAMEWORK TO

STUDY OPINION DYNAMICS

In this section, we present the main ideas of classic

kinetic theory of gases and we focus on the reinter-

pretation of such ideas to study the dynamics of the

opinion in multi-agent systems. Classic kinetic the-

ory of gases assumes that, at each time t, each mole-

cule of the considered gas can be associated with re-

levant parameters, such as its position x and velocity

v, which are both modeled as three-dimensional vec-

tors. The characteristics of the considered gas are des-

cribed in terms of a density function f (x,v,t), which

represents the number of molecules whose position is

in (x,x + dx) and whose velocity is in (v, v + dv) at

ICAART 2018 - 10th International Conference on Agents and Artiﬁcial Intelligence

116

time t. The density function f (x,v,t) evolves follo-

wing a proper balance equation, namely the famous

Boltzmann equation. The Boltzmann equation is an

integro-differential equation whose explicit formula-

tion is based on the analytic description of the ef-

fects of collisions among molecules. The study of

the Boltzmann equation allows deriving macrosco-

pic properties of a gas, concerning, for instance, the

average velocity of its molecules, its temperature, and

its pressure. In summary, kinetic theory of gases stu-

dies the effects of collisions among molecules from

a microscopic point of view and uses such results to

derive macroscopic properties of gases.

The same approach can be used to model opi-

nion dynamics in multi-agent systems by means of

a proper parallelism between molecules and collisi-

ons among them, and agents and interactions among

them, as proposed, for example, in (Schweitzer and

Holyst, 2000) or in (D

uring et al., 2009). It is worth

noting that, since the effects of collisions among mo-

lecules differ from the effects of interactions among

agents, analytic results derived to study the dynamics

of the opinion differ signiﬁcantly from well-known

results of the kinetic theory of gases. However, the

approach of kinetic theory of gases can be effectively

generalized to obtain analytic results in the ﬁeld of

opinion dynamics, as follows.

Each agent is associated with a single parameter

v, which represents its opinion. Note that, in our con-

text, the term opinion is used to denote the level of

agreement on a single topic or the level of apprecia-

tion of a single item. Therefore, we assume that opi-

nion v is valid only in a closed interval

I = [−1, 1] (1)

where −1 stands for strong disagree or strong dislike,

1 stands for strong agree or strong like, and values

close to 0 are considered moderate opinions. We re-

mark that the choice of the interval I in (1) is discreti-

onal and any other closed interval can be used, provi-

ded that the following analytic developments are co-

herently changed. In agreement with the general ideas

of kinetic theory of gases, we also postulate the exis-

tence of a density function f (v,t), which represents

the number of agents with opinion in (v, v + dv) at

time t. The deﬁnition of the density function f (v,t) al-

lows computing proper macroscopic properties of the

system from an analytic point of view. For instance,

the number of agents at time t, denoted as n(t), can be

computed as

n(t) =

f (v,t)dv. (2)

From (2), it is evident that the number of agents at

time t is obtained by integrating the density function

with respect to all values of v ∈I. In kinetic theory of

gases, a similar integral is used to compute the mass

of the considered gas. Similarly, the average opinion

of the system at time t, denoted as u(t), is obtained

by multiplying f (v,t) by v, dividing by the number of

agents, and integrating with respect to v,

u(t) =

n(t)

f (v,t)vdv. (3)

Observe that the average opinion u(t) is related to the

ﬁrst momentum of the density function f (v,t). In

kinetic theory of gases, a similar integral is used to

compute the average velocity of molecules in the con-

sidered gas. Finally, the variance of the opinion at

time t, denoted as σ

(t), can be obtained by multi-

plying f (v,t) by (v −u(t))

, dividing by the number

of agents, and integrating with respect to v, as follows

(t) =

n(t)

(v −u(t))

f (v,t)dv. (4)

Note that the variance of the opinion σ

(t) is related to

the second-order momentum of the density function

f (v,t). In kinetic theory of gases, similar integrals are

used to compute the pressure and the temperature of

the considered gas.

The temporal evolution of the density function

f (v,t) is governed by a balance equation, whose

explicit expression is inspired from that of the

Boltzmann equation. For this reason, we adopt the

same nomenclature of kinetic theory of gases and we

still use the term Boltzmann equation for such a ba-

lance equation. The homogeneus formulation of the

Boltzmann equation that can be used to study the dy-

namics of the opinion is

∂ f

∂t

(v,t) = Q ( f )(v,t) (5)

where the left-hand side is related to the temporal evo-

lution of the density function and in the right-hand

side Q is an operator meant to account for the effects

of interactions among agents. Keeping the same no-

menclature of the kinetic theory of gases, Q is called

collisional operator and it computes a function of va-

riables v and t by using function f .

In order to analytically study the temporal evolu-

tion of the number of agents n(t), the average opi-

nion u(t), and the variance of the opinion σ

(t), the

weak form of the Boltzmann equation can be used.

In functional analysis, the weak form of a differential

equation is obtained by multiplying both sides of the

equation by a suitable test function and by integrating

with respect to one of the variables. Therefore, the

weak form of the Boltzmann equation with respect to

the generic test function φ(v) can be written as

f (v,t)φ(v)dv =

Q ( f )(v,t)φ(v)dv (6)

A Stochastic Model of Diffusion in Opinion Dynamics

117

where the right-hand side is called weak form of the

collisional operator Q with respect to test function

φ(v). Recalling (2), (3), and (4), it is evident that the

left-hand side of (6) can be used to compute the time

derivative of macroscopic characteristics of the multi-

agent system, provided that speciﬁc test functions are

chosen, as follows:

1. If φ(v) = 1, the left-hand side of (6) represents the

time derivative of the number of agents n(t);

2. If φ(v) = v, the left-hand side of (6) is related to

the time derivative of the average opinion; and

3. If φ(v) = (v −u(t))

, the left-hand side of (6) is

related to the time derivative of the variance of the

opinion.

In next section, the details of the microscopic effects

of the interactions among agents on the opinion are

outlined and, consequently, the explicit expression of

the collisional operator and of its weak form are de-

rived. Such derivations are sufﬁcient to express ordi-

nary differential equations whose unknowns are n(t),

u(t), and σ

(t). The solutions of such equations ex-

press the dynamics of such quantities and they allow

studying analytically their asymptotic behaviours.

3 A STOCHASTIC MODEL OF

DIFFUSION

In order to derive explicit results using the framework

outlined in previous section, the explicit formulation

of the interaction among agents is needed. Note that

in the proposed framework interactions among agents

are not coordinated by any supervising entity, and, in

our assumptions, any agent can freely interact with

any other agent in the system. Moreover, we assume

that interactions among agents are binary, which me-

ans that they involve only two agents. This assump-

tion is not restrictive, since interactions involving a

larger number of agents can be regarded as sequences

of binary interactions. Concerning the sociological

phenomenon considered in this paper, as shortly ex-

plained in the introduction, we focus on diffusion, ac-

cording to which agents can change their opinions due

to external inﬂuence. However, we remark that the

proposed approach can be extended to take into ac-

count more complex interaction rules, including, e.g.,

homophily (Tsang and Larson, 2014), which accounts

for the fact that agents tend to communicate only with

those having similar opinions.

Diffusion can be modeled using the following in-

teraction rules, as proposed, e.g., in (Pareschi and

Toscani, 2013; Toscani, 2006). Such interaction ru-

les model how the opinions of two interacting agents

change after an interaction if only diffusion is consi-

dered relevant to opinion formation and other socio-

logical phenomena are neglected

(

= v + η

D(v)

= w + η

D(w)

(7)

where

1. v and w represent the opinions of the two agents

before the interaction, often called pre-interaction

opinions;

2. v

and w

represent the opinions of the two agents

after the interaction, often called post-interaction

opinions;

3. η

and η

are two independent random variables

that we call diffusion parameters; and

4. D(·) models the effects of diffusion on the opi-

nion of the two interacting agents, and it is called

diffusion function.

In the rest of this paper we assume that diffusion pa-

rameters have the same distribution function, which

is denoted as ϑ(·). We denote the support of ϑ(·) as

S, and we assume that the average values of the two

random variables equal 0

ϑ(η

)η

dη

ϑ(η

)η

dη

= 0. (8)

Note that according to the proposed model of diffu-

sion the post-interaction opinion of an agent depends

only on its pre-interaction opinion, and it does not de-

pend on the pre-interaction opinion of the other agent.

This models the idea that the opinion of an agent can

change simply because an interaction occurred, and

the change of the opinion does not necessarily depend

on the opinion of the other agent, which is how dif-

fusion is studied in opinion dynamics (Pareschi and

Toscani, 2013; Toscani, 2006).

Observe that from (7) it is evident that the post-

interaction opinions are obtained by adding to the

pre-interaction opinions an addend which is propor-

tional to the diffusion function evaluated in the pre-

interaction opinions according to the values of the

parameters η

and η

. Some considerations on the

choice of the diffusion function and on the choice of

the distribution functions of η

and η

are needed.

First, let us observe that different choices of the diffu-

sion function may lead to very different models. Fol-

lowing the literature (Toscani, 2006), we assume that

diffusion functions are nonzero and symmetric with

respect to the central value of the interval where opi-

nions are deﬁned. In our assumptions, since interval

I is symmetric with respect to 0, this corresponds to

state that the diffusion function is even and, for this re-

ason, from now on we assume that D(·) is a function

ICAART 2018 - 10th International Conference on Agents and Artiﬁcial Intelligence

118

of the absolute value of the opinion v. Moreover, we

assume that D(·) is not increasing with respect to |v|,

in agreement with the idea that agents with opinions

close to the bounds of I are conservative. Finally, we

also assume that the following inequalities hold to en-

sure that the magnitude of the effects of diffusion are

controlled only by diffusion parameters η

and η

0 ≤ D(|v|) ≤ 1. (9)

We call admissible any diffusion function that re-

spects mentioned assumptions, and we consider only

admissible diffusion functions in the rest of this paper.

Some considerations need to be made to guaran-

tee that post-interaction opinions v

and w

belong to

the interval of interest I. To this aim, it is necessary

to impose proper conditions on the support of the two

random variables η

and η

. Observe that the follo-

wing inequalities hold

|v + η

D(|v|)| ≤ |v|+ |η

|D(|v|)

|w + η

D(|w|)| ≤ |w|+ |η

|D(|w|).

(10)

Hence, possible sufﬁcient conditions to guarantee that

∈ I and w

∈ I are

|η

| ≤

1 −|v|

D(|v|)

|η

| ≤

1 −|w|

D(|w|)

(11)

If (11) holds, diffusion parameters are said to be ad-

missible for the chosen D(·), and note that we con-

sider only admissible diffusion parameters in the rest

of this paper. In next section, such conditions will

be further discussed in correspondence of a speciﬁc

choice of an admissible diffusion function.

The microscopic interaction rules (7) can be ﬁ-

nally used to derive the explicit expression of the

collisional operator, which is the integral operator

that can be evaluated as the difference between the

gain and the loss of agents whose opinions fall in

(v, v + dv). In order to do so, let us introduce the

transition rate W (v,w,v

)dv

which represents

the probability per unit time that after the interaction

among two agents with opinions v and w, the opinion

of the ﬁrst agent is in (v

+ dv

) and the opinion of

the second agent is in (w

+ dw

). Following the

derivation in (Toscani, 2000), in which a different mi-

croscopic model is considered, the transition rate can

be written as

W (v,w,v

)dv

= ϑ(η

)ϑ(η

)dη

dη

(12)

where, as previously said, ϑ(·) denotes the distri-

bution function of the random variables η

and η

whose support is S. Under all mentioned assumptions,

the difference between the gain and the loss terms in

the explicit formulation of the weak form of the col-

lisional operator can be written as a unique integral,

thus obtaining the following formulation of the right-

hand side of the weak form of the Boltzmann equation

ϑ(η

)ϑ(η

) f (v,t) f (w,t)·

(φ(v + η

D(|v|)) −φ(v))dvdw dη

dη

(13)

Given the explicit formulation of the weak form of the

Boltzmann equation (13), it is ﬁnally possible to ana-

lytically study relevant macroscopic properties of the

system, as explained at the end of previous section. In

the following section we show how the choice of pro-

per test functions in (13) allow the derivation of three

ﬁrst order differential equations whose unknowns are

the number of agents, the average opinion, and the

variance of the opinion, respectively.

4 COLLECTIVE AND

ASYMPTOTIC PROPERTIES

In this section we show relevant analytic results con-

cerning collective and asymptotic properties of multi-

agents systems in which agents interact following the

rules outlined in previous section. In particular, star-

ting from the weak form of the Boltzmann equation

(13), we derive proper differential equations whose

unknowns are:

1. The number of agents n(t);

2. The average opinion of the system u(t); and

3. The variance of the opinion σ

(t).

In the last part of this section, analytic results in

closed form are also derived for a speciﬁc diffusion

function, which is then used to run independent simu-

lations.

Proposition 1. Given a multi-agent system where

agents interact according to (7), the chosen diffusion

function D(·) is admissible, and diffusion parameters

are admissible for D(·), the number of agents n(t) in

the multi-agent system does not depend on time.

Proof. Let us consider the test function

φ(v) = 1 (14)

in (13). Since the test function (14) is a constant, the

difference

φ(v + η

D(|v|)) −φ(v) (15)

inside the integral in (13) equals 0. Hence, the weak

form of the Boltzmann equation relative to the chosen

A Stochastic Model of Diffusion in Opinion Dynamics

119

test function φ(v) = 1 reduces to the following rele-

vant equality

f (v,t)dv = 0. (16)

The left-hand side of (16) represents the time deriva-

tive of the number of agents n(t) and, hence, (16) can

be reformulated as

n(t) = 0, (17)

and it can be ﬁnally concluded that the number of

agents is constant

n(t) = n(0), (18)

which proves the proposition.

Proposition 1 allows dropping the dependance on

time for n(t). In the rest of this paper, we denote the

number of agents as n, thus omitting the dependance

on time. We remark that analogous considerations

can be derived also when considering the Boltzmann

equation in the context of kinetic theory of gases. This

property corresponds to mass conservation in gases.

Another, much more interesting, collective and

asymptotic property of diffusion is captured by the

following proposition.

Proposition 2. Given a multi-agent system where

agents interact according to (7), the chosen diffusion

function D(·) is admissible, and diffusion parameters

are admissible for D(·), the average opinion of the

multi-agent system u(t) does not depend on time.

Proof. Let us consider the test function

φ(v) = v (19)

in (13), so that the right-hand side of the weak form

of the Boltzmann equation can be written as

ϑ(η

)ϑ(η

) f (v,t) f (w,t)·

D(|v|)dvdw dη

dη

(20)

From (3) and recalling that, according to (18), the

number of agents is constant, the left-hand side of the

weak form of the Boltzmann equation equals

u(t) (21)

and it is therefore proportional to the derivative of the

average opinion with respect to time. Moreover, the

integral in (20) can be written as the product of four

terms, as follows

ϑ(η

)dη

ϑ(η

)dη

f (v,t)D(|v|)dv

f (w,t)dw,

(22)

and, since ϑ(·) is a distribution function, the integral

on its support S equals 1. Finally, using (2), (22) can

be simpliﬁed to

f (v,t)D(|v|)dv (23)

where

is the average value of random variable η

Recalling that the average value of random variable

is assumed to be equal to 0, from (8), it can be

concluded that (20) equals 0, regardless of the choice

of the diffusion function D(·). Therefore, the weak

form of the Boltzmann equation corresponding to the

test function φ(v) = v can ﬁnally be written as

u(t) = 0. (24)

Equation (24) implies that the average opinion of the

system is constant

u(t) = u(0), (25)

which proves the proposition.

As already done for the number of agents, from

now on we omit the dependance of the average opi-

nion on time and we simply denote it as u. An analo-

gous property is found in the kinetic theory of gases

and it corresponds to the conservation of momentum.

Note that if the average value of η

is not 0, which

equals to dropping the assumption that diffusion pa-

rameters are admissible for the chosen D(·), then the

following differential equation for the average opi-

nion can be derived

u(t) =

f (v,t)D(|v|)dv (26)

and the validity of Proposition 2 would depend on the

actual choice of the diffusion function D(·).

Regarding the asymptotic properties of the vari-

ance of the opinion, the following proposition holds.

Proposition 3. Given a multi-agent system where

agents interact according to (7), the chosen diffusion

function D(·) is admissible, and diffusion parameters

are admissible for D(·), the variance of the opinion of

the multi-agent system σ

(t) is not constant.

Proof. Let us consider the test function

φ(v) = (v −u)

(27)

in (13), so that the right-hand side of the weak form

of the Boltzmann equation can be written as

ϑ(η

)ϑ(η

) f (v,t) f (w,t)·



(|v|) + 2η

D(|v|)(v −u)



dvdw dη

dη

ICAART 2018 - 10th International Conference on Agents and Artiﬁcial Intelligence

120

where we used the following identity that can be ea-

sily veriﬁed by simple algebraic manipulations

(v + η

D(|v|) −u)

−(v −u)

(|v|) + 2η

D(|v|)(v −u).

Further algebraic manipulations, and the assumption

that the average value of η

equals 0, show that the

right-hand side of the weak form of the Boltzmann

equation can be written as

nσ

f (v,t)D

(|v|)dv (28)

where σ

denotes the variance of the random varia-

ble η

. From the deﬁnition of σ

(t), and using previ-

ously obtained results on the conservation of the num-

ber of agents and of the average opinion, the left-hand

side of the weak form of the Boltzmann equation can

be shown to be proportional to the derivative of the

average opinion with respect to time. Actually, it can

be written as

(t). (29)

Therefore, the weak form of the Boltzmann equation

relative to the test function φ(v) = (v −u)

can be ﬁ-

nally written as

(t) = σ

f (v,t)D

(|v|)dv. (30)

Observe that f (v,t) is a density function, and the-

refore, by deﬁnition, it is non-negative. Moreover,

f (v,t) cannot be identically 0, since its integral on

I equals the number of agent n. Similarly, function

(·) is also nonnegative and it is supposed not to be

identically 0. According to these considerations, the

integral in (30) is necessarily strictly positive and it

depends on the choice of the diffusion function D(·).

Hence, it can be concluded that the variance of opi-

nion σ

(t) is not constant.

Note that no equivalent form of Proposition 3 is

found in the kinetic theory of gases because it follows

from the speciﬁc assumptions that we took to model

the sociological phenomenon of diffusion.

In order to explicitly solve the differential equa-

tion (30), and possibly validate obtained results on the

asymptotic behaviour of the opinion, a given admissi-

ble diffusion function needs to be ﬁxed. We consider

the following diffusion function, which is also used in

next section

D(|v|) =

1 −v

. (31)

Observe that the chosen D(·) is admissible because it

satisﬁes the requirements outlined in Section 3. As

a matter of fact, it is an even function and it is a de-

creasing function of |v|. Moreover, it also satisﬁes

condition (9). Using this diffusion function, diffusion

parameters η

and η

are admissible if the following

conditions, derived from (11), are satisﬁed

|η

| ≤

1 −|v|

√

1 −v

|η

| ≤

1 −|w|

√

1 −w

(32)

The rest of this section is dedicated to the proof of the

following proposition.

Proposition 4. Given a multi-agent system where

agents interact according to (7) with diffusion

function (31), and assuming that diffusion parameters

are admissible for D(·), the variance of the opinion

(t) of the multi-agent system exponentially tends to

1 −u

as t tends to +∞.

Proof. Let us start from the differential equation (30)

relative to the variance of opinion σ

(t). Considering

diffusion function (31), equation (30) becomes

(t) = σ

f (v,t)(1 −v

)dv. (33)

The integral at the right-hand side of (33) can be writ-

ten as the difference

f (v,t)dv −

f (v,t)v

dv. (34)

The ﬁrst integral in (34) equals the number of agents

n, while the second integral in (34) can be associated

to the deﬁnition of the variance of the opinion. As a

matter of fact, from the deﬁnition of σ

(t), the follo-

wing equalities can be easily derived

nσ

(t) =

f (v,t)(v −u)

dv =

f (v,t)v

dv −2nu

+ nu

where we used the deﬁnitions of the number of agents

n and of the average opinion u. Using these equations

in (30), the following differential equation for the va-

riance of the opinion σ

(t) can be derived from the

weak form of the Boltzmann equation

(t) = nσ

(1 −u

−σ

(t)). (35)

Observe that (35) is a non-homogeneous ﬁrst-order

differential equation whose solution can be found

analytically. The solution of (34) is

(t) = Ce

−nσ

+ (1 −u

) (36)

where C is a constant which has to be set in order to

satisfy the initial condition. Denoting as σ

(0) the

A Stochastic Model of Diffusion in Opinion Dynamics

121

initial value of the variance of the opinion, constant C

has to satisfy the following condition

(0) = C + (1 −u

) (37)

and, hence, from (36) it can be concluded that the so-

lution of (34) is

(t) = [σ

(0) −(1 −u

)]e

−nσ

+ (1 −u

). (38)

Let us now observe that the coefﬁcient of t in the ex-

ponential function is negative, since both the number

of agents n and the variance of η

are positive. It

can then be concluded that the exponential function

in (38) tends to 0 as t tends to +∞. As a consequence,

it can be ﬁnally concluded that

lim

t→+∞

(t) = (1 −u

), (39)

which proves the proposition.

5 VERIFICATION BY

SIMULATION

In this section, we show results of independent simu-

lations meant to validate analytic results derived in

previous sections. We consider a system composed

of n = 10

agents and, as stated in the introduction,

we assume that each agent can interact with any other

agent in the system. At each step of the simulation,

two agents are randomly chosen and an interaction

among them is simulated. This means that both of

them change their opinions according to (7) with the

chosen diffusion function D(·) in (31). We consider

three different scenarios, corresponding to different

initial distributions of the opinion and, hence, to dif-

ferent values of the average opinion u. In all consi-

dered scenarios, 1.5 ·10

binary interactions among

randomly chosen agents are simulated, which equals

to 300 interactions per agent on average.

We start by considering a multi-agent system

where the initial distribution of the opinion is uniform

on the entire interval I where the opinion is deﬁned

f (v,0) = U

(−1,1)

(v). (40)

According to this assumption, the average opinion of

the system at time t = 0 is u = 0 and, as shown in the

proof of Proposition 2, it remains constant as agents

interact. Under these assumption, from Proposition 4

it is expected that the variance of the opinion tends

to 1 as t tends to +∞. Figure 1 shows the values of

(t) (solid blue line) for values of t between 0 and

1.5 ·10

and, as expected, the variance tends to 1 as t

increases. Note that plotted line follows very closely

the expected exponential increase. Let us now consi-

0 5 10 15

x 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 1: The variance σ

(t) of the opinion as a function

of the number of interactions t when f (v, 0) = U

(−1,1)

(v)

(solid blue line), f (v,0) = U

(−

,1)

(v) (dashed red line), and

f (v, 0) = U

(−1,0)

(v) (dash-dotted green line).

der a multi-agent system where the initial distribution

of the opinion is

f (v,0) = U

(−

,1)

(v) (41)

so that the value of the average opinion is equal to

u = 1/4. In this case, from Proposition 4, it is ex-

pected that the variance of the opinion tends to 15/16

as t tends to +∞. This result is conﬁrmed in Figure 1,

where the values of σ

(t) are shown as a function of

t (dashed red line). As the number of interactions in-

creases, the variance tends to 15/16 = 0.9375, as ex-

pected. Also in this case, the plot approximates well

the expected exponential increase.

Finally, we now assume that the initial distribution

of the opinion in the considered multi-agent system is

f (v,0) = U

(−1,0)

(v). (42)

In this case, the value of the average opinion equals

u = −1/2 and, hence, it is expected that the variance

of the opinion tends to 3/4 as t tends to +∞. Fi-

gure 1 shows that, as expected, the value of the vari-

ance tends to 3/4 = 0.75 (dash-dotted green line), and

the plot follows the expected exponential function.

6 CONCLUSIONS

This paper presented an analytic model of the social

phenomenon of diffusion, which is normally used as

one of the ingredients to study the dynamics of the

opinion in multi-agent systems. First, the adopted ki-

netic framework for the study of opinion dynamics

was recalled. Then, the framework was completed

with the details needed to study diffusion by intro-

ducing speciﬁc interaction rules. Adopted rules mo-

del diffusion in terms of a diffusion function and of

ICAART 2018 - 10th International Conference on Agents and Artiﬁcial Intelligence

122

two stochastic diffusion parameters. Under the as-

sumption that the diffusion function is admissible and

that diffusion parameters are admissible for the dif-

fusion function, interesting asymptotic properties of

the multi-agent system were proved. In detail, the pa-

per showed proofs of the fact that diffusion does not

change the average opinion of the multi-agent system,

but that it inﬂuences its variance. In the last part of

the paper, a speciﬁc diffusion function is considered,

and expected properties of the average opinion and

of the variance of the opinion for the chosen diffu-

sion function were veriﬁed by independent simulati-

ons. Note that presented simulations do not depend

on the adopted kinetic approach. They are simple im-

plementations of the studied interaction rules.

The work reported in this paper can be extended

by considering multi-agent systems made of agents

with different propensity to change opinion because

of interactions. In (Bergenti and Monica, 2017), we

have already investigated the possibility of having dif-

ferent classes of agents in the same multi-agent sy-

stem, where different classes are associated with dif-

ferent parameters, such as different number of agents

and different values of the parameters of compromise.

Similar considerations are planned as future work for

the study of collective and asymptotic properties of

diffusion in multi-agent systems with multiple clas-

ses of agents. Finally, the model of diffusion stu-

died in this paper could be coupled with similar mo-

dels of other sociological phenomena, such as com-

promise and homophily, to study analytically the col-

lective and asymptotic properties of more complex sy-

stems. We have already studied the combined effects

of compromise and diffusion under speciﬁc assumpti-

ons in (Monica and Bergenti, 2015b), and we plan to

extend such results by modelling other phenomena.

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