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In this paper, we discuss the passage to hydrodynamic equations for kinetic models of opinion formation. The considered kinetic models feature an opinion density depending on an additional microscopic variable, identified with the personal preference. This variable describes an opinion-driven polarisation process, leading finally to a choice among some possible options, as it happens e.g. in referendums or elections. Like in the kinetic theory of rarefied gases, the derivation of hydrodynamic equations is essentially based on the computation of the local equilibrium distribution of the opinions from the underlying kinetic model. Several numerical examples validate the resulting model, shedding light on the crucial role played by the distinction between opinion and preference formation on the choice processes in multi-agent societies.
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Hydrodynamic models of preference formation
in multi-agent societies
Lorenzo PareschiGiuseppe ToscaniAndrea TosinMattia Zanella§
Abstract
In this paper we discuss the passage to hydrodynamic equations for kinetic models of
opinion formation. The considered kinetic models feature an opinion density depending on
an additional microscopic variable, identified with the personal preference. This variable de-
scribes an opinion-driven polarisation process, leading finally to a choice among some possible
options, as it happens e.g. in referendums or elections. Like in the kinetic theory of rarefied
gases, the derivation of hydrodynamic equations is essentially based on the computation of the
local equilibrium distribution of the opinions from the underlying kinetic model. Several nu-
merical examples validate the resulting model, shedding light on the crucial role played by the
distinction between opinion and preference formation on the choice processes in multi-agent
societies.
Keywords: Opinion and preference formation, choice processes, kinetic modelling, hydro-
dynamic equations
Mathematics Subject Classification: 35L65, 35Q20, 35Q70, 35Q91, 82B21
1 Introduction
The mathematical modelling of opinion formation in multi-agent societies has enjoyed in recent
years a growing attention [4,5,6,29,36,46]. In particular, owing to their cooperative nature,
the dynamics of opinion formation have been often dealt with resorting to methods typical of
statistical mechanics [14,16]. Among other approaches, kinetic theory served as a powerful basis
to model fundamental interactions among the so-called agents [8,9,10,17,30,47] and to provide
a sound structure for related applications [1,49]. In kinetic models, analogously to the kinetic
theory of rarefied gases, the mechanism leading to the opinion variation is given by binary, i.e.
pairwise, interactions among the agents. Then, depending on the parameters of such microscopic
rules, the society develops a certain macroscopic equilibrium distribution [37,40], which describes
the formation of a relative consensus about certain opinions.
Two main aspects are usually taken into account in designing the elementary binary interac-
tions. The first is the compromise [20,52], namely the tendency of the individuals to reduce the
distance between their respective opinions after the interaction. The second is the self-thinking [47],
i.e. an erratic individual change of opinion inducing unpredictable deviations from the prescribed
deterministic dynamics of the interactions.
Recently, many efforts have been devoted to include further details in the opinion formation
models, so as to capture more and more realistic phenomena. The usual strategy consists in
Department of Mathematics and Computer Sciences, University of Ferrara, Via Machiavelli 35, 44121 Ferrara,
Italy (lorenzo.pareschi@unife.it)
Department of Mathematics “F. Casorati”, University of Pavia, Via Ferrata 1, 27100 Pavia, Italy
(giuseppe.toscani@unipv.it)
Department of Mathematical Sciences “G. L. Lagrange”, Dipartimento di Eccellenza 2018-2022, Politecnico di
Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy (andrea.tosin@polito.it)
§Department of Mathematical Sciences “G. L. Lagrange”, Dipartimento di Eccellenza 2018-2022, Politecnico di
Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy (mattia.zanella@polito.it)
1
taking additional behavioural aspects into account, such as the stubbornness of the agents [34,45],
the emergence of opinion leaders [25,27], the influence of social networks [2,48], the expertise
in decision making tasks [41], the personal conviction [3,7,11,19]. Generally, the aim of such
additional parameters is to model on one hand the resistance of the agents to change opinion and,
on the other hand, the prominent role played by some individuals in attracting others towards their
opinions. In all these contributions, the additional variables act as modifiers of the microscopic
interactions. This means that they affect the process of opinion formation but are not affected in
turn by the evolving opinions.
In the present paper we aim instead at modelling a parallel process to opinion formation,
namely the formation of preferences. The preference, which is driven by, but need not coincide
with, the opinion, is here understood as representative of the choice that an agent makes among
some possible options, such as e.g. some candidates in an election or yes/no in a referendum [18].
As such, and unlike the opinion, the preference evolves towards necessarily polarised states, which
reflect the available options.
To pursue this goal, we consider a novel class of inhomogeneous kinetic models for the joint
distribution function f(t, ξ, w), where ξis the variable describing the preference and wthe one
describing the opinion. In particular, f(t, ξ, w)dξdw is the proportion of agents who, at time
t0, express a preference in the interval [ξ, ξ +] and simultaneously an opinion in the interval
[w, w +dw]. As far as the modelling of the interactions leading to the evolution of the opinion
is concerned, we take advantage of the well consolidated background recalled before. Conversely,
concerning the evolution of the preference, we assume transport-type dynamics of the form
dt = (wα)Φ(ξ).
Here, in analogy with the classical kinematics, wplays morally the role of the velocity, i.e. it
drags the preference in time, however biased by a perceived social opinion α, which accounts for
the predominant social feeling. Moreover, the zeros of the function Φ define the options where the
preference may polarise.
These ingredients lead us to an inhomogeneous Boltzmann-type kinetic equation of the form
tf+ (wα)ξ(Φ(ξ)f) = Q(f, f ),
where Qis a Boltzmann-type collision operator encoding the opinion formation interaction dynam-
ics. From here, analogously to the classical kinetic theory of rarefied gases, we derive macroscopic
equations for the density ρ=ρ(t, ξ) and the mean opinion m=m(t, ξ) of the agents with pref-
erence ξat time tby means of a local equilibrium closure based on the identification of the local
equilibrium distribution of the opinions – the equivalent of a local “Maxwellian”. The precise type
of hydrodynamic equations that we obtain in this way depends on whether the mean opinion of
the agents is or is not conserved in time by the microscopic dynamics of opinion formation. For
instance, if it is conserved we get the following system of conservation laws:
(tρ+ξ(Φ(ξ)ρ(mα)) = 0,
t(ρm) + ξ(Φ(ξ)(M2,αρm)) = 0,
where M2,denotes the energy of the local Maxwellian. For special classes of local equilibrium
distributions, that we compute from the collisional kinetic equation by an asymptotic procedure
reminiscent of the grazing collision limit of the classical kinetic theory [47,50], we can express
such an energy analytically in terms of the hydrodynamic parameters ρ,m. We recover therefore a
self-consistent macroscopic model, which we show to be hyperbolic for all the physically admissible
values of (ρ, m) and able to reproduce the preference polarisations, viz. the choices, discussed
above.
In more detail, the paper is organised as follows. In Section 2we give preliminary microscopic
insights into the joint process of opinion and preference formation, stressing in particular the role
of the perceived social opinion α. In Section 3we move to an aggregate analysis of the opinion
2
formation by means of Boltzmann-type kinetic models, studying in particular their steady states
which, as set forth above, pave the way for the identification of the local equilibrium distributions
needed in the passage to the hydrodynamic equations. In Section 4we derive various types of
macroscopic models of preference formation out of the aforementioned inhomogeneous Boltzmann-
type equation and we link them precisely to key features of the microscopic interactions among the
agents. In Section 5we present several numerical tests, both at the kinetic and at the macroscopic
scales, which exemplify the distinction between the preference formation and the opinion formation
processes and show how much such a distinction enhances the interpretation of the social dynamics.
Finally, in Section 6we discuss further developments and research prospects.
2 A microscopic look at the opinion-preference interplay
The first mathematical models of consensus formation in opinion dynamics were proposed in
the form of systems of ordinary differential equations (ODEs) describing the behaviour of a finite
number of agents. After the pioneering works [21,28], which introduced simple agent-based models
to understand the effects of the influence among connected individuals, many research efforts have
been devoted to the construction of sophisticated differential models of opinion formation. Besides
those dealing with simple consensus dynamics, in recent years new models of more realistic social
phenomena have been proposed to capture additional aspects. Without intending to review all
the literature, we give here some references on certain classes of models for finite systems: the
celebrated Hegselmann-Krause model [32], which considers bounded-confidence-type interactions
to stress the impact of homophily in learning processes, see also [15,35]; models incorporating
leader-follower effects [38]; models of social interactions on realistic networks [51]; models of opinion
control [12].
An aspect which, to our knowledge, has been so far basically disregarded in mathematical
models of opinion dynamics, in spite of its realism, is the distinction between the opinion in the
strict sense of the individuals about single issues and their overall preference, which, in some cases,
is actually mainly responsible for their choices. For instance, in case of referendums or political
elections, the preference of a voter can be identified with his/her voting intention, which may
not always coincide with his/her opinion on every topic debated during the election campaign.
Obviously, the preference evolves in time with the opinion, in such a way that if a certain opinion
persists for a sufficiently long time it can affect the preference considerably. For example, a
voter with a voting intention biased towards right (left, respectively) parties, who however finds
him/herself frequently in agreement with the positions taken by left (right, respectively) parties
on key topics of the election campaign, may end up with a final vote opposite to his/her original
intention.
As a foreword to the subsequent contents of the paper, in this section we present some pre-
liminary considerations about the effect of the interplay between opinion and preference, taking
advantage of a deterministic microscopic model for a finite number of agents. Let us consider then
a system composed by Nagents with microscopic state given by a pair (ξi, wi)[1,1]2, where
wiis the opinion of the ith agent, ξiis his/her preference and i= 1, . . . , N . Sticking to a standard
custom in the literature of models of opinion dynamics, we describe mathematically the opinion of
an agent as a bounded scalar variable wiconventionally taken in the interval [1,1]. In particular,
we understand the values wi=±1 as the two extreme opinions, while wi= 0 as the maximum of
indecisiveness. Since, from the physical point of view, the preference is commensurable with an
opinion, we adopt the same mathematical conventions also for the variable ξi.
In order to highlight the different roles of the opinion and the preference variables, we rely on
the analogy with the classical kinetic theory of rarefied gases. There, a molecule moving on a line
is characterised by its position xRand its velocity vR. In the absence of external forces, the
velocity remains constant, whereas the position varies according to the kinematic law
dx
dt =v. (1)
3
In practice, a particle with positive velocity will move rightwards, while a particle with negative
velocity will move leftwards. In first approximation, it seems natural to assume that the opinion
plays the role of the velocity and the preference that of the position. Indeed, at least in the case
in which an agent has to end up with one of the two preferences ±1, like e.g. in a referendum,
one can assume that a large part of the agents with positive opinion will move their preferences
rightwards, while agents with negative opinion will move their preferences leftwards. Clearly, one
cannot resort to a law like (1), which allows the position to increase or decrease indefinitely in
time: a correction is required in order to maintain the preference variable in the allowed interval
[1,1].
A primary example of the analogy just discussed is provided below, where the time evolution
of the opinions and the preferences of the agents is modelled by the ODE system
i
dt = (wiα)Φ(ξi)
dwi
dt =1
N
N
X
j=1
P(wi, wj)(wjwi)
(2)
(3)
for i= 1, . . . , N, supplemented with initial conditions (ξi(0), wi(0)) = (ξ0,i, w0,i )[1,1]2. The
second equation describes standard alignment dynamics among the opinions of the agents, i.e.
consensus, driven by the interaction/compromise function 0 P(·,·)1, see e.g., [40,47]. The
first equation describes instead the evolution of the preference of the ith agent based on the signed
distance between its true opinion wiand a reference opinion α[1,1] perceived in the society,
which we will refer to as the perceived social opinion. The function Φ : [1,1] [0,1] has to be
primarily chosen so as to guarantee that ξi(t)[1,1] for all t > 0. However, as we will see in
a moment, this function will be also useful to take into account meaningful polarisations of the
preferences.
In model (2)-(3), the coupling between opinion and preference is actually one-directional,
indeed the evolution of ξidepends on that of the wi’s but not vice versa. In particular, the
system (3) for the wi’s can be solved a priori, before analysing the dynamics (2) of the ξi’s. Let
us consider, in particular, the case of interactions with bounded confidence, which are described
by taking
P(wi, wj) = χ(|wjwi| ≤ ∆),(4)
where χdenotes the characteristic function and ∆ [0,2] is a given confidence threshold, above
which agents do not interact because their opinions are too far away from each other. If ∆ = 0
then only agents with the very same opinion interact, whereas if ∆ = 2 then we speak of all-to-all
interactions, considering that |wjwi| ≤ 2 for all wi, wj[1,1]. The latter case is actually
equivalent to choosing P1.
Depending on the value of ∆, one can observe a loss of global consensus. Asymptotically,
the opinions may form several clusters, whose number is dictated by ∆ and by the initial condi-
tions w0,1, . . . , w0,N , see Figure 1. However, since the function Pgiven in (4) is symmetric, i.e.
P(wi, wj) = P(wj, wi) for all i, j = 1, . . . , N, the mean opinion 1
NPN
i=1 wiis conserved in time
and for all t > 0 coincides, in particular, with the mean opinion at t= 0.
Now we give some insights into the preference dynamics modelled by (2), at least under special
forms of the function Φ. In order to avoid that the preference ξileaves the interval [1,1], a very
natural condition is Φ(±1) = 0. This implies that the constant functions ξi(t) = 1 and ξi(t)=1
are indeed stationary solutions to (2) and may therefore represent attractive or repulsive equilibria
of the system, depending on the sign of wiα.
For instance, we may choose
Φ(ξ) = 1 − |ξ|.
Taking for granted from (3) that wi(t)1 for all t > 0, if we integrate (2) starting from an initial
condition ξ0,i [0,1] then for all times t > 0 in which ξiremains non-negative we find
ξi(t)1(1 ξ0,i)e(1α)t1.
4
(a) ∆ = 1 (b) ∆ = 0.4 (c) ∆ = 0.2
Figure 1: Solution of (3) with N= 50 agents and Pgiven by (4) for decreasing values of the
confidence threshold ∆. The initial opinions w0,i have been sampled uniformly in [1,1]. The
ODE system has been integrated numerically via a standard fourth order Runge-Kutta method.
Likewise, taking for granted from (3) that wi(t)≥ −1 for all t > 0, if we start from an initial
condition ξ0,i [1,0] then for all times t > 0 in which ξiremains non-positive we deduce
ξi(t)≥ −1 + (1 + ξ0,i)e(1+α)t≥ −1.
This argument, applied to the various time intervals in which ξihas constant sign, shows indeed
that ξi(t)[1,1] for all t0. Nevertheless, we cannot solve (2) exactly, because from (3) we
cannot calculate exactly the function t7→ wi(t). On the other hand, we can get a useful idea at
least of the large time dynamics of (2) by fixing wito its asymptotic value, say w,i [1,1],
and considering the equation i
dt = (w,i α) (1 − |ξi|),
whose solution reads
ξi(t) = (1 + (1 + ξ0,i)e(w,i α)tif ξi(t)0
1(1 ξ0,i)e(w,i α)tif ξi(t)0.
From here we easily deduce that:
if ξ0,i <0 and w,i < α then ξi→ −1 for t+;
if ξ0,i >0 and w,i > α then ξi1 for t+.
In both cases, the final preference confirms and consolidates the initial one, because w,i αhas
the same sign as ξ0,i. Conversely,
if ξ0,i <0 but w,i > α then ξi1 for t+;
if ξ0,i >0 but w,i < α then ξi→ −1 for t+.
In these cases, the final preference reverses the initial one, because w,i αhas opposite sign
with respect to ξ0,i.
Another possible choice of the function Φ is:
Φ(ξ) = |ξ|1ξ2,(5)
which vanishes also at ξ= 0. Thus we are led to consider the equation
i
dt = (w,i α)|ξi|1ξ2
i,
whose solution reads
ξi(t) = ξ0,i
qξ2
0,i +1ξ2
0,ie2 sgn ξ0,i (w,iα)t
.
Now the asymptotic trend of the preference can be summarised as follows.
5
(a) ∆ = 1, α =0.3 (b) ∆ = 1, α = 0.3
Figure 2: The curves t7→ ξi(t) generated by the coupled system (2)-(3) with N= 50 agents and
P, Φ like in (4), (5) with ∆ = 1 and α=±0.3. The initial values w0,i,ξ0,i have been sampled
uniformly in the interval [1,1]. The ODE system has been integrated numerically via a standard
fourth order Runge-Kutta method.
For w,i < α:
if ξ0,i <0 then ξi→ −1 for t+;
if ξ0,i >0 then ξi0 for t+.
For w,i > α:
if ξ0,i <0 then ξi0 for t+;
if ξ0,i >0 then ξi1 for t+.
We observe that the large time behaviour of the preference is again a polarisation in poles coin-
ciding with the zeroes of the function Φ. Unlike the previous case, however, the presence of an
intermediate pole at ξ= 0 prevents a complete reversal of the initial preference when the latter has
opposite sign with respect to w,i α. In such a situation, the agents simply become indecisive,
their preference tending indeed to zero.
In order to illustrate the actual coupled dynamics of (2)-(3), we solve numerically the coupled
system of equations with N= 50 agents and with the functions P, Φ given in (4), (5), respectively.
In Figure 2we present the curves t7→ ξi(t) in the case ∆ = 1 and for α=±0.3. Since
the agents reach a global consensus around the centrist opinion w= 0, cf. Figure 1(a), with a
leftward-biased perceived social opinion α < 0 we observe the polarisation of the preferences either
towards the indecisiveness ξ= 0, if the initial preference was in turn leftward-biased, i.e. ξ0,i <0,
or in ξ= 1, if the initial preference was rightward-biased, i.e. ξ0,i >0, cf. Figure 2(a). Conversely,
with a rightward-biased perceived social opinion α > 0 we observe indecisiveness if ξ0,i >0 and
consolidation in ξ=1 if ξ0,i <0, cf. Figure 2(b).
Such rather simple dynamics may become more complex under the formation of multiple
opinion clusters. To exemplify this case, we consider now ∆ = 0.4, like in Figure 1(b), and again
the two cases α=±0.3, cf. Figure 3(a, b) along with also α=±0.6, cf. Figure 3(c, d). In this
case, simultaneous polarisations in ξ=±1 can also be observed, depending on the distribution of
the pairs (ξ0,i, w0,i ) at the initial time.
6
(a) ∆ = 0.4, α =0.3 (b) ∆ = 0.4, α = 0.3
(c) ∆ = 0.4, α =0.6 (d) ∆ = 0.4, α = 0.6
Figure 3: The same as in Figure 2but with ∆ = 0.4, cf. Figure 1(b), and α=±0.3 (top row),
α=±0.6 (bottom row).
3 Aggregate analysis of opinion dynamics
The discussion set forth in the previous section shows that it is in general quite hard to analyse
exactly the interplay between opinions and preferences from a strictly microscopic point of view.
Due to the severe dependence of the microscopic system on the particular initial state and tra-
jectory of each agent, the main difficulty is, as usual, to grasp the essential facts able to explain
the big picture, namely to depict the collective behaviour. For this reason, from this section we
move to a more aggregate analysis, which, starting from a description of opinion dynamics by
methods of statistical physics and kinetic theory, will finally lead us to macroscopic equations for
the preference dynamics written in terms of hydrodynamic parameters such as the density of the
agents and their mean opinion.
3.1 Microscopic binary interactions
In order to approach the opinion dynamics (3) from the point of view of kinetic theory, we need to
set up a consistent scheme of binary, i.e. pairwise, interactions among the agents. To this purpose,
inspired by [13], we consider (3) for just two agents, say i,j, and we discretise the differential
equation with the forward Euler formula during a small time step 0 < γ < 1. Setting
w:= wi(t), w:= wj(t), w0:= wi(t+γ), w0
:= wj(t+γ)
7
we obtain the binary rules
w0=w+γP (w, w)(ww) + D(w)η,
w0
=w+γP (w, w)(ww) + D(w)η, (6)
where we have also added a random contribution, given by a centred random variable η, modelling
stochastic fluctuations induced by the self-thinking of the agents. Here, D(·)0 is an opinion-
dependent diffusion coefficient modulating the amplitude of the stochastic fluctuations, that is the
variance of η.
In general, the binary interactions (6) are such that
hw0+w0
i=w+w+γ(P(w, w)P(w, w)) (ww),(7)
where h·i denotes the expectation with respect to the distribution of η. Hence the mean opinion
is in general not conserved on average in a single binary interaction unless Pis symmetric, i.e.
P(w, w) = P(w, w) for all w, w[1,1]. Furthermore, at leading order for γsmall enough
we have
h(w0)2+ (w0
)2i=w2+w2
+ 2γ(wP (w, w)wP(w, w)) (ww)
+D2(w) + D2(w)σ2+o(γ),(8)
where σ2>0 denotes the variance of η. Therefore, in general, also the energy is not conserved on
average in a single binary interaction, not even for a symmetric function P.
Equations (7), (8) show that a particularly interesting case is when Pis constant, for then
from (7) we deduce that the mean opinion is conserved in each binary interaction, while from (8)
we see that, at least in the absence of stochastic fluctuations (i.e. formally for σ2= 0), the average
energy is dissipated:
h(w0)2+ (w0
)2i=w2+w2
2γ(ww)2+o(γ)w2+w2
+o(γ).
In order to be physically admissible, the interaction rules (6) have to be such that |w0|,|w0
| ≤ 1
for |w|,|w| ≤ 1. Observing that
|w0|=|(1 γP (w, w))w+γP (w, w)w+D(w)η|
(1 γP (w, w)) |w|+γP (w, w) + D(w)|η|,
where we have used the fact that |w| ≤ 1, we see that a sufficient condition for |w0| ≤ 1 is
D(w)|η| ≤ (1 γP (w, w))(1 − |w|),
which is satisfied if there exists a constant c > 0 such that
(|η| ≤ c(1 γP (w, w))
cD(w)1− |w|,w, w[1,1].(9)
Considering that P(w, w)1 by assumption, the first condition can be further enforced by
requiring |η| ≤ c(1 γ), which implies that ηhas to be chosen as a compactly supported random
variable. The second condition forces instead D(±1) = 0. Taking inspiration from [47], possible
choices are: D(w)=1− |w|and c= 1, which produces |η| ≤ 1γ; or D(w) = 1 w2and c=1
2,
which yields |η| ≤ 1
2(1 γ). Another less obvious option is
D(w) = q(1 (1 + γs)w2)+and c=γs/2
1 + γs, s > 0,(10)
where (·)+:= max{0,·} denotes the positive part, which produces |η| ≤ γs/2(1γ)
1+γs. This function
Dconverges uniformly to 1w2in [1,1] when γ0+. Notice, however, that such a uniform
limit does not comply with (10) regardless of choice of c > 0, because of the infinite derivative at
w=±1.
Exactly the same considerations hold true for the second interaction rule in (6).
8
3.2 Kinetic description and steady states
Introducing the distribution function f=f(t, w) : R+×[1,1] R+, such that f(t, w)dw is the
fraction of agents with opinion in [w, w +dw] at time t, the binary rules (6) can be encoded in a
Boltzmann-type kinetic equation, which, in weak form, writes:
d
dt Z1
1
ϕ(w)f(t, w)dw
=1
2Z1
1Z1
1hϕ(w0) + ϕ(w0
)ϕ(w)ϕ(w)if(t, w)f(t, w)dw dw,(11)
where ϕ: [1,1] Ris an arbitrary test function, i.e. any observable quantity depending on the
microscopic state of the agents. Choosing ϕ(w) = 1, we obtain that the integral of fwith respect
to wis constant in time, i.e. that the total number of agents is conserved. This also implies that,
up to normalisation at the initial time, fcan be thought of as a probability density for every
t > 0. Choosing instead ϕ(w) = wwe discover
d
dt Z1
1
wf (t, w)dw =γ
2Z1
1Z1
1
(P(w, w)P(w, w))(ww)f(t, w)f(t, w)dw dw,(12)
therefore the mean opinion M1:= R1
1wf (t, w)dw is either conserved in time, if Pis symmetric so
that the right-hand side of the previous equation vanishes, or not conserved, if Pis non-symmetric.
This difference has important consequences on the steady distributions of (11), which in turn will
impact considerably on the equations describing the formation of the preferences. Therefore, in
what follows we investigate it in some detail.
3.2.1 Symmetric P
The prototype of a symmetric Pis the constant function P1. In this case, from (11) we
can recover an explicit expression of the asymptotic distribution function at least in the so-called
quasi-invariant regime, i.e. the one in which the variation of the opinion in each binary interaction
is small. To describe such a regime, we scale the parameters γ,σ2in (6) as
γγ, σ2σ2,(13)
where  > 0 is an arbitrarily small scaling coefficient. Parallelly, in order to study the large time
behaviour of the system, we introduce the new time scale τ:= t and we scale the distribution
function as g(τ, w) := f(τ
, w). In this way, it is clear that, at every fixed τ > 0 and in the limit
0+,gdescribes the large time trend of f. Since τg=1
tf, substituting in (11) and using
the symmetry of the interactions (6) with P1 we see that the equation satisfied by gis
d
Z1
1
ϕ(w)g(τ, w)dw =1
Z1
1Z1
1hϕ(w0)ϕ(w)ig(τ, w)g(τ, w)dw dw.(14)
Now, because of the scaling (13), if ϕis sufficiently smooth then the difference hϕ(w0)ϕ(w)i
is small and can be expanded about wto give:
hϕ(w)ϕ(w0)i=ϕ0(w)hw0wi+1
2ϕ00(w)h(w0w)2i+1
6ϕ000( ¯w)h(w0w)3i,
where min{w, w0}<¯w < max{w, w0}. Plugging into (14) this produces
d
Z1
1
ϕ(w)g(τ, w)dw =γZ1
1
ϕ0(w)(mw)g(τ, w)dw
+σ2
2Z1
1
ϕ00(w)D2(w)g(τ, w)dw +Rϕ(g, g),
9
Figure 4: Asymptotic opinion distribution (16) with mean m= 0.25 and four different values of
the parameter λ.
where we have denoted by m[1,1] the constant mean opinion and where Rϕ(g, g) is a reminder
such that |Rϕ(g, g)|=O() under the assumption that ηhas finite third order moment, i.e.
h|η|3i<+, cf. [47] for details. Hence for 0+it results Rϕ(g, g)0 and we get
d
Z1
1
ϕ(w)g(τ, w)dw =γZ1
1
ϕ0(w)(mw)g(τ, w)dw +σ2
2Z1
1
ϕ00(w)D2(w)g(τ, w)dw.
Integrating back by parts the terms on the right-hand side and assuming ϕ(±1) = ϕ0(±1) = 0,
due to the arbitrariness of ϕthis can be recognised as a weak form of the Fokker-Planck equation
τg=σ2
22
wD2(w)g+γ∂w((wm)g).(15)
Fixing1D(w) = 1w2, the unique asymptotic (τ+) solution with unitary mass, say
g(w), to (15) reads
g(w) = (1 + w)1+m
λ1(1 w)1m
λ1
22
λ1B1+m
λ,1m
λ, λ := σ2
γ,(16)
where B(·,·) denotes the Beta function. Notice that such a gis a Beta probability density
function on the interval [1,1]. Using the known formulas for the moments of Beta random
variables, we easily check that its mean is indeed mand we compute its energy as
M2,:= Z1
1
w2g(w)dw =2m2+λ
2 + λ.(17)
In Figure 4we illustrate some typical trends of the distribution function (16) with positive
mean, m= 0.25 in this example. We observe that, depending on the value of λ, such a distribution
may depict a transition from a strong consensus around the mean (λ= 0.1) to a milder consensus
(λ= 0.4) and further to a radicalisation in the extreme opinion w= 1 (λ= 1) up to the appearance
of a double radicalisation in the two opposite extreme opinions w=±1 (λ= 2).
3.2.2 Non-symmetric P
A natural prototype of a non-symmetric function Pis a linear perturbation of a constant P
depending on only one of the two variables w,w. More specifically, we consider
P(w, w) = P(w) = pw+q, (18)
1In view of the scaling (13), as 0+the function (10) converges uniformly to 1w2, which can therefore
be chosen as diffusion coefficient in the Fokker-Planck equation (15)after performing the quasi-invariant limit.
10
where p, q Rhave to be chosen in such a way that pw+q[0,1] for all w[1,1]. This is
obtained if
0q1,|p| ≤ min{q, 1q}.
With respect to model (6), such a function Pdescribes a situation in which agents with opinion
w>0 are more persuasive than agents with opinion w<0 if p > 0 and vice versa if p < 0.
Using (18) in (12) we obtain that the evolution of the mean opinion M1=M1(t) is ruled by
dM1
dt =
2Z1
1Z1
1
(ww)2f(t, w)f(t, w)dw dw,
whence we see that the sign of the time derivative dM1
dt coincides with that of p. Thus, if p > 0
the mean opinion is non-decreasing, while if p < 0 the mean opinion is non-increasing. Continuing
the previous calculation, we further find:
dM1
dt =
2M2M2
1,
which indicates that at the steady state (t+) it results invariably M2,=M2
1,. This
implies that the asymptotic distribution has zero variance, thus that it is necessarily a Dirac delta
centred in the asymptotic mean opinion, i.e. f(w) = δ(wM1,). Plugging this into (11) we
discover
hϕ(M1,+D(M1,)η)i − ϕ(M1,)=0,
which has to hold for every test function ϕ. As a consequence, we deduce D(M1,) = 0, whence
M1,=±1 if the only zeroes of the diffusion coefficient are w=±1 like in the examples considered
in Section 3.1.
In conclusion, with the non-symmetric function Pgiven by (18) we fully characterise the
asymptotic distribution function as:
f(w) = δ(w+ 1) if p < 0, the mean opinion decreasing from its initial value to M1,=1;
f(w) = δ(w1) if p > 0, the mean opinion increasing from its initial value to M1,= 1.
The considerations above can be generalised to the following function P:
P(w, w) = rw +pw+q, (19)
where p6=r, so that Pis non-symmetric, and where the coefficients p, q, r Rhave to be chosen
in such a way that rw +pw+q[0,1] for all (w, w)[1,1]2. Repeating the previous
calculations, we conclude that:
f(w) = δ(w+ 1) if pr < 0; in this case, the mean opinion decreases from its initial value
to M1,=1;
f(w) = δ(w1) if pr > 0; in this case, the mean opinion increases from its initial value
to M1,= 1.
From the modelling point of view, we may interpret the difference pras a balance between
the persuasion ability of the agents, expressed by p, and their tendency to be persuaded, expressed
by r. Notice indeed that for p= 0 and r6= 0 we obtain the mirror case of (18), in which agents
with opinion w > 0 are more inclined to change their opinion than agents with opinion w < 0 if
r > 0 and vice versa if r < 0.
The discussion above clearly shows that an arbitrarily small perturbation of a constant P, by
destroying the conservation of the mean opinion, may drag the system towards asymptotic config-
urations much less variegated than (16) independently of the parameters γ,σ2of the interactions.
11
4 Macroscopic description of preference formation
According to model (2)-(3), the opinions of the agents evolve through mutual interactions inde-
pendent of the preferences; on the other hand, the preference of each agent is transported in time
by his/her opinion. This suggests that a proper way to account for the interplay between opinion
and preference in an aggregate manner is by means of an inhomogeneous Boltzmann-type kinetic
equation, whose transport term describes the evolution of the preference and whose “collisional”
term accounts simultaneously for the changes in the opinions.
4.1 Inhomogeneous Boltzmann-type description and hydrodynamics
A Boltzmann-type description of the opinion dynamics in the form of binary interactions (6)
coupled to the transport of the preference (2) is obtained by introducing the kinetic distribution
function
f=f(t, ξ, w) : R+×[1,1] ×[1,1] R+,
such that f(t, ξ, w)dξ dw is the proportion of agents that at time thave a preference in [ξ, ξ +]
and an opinion in [w, w +dw]. The distribution function fsatisfies the following weak Boltzmann-
type equation:
tZ1
1
ϕ(w)f(t, ξ, w)dw +ξΦ(ξ)Z1
1
(wα)ϕ(w)f(t, ξ, w)dw
=1
2Z1
1Z1
1hϕ(w0) + ϕ(w0
)ϕ(w)ϕ(w)if(t, ξ, w)f(t, ξ, w)dw dw,(20)
where the transport term (second term on the left-hand side) has been written taking into account
that, according to (2), the transport velocity of the preference ξis (wα)Φ(ξ) and where w0,w0
on the right-hand side are given by (6).
From the distribution function f, by integration with respect to the opinion w, we can compute
macroscopic quantities in the space of the preferences, such as the density of the agents with
preference ξat time t:
ρ(t, ξ) := Z1
1
f(t, ξ, w)dw
and the mean opinion of the agents with preference ξat time t:
m(t, ξ) := 1
ρ(t, ξ)Z1
1
wf (t, ξ, w)dw.
The interest in (20) is that it allows one to obtain evolution equations directly for the quantities
ρ,m, provided one is able to characterise the large time statistical trends of the opinions, like in
Section 3. The underlying key idea is to consider a so-called hydrodynamic regime, in which the
opinions reach a local equilibrium much more quickly than the preferences, pretty much in the
spirit of the microscopic investigations performed in Section 2.
Let 0 < δ 1 be a small parameter, which we use to define a macroscopic time scale τ:= δt,
i.e. the time scale of the evolution of the preferences, which then turns out to be much larger, viz.
slower, than the characteristic one of the binary interactions among the agents. If we want that,
on this new scale, the preference dynamics remain the same, from (2) we see that we need to scale
simultaneously the transport speed of the preference by letting Φ(ξ)δΦ(ξ).
Let g(τ, ξ, w) := f(τ
δ, ξ, w), whence τg=1
δtf. Plugging into (20) we find that gsatisfies
the equation
τZ1
1
ϕ(w)g(τ, ξ, w)dw +ξΦ(ξ)Z1
1
(wα)ϕ(w)g(τ, ξ, w)dw
12
=1
2δZ1
1Z1
1hϕ(w0) + ϕ(w0
)ϕ(w)ϕ(w)ig(τ, ξ, w)g(τ, ξ, w)dw dw.(21)
Basically, the aforesaid scaling produces the coefficient 1in front of the interaction term,
hence δis analogous to the Knudsen number in classical fluid dynamics. Since we are assuming
that δis small, a hydrodynamic regime is justified and, in particular, it can be described by a
splitting of (21), cf. [26], totally analogous to the one often adopted in the numerical solution of
the inhomogeneous Boltzmann equation, see e.g. [22,23,39]. One first solves the fast interactions:
τZ1
1
ϕ(w)g(τ, ξ, w)dw
=1
2δZ1
1Z1
1hϕ(w0) + ϕ(w0
)ϕ(w)ϕ(w)ig(τ, ξ, w)g(τ, ξ, w)dw dw,(22)
which, owing to the high frequency 1, reach quickly an equilibrium described by a local (in ξand
τ) asymptotic distribution function playing morally the role of a local Maxwellian. Notice indeed
that (22) is actually an equation on the time scale of the microscopic interactions, because τcan
be scaled back to tusing the factor 1. Next, one transports such a local equilibrium distribution
according to the remaining terms of (21) on the slower hydrodynamic scale:
τZ1
1
ϕ(w)g(τ, ξ, w)dw +ξΦ(ξ)Z1
1
(wα)ϕ(w)g(τ, ξ, w)dw= 0.(23)
Due to (22), and taking the definition of ρinto account, the local “Maxwellian” can be given the
form g(τ, ξ, w) = ρ(τ, ξ)g(w), where gis one of the asymptotic opinion distribution functions
found in Section 3. This is the distribution transported by (23), hence we finally obtain
τρZ1
1
ϕ(w)g(w)dw+ξΦ(ξ)ρZ1
1
(wα)ϕ(w)g(w)dw= 0 (24)
and we can use the knowledge of gto compute explicitly the remaining integral terms.
4.2 First order hydrodynamic models
Let us consider at first the case of the non-symmetric functions P(18), (19) discussed in Sec-
tion 3.2.2. The asymptotic opinion distribution is either g(w) = δ(w+ 1) or g(w) = δ(w1),
depending on the asymmetry of P. Plugging into (24) along with the choice ϕ(w) = 1 we find
therefore either
τρ(1 + α)ξ(Φ(ξ)ρ) = 0 (25)
or
τρ+ (1 α)ξ(Φ(ξ)ρ)=0.(26)
In both cases, we get a self-consistent equation for the sole density ρand we speak thus of first
order hydrodynamic model.
Unlike typical conservation laws, in (25) and (26) the flux does not only depend on the variable
ξthrough the conserved quantity ρbut also explicitly through the function Φ. An analogous
characteristic is found, for instance, in conservation-law-based macroscopic models of vehicular
traffic featuring different flux functions in different roads, see [31].
We observe that both (25) and (26) admit the family of stationary distributional solutions
ρ(ξ) =
M
X
k=1
ρkδ(ξξk), ρk0,(27)
where the ξk’s are the zeroes of the function Φ. This indicates that models (25) and (26) reproduce
the asymptotic polarisation of the agents in the preference poles individuated by the points where
13
Φ vanishes. The coefficients ρkrepresent the masses concentrating in each pole. Furthermore, (25)
describes invariably a leftward transport of ρin the space of the preferences, because (1 + α)<0
for all α(1,1] (if α=1 the density is simply not transported). Conversely, (26) describes
invariably a rightward transport of ρ, since 1 α > 0 for all α[1,1) (now the density is not
transported if α= 1).
4.3 Second order hydrodynamic model
We now consider the symmetric case P1 discussed in Section 3.2.1, which produces the asymp-
totic opinion distribution ggiven by (16). Notice that this distribution is parametrised by the
(local) mean opinion m=m(τ, ξ), because the latter is conserved by the opinion dynamics. This
implies that, if we plug such a ginto (24) together with the choice ϕ(w) = 1, we do not get a
self-consistent equation for the density ρ. In fact, we find:
τρ+ξ(Φ(ξ)ρ(mα)) = 0,
with both hydrodynamic parameters ρ,munknown. In order to close the macroscopic equations,
we need a further equation relating ρand m, which we can obtain from (24) with ϕ(w) = wand
recalling also (17):
τ(ρm) + ξΦ(ξ)ρ2m2+λ
2 + λαm= 0.
On the whole, we get the second order (i.e., composed of a self-consistent pair of equations)
hydrodynamic model
τρ+ξ(Φ(ξ)ρ(mα)) = 0
τ(ρm) + ξΦ(ξ)ρ2m2+λ
2 + λαm= 0,
(28)
where the parameter λ=σ2, which here enters the game through the energy of the stationary
opinion distribution (16), is reminiscent of the self-thinking (diffusion) of the agents.
Also in this case, (27) is a family of admissible stationary distributional solutions. Hence
model (28) can in turn reproduce the asymptotic polarisation of the preferences already observed
in the microscopic model.
It is useful to ascertain under which conditions system (28) is hyperbolic in the natural state
space {(ρ, m)R+×[1,1]}. To this purpose, we rewrite it in the quasilinear matrix form
τU+ Φ(ξ)A(U)ξU+ Φ0(ξ)F(U)=0,
where U:= (ρ, m)T,A(U) is the matrix
A(U) := mα ρ
λ(1m2)
ρ(2+λ)
(2λ)m
2+λα!(29)
and F(U) denotes lower order terms, which is not important to write explicitly. Since Φ is real-
valued, system (28) is hyperbolic if both eigenvalues of A(U) are real. To check this, we compute
the discriminant ∆(U) of the characteristic polynomial of A(U):
∆(U) := tr2A(U)4 det A(U) = 4λ
λ+ 2 12m2
λ+ 2.
Since m[1,1], thus m2[0,1], and λ0, we easily see that ∆(U) is always non-negative.
Therefore, we conclude:
Proposition 4.1. System (28)is hyperbolic in the whole state space {(ρ, m)R+×[1,1]}for
every choice of the parameters α[1,1],λ0and for every function Φ:[1,1] [0,1].
14
4.4 General first and second order hydrodynamic models
If the asymptotic opinion distribution gis not known analytically, like e.g. in the significant
case (4), the hydrodynamic models can still be written from (24), although only in a semi-analytical
form.
Assume that the microscopic dynamics (6) do not conserve the mean opinion. Then the sole
conserved quantity is the mass of the agents and from (24) with ϕ(w) = 1 we obtain the first order
model
τρ+ (M1,α)ξ(Φ(ξ)ρ)=0
in the unknown ρ=ρ(τ, ξ), where M1,:= R1
1wg(w)dw is the asymptotic mean opinion. The
latter may be computed e.g. from (12), by means of an appropriate numerical approach.
Conversely, if the microscopic dynamics (6) conserve the mean opinion then gis parametrised
by mand from (24) with ϕ(w)=1, w we obtain the second order model
τρ+ξ(Φ(ξ)ρ(mα)) = 0
τ(ρm) + ξ(Φ(ξ)ρ(M2,(m)αm)) = 0
(30)
in the unknowns ρ=ρ(τ, ξ), m=m(τ, ξ). Here, M2,(m) := R1
1w2g(w)dw is the energy of
the asymptotic opinion distribution, expressed as a function of the conserved quantity m.
The precise calculation of M2,requires, in general, an accurate numerical reconstruction of
g. The latter is a stationary solution to the Fokker-Planck equation
τg=σ2
22
wD2(w)g+γ∂w(B[g]g) (31)
with
B[g](τ, w) := Z1
1
P(w, w)(ww)g(τ, w)dw,(32)
which is obtained in the quasi-invariant regime starting from the binary interactions (6) with
a symmetric but not necessarily constant compromise function P. In particular, the following
implicit representation of gcan be given:
g(w) = C
D2(w)exp 2
λZB[g](w)
D2(w)dw,(33)
where C > 0 is a normalisation constant and the integral on the right-hand side denotes any
antiderivative of the function w7→ B[g](w)/D2(w). For instance, if Pis the function (19) with
r=p, 0q1,|p| ≤ 1
2min{q, 1q},
so that Pis symmetric and P(w, w)[0,1] for all (w, w)[1,1]2, then from (33) we find
the semi-explicit expression
g(w) = Ce 2p
λw(1 + w)
q(1+m)p(1M2,)
λ1(1 w)
q(1m)+p(1M2,)
λ1,
which brings the calculation of M2,back to the numerical solution of the non-linear system of
equations
R1
1g(w)dw = 1
R1
1w2g(w)dw =M2,
parametrised by m.
In general, however, we observe that the type of dependence of M2,on mvalid for P1,
cf. (17), is somewhat paradigmatic. In fact, let us consider (14) in the quasi-invariant limit 0+
15
for binary interactions (6) with a symmetric P. Fixing D(w) = 1w2, we obtain the following
equation for M2:
dM2
= 2γZ1
1Z1
1
w(ww)P(w, w)g(τ, w)g(τ, w)dw dw+σ2(1 M2).
Set a:= infw,w[1,1] P(w, w), 0 a1. Then
(2 + λ)M2+ 2am2+λ1
γ·dM2
≤ −(2a+λ)M2+ 2m2+λ,
which produces asymptotically
2am2+λ
2 + λM2,2m2+λ
2a+λ.
This suggests that a perhaps rough but possibly useful approximation of M2,, to be used in (30),
is the average of these lower and upper bounds, i.e.:
M2,a
λ+ 2 +1
2a+λm2+λ(1 + a+λ)
(λ+ 2)(2a+λ),
which for a= 0, like e.g. in case (4), yields M2,m2
λ+λ+1
λ+2 .
5 Numerical tests
In this section we exemplify, by means of several numerical tests, the main features of the formation
of preferences at the kinetic and hydrodynamic scales as described by the models presented in the
previous sections.
The numerical approach is essential, in particular, to investigate the cases in which the com-
promise function Pdoes not allow for an explicit computation of the asymptotic opinion distri-
bution g. Therefore, first we will briefly review Structure Preserving (SP) numerical methods,
which are able to capture the large time solution to possibly non-local Fokker-Planck equations
with non-constant diffusion, such as those introduced in Sections 3.2.1 and 4.4, see [42,43]. Next,
we will compare the large time distributions so computed with those obtained from the numer-
ical solution of the original Boltzmann-type equation (14) in the quasi-invariant limit (1) by
means of classical Monte Carlo (MC) methods for kinetic equations [23,40]. After validating in
this way the accuracy of the numerical solver for the sole opinion dynamics (homogeneous kinetic
model), we will investigate the inhomogeneous kinetic model (20) as well as the hydrodynamic
models derived therefrom.
5.1 MC and SP methods for the homogeneous kinetic equation (14)
We begin we rewriting the Boltzmann-type equation (14) in strong form:
τg=1
Q+(g, g)g,(34)
where Q+is the gain part of the kinetic collision operator:
Q+(g, g)(τ , w) := Z1
1
1
0Jg(τ, 0w)g(τ, 0w)dw.
Here, (0w, 0w) are the pre-interaction opinions generating the post-interaction opinions (w, w)
according to the binary interaction rule (6) and 0Jis the Jacobian of the transformation from the
former to the latter.
16
To compute the solution of (34), we adopt a direct MC scheme based on the Nanbu algorithm
for Maxwellian molecules [40]. We introduce a uniform time grid τn:= nτwith fixed step
τ > 0 and we denote gn(w) := g(τn, w). A forward discretisation of (34) on such a mesh reads
then
gn+1 =1τ
gn+τ
Q+(gn, gn).(35)
From (34), owing to mass conservation, we see that R1
1Q+(g, g)(τ , w)dw = 1 for all τ > 0 if
R1
1g(0, w)dw = 1, therefore Q+(g, g)(τ, ·) can be regarded as a probability density function at
all times. From (35), under the restriction ∆τ, we obtain therefore that gn+1 is a convex
combination of two probability density functions and is therefore in turn a probability density
function. The probabilistic interpretation of (35) is clear: with probability τ
any two particles
interact during the time step ∆τ; with complementary probability 1 τ
they do not. This is
the basis on which to ground an MC-type numerical method for the approximate solution of (34).
However, it is in general numerically demanding to obtain from (35) an accurate reconstruction
of the asymptotic distribution g. To obviate this difficulty, one can take advantage of the fact
that, for sufficiently small, the large time trend of (34) is well approximated by the Fokker-
Planck equation (31). In [43], an SP numerical scheme has been specifically designed to capture
the large time behaviour of the solution to (31) with arbitrary accuracy and no restriction on the
w-mesh size. Moreover, in the transient regime that scheme is second order accurate, preserves the
non-negativity of the solution and is entropic for specific problems with gradient flow structure.
See also [24,33,42] for further applications.
To derive SP schemes for the Fokker-Planck equation (31), we rewrite the latter in flux form:
τg=F[g],(36)
where the flux is
F[g](τ, w) := C[g](τ, w)g(τ, w) + σ2
2D2(w)wg(τ, w)
and
C[g](τ, w) := γZ1
1
P(w, w)(ww)g(τ, w)dw+σ2
2(D2)0(w).
Next, we introduce a uniform grid {wi}N
i=1 [1,1] such that wi+1 wi= ∆w > 0, we denote by
gi(τ) an approximation of the grid value g(τ, wi) and we consider the conservative discretisation
of (36)
dgi
=Fi+1/2− Fi1/2
w,(37)
where Fi±1/2is an approximation of Fat wi±1/2:= wi±w
2. In particular, we choose a numerical
flux of the form
Fi+1/2:= ˜
Ci+1/2˜gi+1/2+σ2
2D2
i+1/2
gi+1 gi
w,
with ˜gi+1/2defined as a convex combination of gi,gi+1:
˜gi+1/2:= 1δi+1/2gi+1 +δi+1/2gi.
The coefficient δi+1/2[0,1] has to be properly chosen. Setting in particular
˜
Ci+1/2:= σ2D2
i+1/2
wγ
σ2Zwi+1
wi
B[g](τ, w)
D2(w)dw + log Di+1
Di,(38)
where B[g] is given by (32), we obtain explicitly
δi+1/2:= 1
λi+1/2
+1
1exp(λi+1/2)with λi+1/2:= 2∆w˜
Ci+1/2
σ2D2
i+1/2
.
17
The order of this scheme for large times coincides with that of the quadrature formula employed
for computing the integral contained in (38). In particular, if a standard Gaussian quadrature
rule is used then spectral accuracy is achieved in the wvariable. In the transient regime, instead,
the scheme is always second order accurate.
5.1.1 Comparison of the numerical solutions for large times
We now compare the large time numerical solution of the Boltzmann-type equation (34), obtained
by means of the MC scheme with the following specifications:
105particles;
quasi-invariant regime approximated by taking either = 101or = 102,
with the numerical solution of the Fokker-Planck equation (31), obtained by means of the SP
scheme with the following specifications:
N= 81 grid points for the mesh {wi}N
i=1 [1,1], yielding a mesh step ∆w= 2.5·102;
fourth order Runge-Kutta method for the time integration of (37);
Gaussian quadrature rule, with 10 quadrature points in each cell [wi, wi+1], for the approx-
imation of the integral in (38).
We use the symmetric bounded-confidence-type compromise function Pgiven by (4) with several
choices of the confidence threshold ∆ [0,2]. At the initial time τ= 0, we prescribe the uniform
distribution in [1,1], i.e.
g(0, w) = 1
2χ(w[1,1]).
In Figure 5we fix ∆ = 1 (left panels) and ∆ = 0.4 (right panels) and we take λ=σ2=
5·103. We observe that, as expected, the smaller the more the MC solution coincides with
the SP solution of the Fokker-Planck equation for either value of the confidence threshold ∆.
Furthermore, the asymptotic profiles compare qualitatively well with those obtained with the
deterministic microscopic model (3), cf. Figure 1(a, b), in terms of number and location of the
opinion clusters.
In Figure 6we repeat the same comparisons between the MC and SP numerical solutions
but with ∆ = 0.2. For λ= 104(left panels) we recover both a transient behaviour and an
asymptotic trend of the solution fully consistent with those already observed with the deterministic
microscopic model (3). In particular, four opinion clusters emerge in the long run. Interestingly, for
a slightly larger parameter λ= 103, indicating a higher relevance of the self-thinking (stochastic
fluctuation) in the behaviour of the individuals, two opinion clusters merge, thereby giving rise
to just three clusters in the long run. This aggregate phenomenon can only be observed if some
microscopic randomness is duly taken into account in the model.
5.2 Inhomogeneous kinetic equation (21)
We now pass to the inhomogeneous kinetic model, in which the formation of the preferences is
driven by an interplay with the opinion dynamics studied before.
We start by outlining the procedure by which we solve the inhomogeneous Boltzmann-type
equation (21). Since the Knudsen-like number δis assumed to be small, at each time step we adopt
the very same splitting procedure already discussed in Section 4.1. Therefore, upon introducing a
time discretisation τn:= nτ, with ∆τ > 0 constant, we proceed as follows.
Interaction step. At time τ=τn, we solve the interactions towards the equilibrium during half
a time step:
τG(τ, ξ, w) = 1
δQ(G, G)(τ, ξ, w), τ (τn, τ n+1/2]
G(τn, ξ, w) = g(τn, ξ, w)
(39)
18
-1 -0.5 0 0.5 1
0
1
2
3
4
5
-1 -0.5 0 0.5 1
0
0.5
1
1.5
2
2.5
Figure 5: Top row: contours of the distribution function gcomputed numerically for τ(0, T ],
T= 50, from the Fokker-Planck equation (31) with the SP scheme. Bottom row: comparison
of the numerical approximations at τ=Tof the large time distribution gobtained with the
previous SP scheme and with the MC scheme for the Boltzmann-type equation (34) with two
decreasing values of the parameter simulating the quasi-invariant regime. In both rows, the
confidence thresholds are ∆ = 1 (left) and ∆ = 0.4 (right).
for all ξ=ξibelonging to a suitable mesh {ξi}i[1,1]. In this step, we take advantage
of the MC scheme introduced in Section 5.1, which has proved to give asymptotic solutions
comparable to those of the more accurate SP scheme, provided the parameter δis sufficiently
small. In particular, we use a sample of 106particles and we fix δ= 102.
In (39), Qdenotes the collision operator that appears on the right-hand side of (21) once
this equation has been written in strong form.
Transport step. Next, we take the asymptotic distribution obtained in the interaction step as
the input of a pure transport towards the next time step τn+1:
τg(τ, ξ, w)+(wα)ξ(Φ(ξ)g(τ, ξ, w)) = 0, τ (τn+1/2, τ n+1 ]
g(τn+1/2, ξ, w) = G(τn+1/2, ξ, w).
In the tests of this section, unless otherwise specified, we prescribe the uniform distribution in
the variables ξ,was initial datum:
g(0, ξ, w) := 1
4χ((ξ, w)[1,1]2) (40)
we fix λ= 103and we take the function Φ given in (5).
19
-1 -0.5 0 0.5 1
0
2
4
6
8
10
-1 -0.5 0 0.5 1
0
1
2
3
4
Figure 6: The same as Figure 5but with ∆ = 0.2. In the left panels we use λ= 104, in the
right panels λ= 103. In the latter case, the asymptotic distribution features only three opinion
clusters, well reproduced by both the SP Fokker-Planck solution and the MC Boltzmann solution
(especially with = 102), because the two central clusters merge during the transient due to
a higher relevance of the self-thinking (diffusion) with respect to the tendency to compromise
(transport).
(a) τ= 1 (b) τ= 3 (c) τ= 5
Figure 7: Contours of the inhomogeneous kinetic distribution g(τ, ξ, w) at different times with
∆ = 1 and α=0.3.
5.2.1 Symmetric P
First, we consider symmetric interactions described again by the bounded confidence compromise
function (4) with ∆ = 1. In Figures 7,8we show the evolution of the inhomogeneous kinetic model
20
(a) τ= 1 (b) τ= 3 (c) τ= 5
Figure 8: The same as Figure 7but with α= 0.3.
(a) τ= 1 (b) τ= 3 (c) τ= 5
Figure 9: The same as Figure 7but with ∆ = 0.4 and α= 0.3.
-1 -0.5 0 0.5 1
0
2
4
6
(a) w-marginal at T= 10
-1 -0.5 0 0.5 1
0
2
4
6
8
(b) ξ-marginal at T= 10
Figure 10: Marginal distributions of the opinions (a) and of the preferences (b) for the numerical
test of Figure 9.
for two different choices of the perceived social opinion, α=±0.3 respectively. We clearly observe
that while the opinions distribute around the conserved mean opinion m= 0, as expected, the
preferences polarise in two possible ways. For α=0.3, cf. Figure 7, polarisations emerge in ξ= 0
and ξ= 1. Specifically, individuals with an initial preference in [1,0] tend to polarise in ξ= 0,
whereas individuals with an initial preference in (0,1] tend to polarize in ξ= 1. For α= 0.3, cf.
Figure 8, the mirror trends emerge. These polarisation patterns of the preferences are very much
consistent with those observed in Section 2with the deterministic microscopic model (2)-(3), cf.
Figure 2.
Next, we consider the same symmetric compromise function Pas before but now we fix ∆ = 0.4.
21
In Figure 9we depict the evolution of the inhomogeneous kinetic model for α= 0.3. As far as
the opinion dynamics are concerned, we recognise that individuals tend to cluster in two well
distinct positions, see Figure 10(a), directly comparable with the emerging clusters shown in
Figure 5(right) and also, up to diffusion, in Figure 1(b). Nevertheless, the social detail is now
higher, because we clearly distinguish that individuals with the same asymptotic opinion may
actually polarise in different preferences. More specifically, the opinion cluster near w=0.5 is
formed by individuals with preferences polarised in either ξ=1 or ξ= 0, while the opinion
cluster near w= 0.5 is formed by individuals with preference polarised in either ξ= 0 or ξ= 1,
see Figure 9(c). Remarkably, three polarisations of the preference emerge on the whole in the long
run, see Figure 10(b), because the opinions do not reach a global consensus.
Also these polarisation patterns of the preferences are consistent with those discussed in Sec-
tion 2, indeed the deterministic microscopic model can account in principle for three preference
poles. The fact that Figure 3(b) shows asymptotically only two of them depends essentially on
the choice of the initial conditions, which in a particle model hardly allow one to observe the
representative average trend in a single realisation.
The case α=0.3 is qualitatively analogous to the one just discussed, therefore we do not
report it in detail.
5.2.2 Non-symmetric P
Finally, we investigate the effect of a non-symmetric compromise function P. As already discussed
in Section 3.2.2, we recall that the asymmetry of Pcan be understood as a systematic bias of
the individuals, who for some reason are more prone to change opinion in a specific direction.
In this numerical example, we remain in the class of the bounded confidence models and, taking
inspiration from [32], we consider
P(w, w) = χ(LwwR),(41)
where ∆L,R[0,2] are two confidence thresholds.
In order to understand the effect of function (41), we observe that if wwthen interactions
are allowed provided |ww|=wwR. Otherwise, if wwthen interactions are allowed
provided |ww|=wwL. Thus, if e.g. ∆R>Lthen an individual with opinion wis
more incline to interact with other individuals with opinion ww. The converse holds if instead
R<L.
Remark 5.1.If ∆L= ∆Rthen (41) actually reduces to (4) with ∆ = ∆R.
We choose ∆L= 0.3 and ∆R= 0.7, meaning that individuals compromise preferentially with
other individuals with an opinion located on the right of their own. Moreover, we consider the
perceived social opinion α= 0.3. In Figure 11 we show the evolution of the inhomogeneous kinetic
model starting from the uniform distribution (40).
We observe that initially the mean opinion is neutral at any preference, indeed
Z1
1
wg(0, ξ, w)dw = 0,ξ[1,1].
Nevertheless, due to the non-symmetric interactions, the mean opinion is not conserved in time,
cf. Figure 11(d). In particular, owing to the bias induced by ∆R>L, the opinions tend to
shift on the whole rightwards, cf. Figure 11(e), while the preferences polarise in the three poles
ξ=1,0,1, cf. Figure 11(f). Again, we notice that the joint picture preference-opinion is a
lot more informative than the sole opinion dynamics, because it allows us to observe e.g. that
two clusters with nearly the same asymptotic opinion about w0.5 actually include individuals
expressing strongly different preferences (ξ= 0,1), cf. Figures 11(b, c).
22
(a) τ= 1 (b) τ= 5 (c) τ= 10
0246810
-1
-0.5
0
0.5
1
(d) Mean opinion trend
-1 -0.5 0 0.5 1
0
2
4
6
8
10
(e) w-marginal at T= 10
-1 -0.5 0 0.5 1
0
2
4
6
8
(f) ξ-marginal at T= 10
Figure 11: Top row: Contours of the inhomogeneous kinetic distribution g(τ, ξ, w) at different
times with the non-symmetric compromise function (41) featuring ∆L= 0.3, ∆R= 0.7. Bottom
row: Time trend of the mean opinion (the symmetric case is plotted for duly comparison) and
marginal distributions of opinions and preferences at time T= 10.
5.3 Hydrodynamic model
Now we test the hydrodynamic model of preference formation derived in Section 4. In particular,
since the dynamics predicted by the first order models of Section 4.2 are quite well understood
analytically, we focus on the second order model presented in Section 4.3, cf. (28).
To discretise the system of conservation laws (28), we introduce a uniform mesh in the pref-
erence domain [1,1] made of 300 grid points. Furthermore, we choose a time step ∆τ > 0 such
that the following CFL condition is met at each computational time:
τ
ξmax
ξ[1,1] [Φ(ξ) max{|µ1(τ, ξ)|,|µ2(τ, ξ)|}]1,
where µ1,µ2are the eigenvalues of the matrix (29). Then we use a WENO reconstruction in
the variable ξwith a Godunov-type numerical flux, coupled with a third order Runge-Kutta
integration in time. See [44] for a detailed description of the numerical scheme.
In all the numerical tests of this section we consider the following initial condition:
ρ(0, ξ) = 1
2χ(ξ[1,1]), m(0, ξ)=0,(42)
which represents a uniform distribution of the density over the whole range of preferences with a
null mean opinion, denoting initial indecisiveness, at all preferences.
In the first test, whose results are displayed in Figure 12 at time τ= 3, we fix the perceived
social opinion α= 0 and we consider three increasing values of the parameter λ, in particular
λ= 102,101,1, denoting a progressively stronger influence of the self-thinking. The numerical
solution clearly shows that, due to α= 0, the symmetry of the initial density is preserved,
cf. Figure 12(a). In other words, the system is not driven spontaneously towards a specific
preference, hence three symmetric polarisations emerge in ξ=1,0,1. The inspection of the
23
-1 -0.5 0 0.5 1
0
2
4
6
(a) Density
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
(b) Mean opinion
012345
0
0.1
0.2
0.3
(c) λ= 102
0 1 2 3 4 5
0
0.2
0.4
0.6
(d) λ= 1
Figure 12: Top row: density (a) and mean opinion (b) at time τ= 3 computed from the second
order hydrodynamic model (28) with α= 0 and increasing values of λ, starting from the initial
condition (42). Bottom row: time evolution of the mass of agents concentrating about the three
preference poles ξ=1,0,1 for (c) small and (d) large λ.
trend of the mean opinion, cf. Figure 12(b), reveals that individuals ending in one of the two
choices ξ=±1 tend to develop opinions in agreement with their preference, while individuals
ending in the neighbourhood of the choice ξ= 0 tend to develop opinions markedly opposite to
their preference. Such shapes of the solution are more and more evident for increasing λ. The
role of the parameter λis further stressed by Figures 12(c), (d), which show the time trend of
the mass of agents concentrating in suitable neighbourhoods of the preference poles ξ=1,0,1,
specifically the intervals [1,3
4], [1
8,1
8], [3
4,1]. For small λ, cf. Figure 12(c), there is a perfect
equilibrium among the concentrations in the three poles. For relatively larger λ, cf. Figure 12(d),
denoting an increased relevance of the self-thinking, the individuals who concentrate about ξ= 0
raise considerably with respect to those who concentrate instead in ξ=±1. On the other hand,
the latter remain symmetric.
In the second test, cf. Figure 13, we consider the same situation described before but we set
the perceived social opinion to α= 0.3, which induces a bias in the transport of the preference.
The system exhibits again three polarisations, but now with an overall trend towards ξ=1,0
and a residual polarisation in ξ= 1, cf. Figure 13(a). The reason is that now the perceived social
opinion is higher than the initial mean opinion of the individuals, therefore initially the dominant
drift is leftwards. During time, mean opinions higher than αemerge, cf. Figure 13(b), which give
24
-1 -0.5 0 0.5 1
0
5
10
(a) Density
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
(b) Mean opinion
0 1 2 3 4 5
0
0.2
0.4
0.6
(c) λ= 102
0 1 2 3 4 5
0
0.2
0.4
0.6
(d) λ= 1
Figure 13: The same as Figure 12 but with α= 0.3.
rise to the polarisation in ξ= 1 and to a further contribution to the polarisation in ξ= 0. The
latter subtracts mass to the polarisation in ξ=1, especially for large λ, cf. Figures 13(c), (d).
In order to validate the macroscopic model, in Figure 14 we compare the hydrodynamic quant-
ities ρ,mwith the numerical marginals R1
1g(τ, ξ, w)dw,R1
1wg(τ, ξ, w)dw computed from the
solution gto the inhomogeneous Boltzmann model (21). In particular, we solve the kinetic equa-
tion via the numerical procedure outlined in Section 5.2 with 50 grid points in the ξ-mesh. Starting
from the initial condition (40) for the Boltzmann model, to which there corresponds the initial
condition (42) at the hydrodynamic level, we observe that the macroscopic model describes con-
sistently the time evolution of the density of agents and of their mean opinion, as expected.
Finally, in Figure 15 we show that the mirror behaviour is observed with α=0.3, with an
overall trend towards ξ= 0,1 and a residual polarisation in ξ=1.
6 Summary and outlook
In this paper we have proposed a development of classical consensus-based opinion formation
models by including an additional variable, that we have called preference, which is transported
in time by the evolving opinions of the agents. Although commensurable with an opinion, the
preference does not simply replicate the opinion dynamics. It is rather the expression of a final
choice, which is often necessarily much sharper, i.e. more polarised, than the opinion, like for
25
-1 -0.5 0 0.5 1
0
1
2
3
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
0
1
2
3
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
Figure 14: Comparison of the numerical solution obtained in Figure 13 in the case λ= 102with
the numerical solution of the Boltzmann-type equation (21) in the quasi-invariant scaling with
δ= 102at two successive times: τ= 1 (top row) and τ= 3 (bottom row).
-1 -0.5 0 0.5 1
0
5
10
(a) Density
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
(b) Mean opinion
Figure 15: The same as Figure 12 but with α=0.3.
instance in case of polls, referendums, elections.
At the agent-based level, the inspiration to model the interplay between opinion and prefer-
ence is brought from the classical kinematic relation between position and velocity in mechanics:
roughly speaking, the time variation of the preference ξis dictated by the sign of the instantaneous
opinion w. However, here we introduce two main differences: (i) a polarisation mechanism of the
preference on some values denoting the choices available to the agents; (ii) a bias linked to the
26
general feeling perceived in the society, that we identify with a perceived social opinion α. Hence,
ultimately, the time variation of ξdepends on the signed distance wα. The meaning is clear:
an agent will tend to move his/her preference, hence to orient his/her choice, according to the rel-
ative collocation in the society that s/he perceives for him/herself. For instance, if there are three
possible choices, viz. preference poles, say ξ=1,0,1, and the perceived social opinion is α > 0,
an agent with preference ξ > 0 and opinion w > α will move his/her preference towards ξ= 1,
because wα > 0; conversely, if the opinion is w < α, the agent will move his/her preference
towards ξ= 0, because wα < 0. Such a distinction between opinion and preference is crucial to
explain how the choice processes, although originating from the opinion dynamics, produce finally
outcomes different from those of the opinion formation processes.
Taking advantage of the methods of statistical physics, in particular of the kinetic theory, we
have given an aggregate description of the interplay between opinion and preference by means of
an inhomogeneous Boltzmann-type equation, in which wplays morally the role of the “velocity”
and ξthat of the “position”. Specifically, the collision term of this equation accounts for the opin-
ion formation due to binary interactions among the agents, while the transport term accounts for
the opinion-driven preference formation. From here, we have finally derived macroscopic hydro-
dynamic models of preference formation with the technique of the local equilibrium closure, thanks
to the possibility to identify precisely, from the kinetic model, the local equilibrium distribution
of the opinions.
The analytical investigation of our models, and the related numerical results further extending
the scope of the qualitative analysis, have shown the soundness and consistency of our hierarchical
approach across all the considered scales. Moreover, they have highlighted the importance of
the parameter αin shaping the collective distribution of the preferences. In this work, we have
considered αsimply as a given constant. Future research may instead address a variable perceived
social opinion, which changes in time driven by various social forces. For instance, the control
of αcould be the goal of antithetical communication strategies commonly seen in contemporary
political scenarios. Majority parties might try to weaken the opposition parties by creating the
possibly exaggerated perception of a social opinion strongly oriented in their favour. At the same
time, opposition parties might try to exaggerate the social bias towards the majority parties, in
order to guard the electorate against the risks of too extremist positions. On the other hand,
truly extremist parties might prefer to falsely soften the perceived social opinion, in order to
avoid scaring the electorate and losing consensus. All these competitive strategies amount to
controlling αwith the aim of optimising a certain cost functional. This implies controlling the flux
of the macroscopic models of preference formation proposed in this paper, taking however into
account that the control strategy depends necessarily on the instantaneous opinion distribution,
because αis itself an opinion. From the mathematical point of view, this requires to control the
inhomogeneous Boltzmann-type kinetic equation and then to study the passage from the controlled
kinetic model to hydrodynamic equations. Suitable techniques need to be explored, in order to
make such a passage feasible from both the analytical and the numerical point of view.
Acknowledgements
This work has been written within the activities of GNFM (Gruppo Nazionale per la Fisica Mate-
matica) and GNCS (Gruppo Nazionale per il Calcolo Scientifico) of INdAM (Istituto Nazionale di
Alta Matematica), Italy. This work is also part of the activities of the Starting Grant “Attracting
Excellent Professors” funded by “Compagnia di San Paolo” (Torino) and promoted by Politecnico
di Torino.
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