# Asymptotic and stability analysis of kinetic models for opinion formation on
networks: an Allen-Cahn approach

Martin Burger , Nadia Loy, Alex Rossi Helmholtz Imaging, Deutsches Elektronen-
Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany.
Fachbereich Mathematik, Universität Hamburg, Bundesstrasse 55, Hamburg, 20146,
Germany<EMAIL_ADDRESS>di Scienze Matematica “G.L.
Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino,
Italy<EMAIL_ADDRESS>
###### Abstract

We present the analysis of the stationary equilibria and their stability in
case of an opinion formation process in presence of binary opposite opinions
evolving according to majority-like rules on social networks. The starting
point is a kinetic Boltzmann-type model derived from microscopic interactions
rules for the opinion exchange among individuals holding a certain degree of
connectivity. The key idea is to derive from the kinetic model an Allen-Cahn
type equation for the fraction of individuals holding one of the two opinions.
The latter can be studied by means of a linear stability analysis and by
exploiting integral operator analysis. While this is true for ternary
interactions, for binary interactions the derived equation of interest is a
linear scattering equation, that can be studied by means of General Relative
Entropy tools and integral operators.

Keywords: opinion formation, statistical connectivity, Boltzmann equation,
Allen-Cahn equation, linear scattering equation, General Relative Entropy

Mathematics Subject Classification: 35Q20, 35Q70, 92D50.

##  1 Introduction

Opinion dynamics has raised a relevant interest in recent years, and has been
extensively studied by means of mathematical models [17, 7, 18, 19, 10]. A
popular approach is to model the opinion as a binary value representing two
different opposite and competing opinions. The opinion formation and evolution
process is then described as a consequence of interactions between
individuals. In this sense, the possible connections among individuals are of
utmost interest, as people may be affected by those who are connected to them
in such a way to match the opinion of the majority. This is the reason why a
prominent interest is represented by opinion dynamics on complex social
networks, where each individual holds a binary opinion that may change in
consequence of majority-like rules. In this framework, both the opinion
dynamics and the network may lead the system to an equilibrium in the final
steady state, where a certain opinion dominates the system or coexists with
the other one. A large number of theoretical investigations rely on the
construction of graph models for complex networks [16, 26]. In this direction
there are works relying on physical approaches or probabilistic ones, in which
from majority-like rules on graphs, equations for aggregate quantities are
approximated or proposed by means of mainly euristic approaches [17, 7, 18,
19, 10]. In these works, the model relies on the description of the evolution
of the probability of having one of the two opposite opinions in a node of a
graph that can be characterized by its degree of connectivity (that in a
finite graph is given by the number of connections). Such a probability is
assumed to evolve according a markovian process described by a master
equation. From the latter, as done for example in [18], a Fokker-Planck
equation and the corresponding stochastic differential equation are derived
and, assuming a small negligible diffusion, the model becomes a system of ODEs
whose linear stability is analysed. Other works, such as [19], rely on linear
stability analysis of an equilibrium determined using the Perron-Frobenius
theory and numerics, or fully rely on numerical simulations [17]. In these
works, the network is often modelled as a finite graph, but also as an
hypergraph or as a multiplex network, in order to describe hierarchies in the
connections.

Concerning the description of the network, other works rely on the
characterisation of the statistical structure of social networks [1, 4, 5].
The reason for a statistical approach is due to the fact that the number of
nodes and links of a social network is so large that a detailed description by
means of classical graphs can be hardly manageable. In the present work, we
consider mesoscopic models which incorporate a statistical description of the
connectivity of the individuals. In this description, the individuals are
characterised by a discrete binary opinion ($\pm 1$) and by a positive degree
of connectivity. The social network is described statistically by means of a
probability density function for the degree of connectivity, as done for
example in [3, 25, 2, 11]. Microscopic stochastic processes describing the
interactions among individuals and leading to an exchange of the opinion given
a certain distribution of the connectivity can be stated and allow to derive a
corresponding kinetic model [11]. Such kinetic models describe the time
evolution of a distribution function of the opinion and of the degree of
connectivity. Kinetic models also allow to derive evolution equations for
collective averaged quantities, such as the average opinion on the network.

Specifically, we consider the Boltzmann-type kinetic model studied in [11].
This kinetic model implements a microscopic interaction that is a Two-against-
one mechanism (that is a kind of majority rule, in the same spirit as [21]),
that is ruled by a specific interaction kernel, while the social network is
described statistically by means of a probability density function for the
degree of connectivity. In [11] the authors consider one specific interaction
kernel that is proportional to the connectivity of the influencing agents, and
manage to derive, from the kinetic Boltzmann-type description, an evolution
equation of the average opinion on the network. The latter allows to determine
the stationary asymptotic equilibria and to characterise them. Here, we
consider a generic interaction kernel. The aim is to determine the asymptotic
states and their stability for each interaction kernel. This is not banal,
specially far from a linear stability regime, for a Boltzmann-type equation.
Then, our leading idea is that determining the stationary sates and their
stability through the analysis of an Allen-Cahn equation is easier, even for a
generic interaction kernel and at least in a linear regime, than for the
Boltzmann-type equation. As a consequence, we formally derive from the kinetic
model an Allen-Cahn equation for the evolution of the opinion distribution
over the network and study its stationary equilibria and their stability.

In Section 2, we revise the kinetic model proposed in [11] implementing the
Two-against-one intereaction, and the related essential statistical properties
of the network and of the opinion on the network. Then, we derive an
appropriate Allen-Cahn equation and perform a linear stability analysis. In
Section 3 we consider the Ochrombel simplified model and perform a similar
analysis. While in Section 2 and 3 the network is considered to be fixed and
constant in time, in Section 4, both the Two-against-one and the Ochrombel
simplified model are considered on a co-evolving network, and similar results
on the stationary equilibria are presented. In Section 5 we show some
numerical tests, while in Section 6 we draw some conclusions.

##  2 Opinion formation models and their Allen-Cahn description

In this section, firstly we briefly revise the kinetic model for opinion
formation as proposed in [11], then we derive from the kinetic description an
Allen-Cahn equation for the opinion dynamics, and, eventually, we perform the
linear stability analysis of the latter model.

###  2.1 Preliminaries on the Boltzmann-type description

Let us consider a large number of agents interacting through a social network
and exchanging their opinion. The microscopic state of a generic user is
defined by their opinion-connectivity pair $(w,\,c)$, where $w\in\\{-1,\,1\\}$
is a discrete binary variable and $c\in\mathbb{R}_{+}$ is a continuous
positive one (being $c\equiv 0$ a degenerate case). The values $w=\pm 1$
conventionally denote two opposite opinions in the same spirit as the Sznajd
model [22, 23]. The quantity $c$ is assumed to be a representative measure of
the degree of connectivity of a given individual, namely of the number of
users who may be exposed to the opinion expressed by that individual. We
consider this variable to be continuous in accordance with the reference
literature on the connectivity distribution of social networks, see e.g., [4,
5, 8, 16, 26].

Let $f=f(w,c,t)$ be the kinetic distribution function of the pair $(w,\,c)$ at
time $t\geq 0$. Owing to the discreteness of $w$, we may represent it as

$f(w,c,t)=p(c,t)\delta(w-1)+q(c,t)\delta(w+1),$ (1)

where $\delta(w-w_{0})$ denotes the Dirac delta distribution centered at
$w=w_{0}$ and $p,\,q\geq 0$ are coefficients which depend in general on $c$
and $t$ and represent the portion of agents with opinion $w=\pm 1$,
respectively, having connectivity $c$ at time $t$. Moreover, we assume
$p(\cdot,t),\,q(\cdot,t)\in L^{1}(\mathbb{R}_{+})$ for all $t\geq 0$.
Considering a constant-in-time number of users and connections of the social
network, we may impose the normalisation condition

$\int_{\mathbb{R}_{+}}\int_{\\{-1,\,1\\}}f(w,c,t)\,dw\,dc=1\qquad\forall\,t\geq
0,$ (2)

and consequently think of $f$ as the probability density of the pair $(w,c)$.
The probability density distribution of the opinion is then given by the
marginal

$h(w,t):=\int_{\mathbb{R}_{+}}f(w,c,t)\,dc=\hat{p}(t)\delta(w-1)+\hat{q}(t)\delta(w+1),$

where we have set

$\hat{p}(t):=\int_{\mathbb{R}_{+}}p(c,t)\,dc,\qquad\hat{q}(t):=\int_{\mathbb{R}_{+}}q(c,t)\,dc.$

Notice that $\hat{p}(t)$, $\hat{q}(t)$ are the probabilities that an
individual expresses the opinion $w=1$ or $w=-1$, respectively, at time $t$.
Consistently with (2), it results

$\hat{p}(t)+\hat{q}(t)=1,\qquad\forall\,t\geq 0.$ (3)

The connectivity density of the social network is instead given by the
marginal

$g(c,t):=\int_{\\{-1,\,1\\}}f(w,c,t)\,dw=p(c,t)+q(c,t).$

In particular, $g$ will be an assigned probability density function satisfying

$\int_{\mathbb{R}_{+}}g(c,t)\,dc=1,\qquad\forall t\geq 0.$ (4)

We also define

$m_{C}(t):=\int_{\mathbb{R}_{+}}c\leavevmode\nobreak\ g(c,t)dc$ (5)

being the average connectivity, that we assume to be finite.

If we assume that the number of followers of a generic individual possibly
varies in time more slowly than their opinion, and we actually do not specify
a dynamics for the connectivity, then the marginal distribution $g$ may be
well considered constant in time, i.e. $g(c,t)=g(c)$ for all $t\geq 0$.
Consequently, the sum $p(c,t)+q(c,t)$ is constant in $t$ although the single
terms $p(c,t)$, $q(c,t)$ may be not. Of course, when $g$ does not depend on
time, then $m_{C}$ is a constant.

However, a priori, the probability density function $f$ may be factorized as
follows

$f(w,c,t)=g(c,t)h_{c}(w,t)\quad\forall\,t\geq 0,$ (6)

where $h_{c}(w,t)$ is the conditional probability density of the opinions of
the agents given the degree of connectivity $c$. Assuming statistical
independence of $w$ and $c$ corresponds to

$h_{c}(w,t)=h(w,t),$ (7)

i.e. $f$ is totally factorized with respect to the dependence on $w$ and $c$.

Following standard procedures, in [11] the authors derive formally a
continuous-in-time statistical description in terms of kinetic Boltzmann-type
“collisional” equations for the distribution function $f$. Such a kinetic
equation, in weak form, using an arbitrary observable quantity that is a test
function of the microscopic state $(w,c)$, i.e.
$\phi=\phi(w,c):\\{-1,\,1\\}\times\mathbb{R}_{+}\to\mathbb{R}$, reads

$\frac{d}{dt}\int_{\mathbb{R}_{+}}\int_{\\{-1,\,1\\}}\phi(w,c)f(w,c,t)\,dw\,dc\\\
=\frac{1}{3}\int_{\mathbb{R}_{+}^{3}}\int_{\\{-1,\,1\\}^{3}}\bar{B}(w,c,w_{\ast},c_{\ast},w_{\ast\ast},c_{\ast\ast})\bigl{(}\phi(w^{\prime},c)-\phi(w,c)\bigr{)}\\\
\times
f(w,c,t)f(w_{\ast},c_{\ast},t)f(w_{\ast\ast},c_{\ast\ast},t)\,dw\,dw_{\ast}\,dw_{\ast\ast}\,dc\,dc_{\ast}\,dc_{\ast\ast},$
(8)

where $w^{\prime}=w_{\ast\ast}$ and where the collision kernel $\bar{B}$ is

$\bar{B}(w,c,w_{\ast},c_{\ast},w_{\ast\ast},c_{\ast\ast}):=\eta\delta(w_{\ast}-w_{\ast\ast})\omega(c,c_{\ast},c_{\ast\ast}),$
(9)

being $\eta>0$ a proportionality constant (the frequency of the interactions).
The latter implements a majority rule, which means that the opinion changes
according to the two-against-one model, which assumes ternary interactions in
which the first individual ($w$) takes the opinion ($w^{\prime}=w_{\ast\ast}$)
of the second two ones ($w_{\ast},w_{\ast\ast}$) if they have the same opinion
(if $w_{\ast\ast}=w_{\ast}$); otherwise, the interaction takes place but leas
to no exchange. The probability that the first individual is reached by the
common opinion of the second two individuals and gets convinced by them is
proportional to the generic interaction kernel
$\omega(c,c_{\ast},c_{\ast\ast})$, the gives a measure of the interaction rate
depending on the degree of connectivity of the individuals. In this model, the
connectivity does not change and, therefore, it is assumed that $g(c)$ is
constant in time and assigned.

In [11], the authors choose
$\omega(c,c_{\ast},c_{\ast\ast})=c_{\ast}c_{\ast\ast}$, i.e. the probability
that the first agent is reached by the opinion of the second two ones is
proportional to the connectivity of the second two agents. In this case, it is
possible to compute the exact evolution equation of the macroscopic average
opinion

$M_{w}:=\int_{\mathbb{R}_{+}}\int_{\\{-1,1\\}}f(w,c,t)\,wdcdw.$

Bu studying the evolution equation for $M_{w}$, in [11] the authors show that
for the specific choice
$\omega(c,c_{\ast},c_{\ast\ast})=c_{\ast}c_{\ast\ast}$, $M_{w}=\pm 1$ are
stable stationary states, i.e. there is polarization of the social network.
This corresponds to $\hat{p}=0$ or $\hat{p}=1$, respectively. Else, the
vanishing average opinion $M_{w}=0$, that represents coexistence of the two
different opinions and corresponds to $\hat{p}=\frac{1}{2}$, is unstable. It
is also shown that if there is statistical independence, i.e.
$h_{c}(w,t)=h(w)$ does not depend on $c$, then the average opinion has the
same sign as the initial one, i.e. there is no switch in the opinion
polarization. Specifically, it is shown that

$\frac{d\hat{p}}{dt}=\dfrac{\eta}{3}m_{C}^{2}\hat{p}(1-\hat{p})(2\hat{p}-1),$
(10)

which admits $\hat{p}=0,1$ as stable states and $\hat{p}=\frac{1}{2}$ as
unstable ones.

###  2.2 Allen-Cahn description

We now want to derive from the Boltzmann-type description (8) an Allen-Cahn
equation. Let us consider the weighted distributions of the connectivity, i.e.

$\tilde{p}(c,t):=\dfrac{p(c,t)}{g(c,t)},\quad\tilde{q}(c,t):=\dfrac{q(c,t)}{g(c,t)}.$
(11)

Specifically, we remind that, in (8), $g(c)$ is time independent. We remark
that now

$\tilde{p}(c,t)+\tilde{q}(c,t)=1,$ (12)

i.e. $\tilde{p},\tilde{q}$ still feature their dependence on $c$, but their
sum is constantly equal to one, and they may be thus seen as probabilities.
Considering the Boltzmann-equation (8) with $\bar{B}$ defined as in (9) and
the test function of the form $\phi(w,c)=\varphi(w)\psi(c)$, we get the
$c$-strong form of the Boltzmann-equation

$\frac{d}{dt}\int_{\\{-1,\,1\\}}\varphi(w)f(w,c,t)\,dw=\frac{\eta}{3}\int_{\mathbb{R}_{+}^{2}}\int_{\\{-1,\,1\\}^{3}}\omega(c,c_{\ast},c_{\ast\ast})\bigl{(}\varphi(w^{\prime})-\varphi(w)\bigr{)}\\\
\times
f(w,c,t)f(w_{\ast\ast},c_{\ast},t)f(w_{\ast\ast},c_{\ast\ast},t)\,dw\,dw_{\ast}\,dw_{\ast\ast}\,dc_{\ast}\,dc_{\ast\ast},\qquad
w^{\prime}=w_{\ast\ast}.$ (13)

If we now set $\varphi(w)=\dfrac{w+1}{2}$ in (13), divide (13) by $g(c)$ and
augment the integrand of the right-hand side of (13) by
$\dfrac{g(c_{\ast})g(c_{\ast\ast})}{g(c_{\ast})g(c_{\ast\ast})}$, then we
obtain the equation for the weighted distribution of the connectivity
$\tilde{p}$ that is

$\partial_{t}\tilde{p}(c,t)=\mathcal{D}(\tilde{p})+\Gamma(c)\left[\tilde{p}(1-\tilde{p})(2\tilde{p}-1)\right],$
(14)

where

$\mathcal{D}(\tilde{p}):=\int_{\mathbb{R}_{+}^{2}}\omega(c,c_{\ast},c_{\ast\ast})\left[\left(\tilde{p}(c_{\ast},t)\tilde{p}(c_{\ast\ast},t)-\tilde{p}^{2}(c,t)\right)\left(1-\tilde{p}(c,t)\right)\right.\\\
\left.-\tilde{p}(c,t)\left((1-\tilde{p}(c_{\ast},t))(1-\tilde{p}(c_{\ast\ast},t))-(1-\tilde{p}(c,t))^{2}\right)\right]g(c_{\ast})g(c_{\ast\ast})\,dc_{\ast}\,dc_{\ast\ast},$
(15)

and

$\Gamma(c):=\int_{\mathbb{R}_{+}^{2}}\omega(c,c_{\ast},c_{\ast\ast})g(c_{\ast})g(c_{\ast\ast})\,dc_{\ast}\,dc_{\ast\ast}.$
(16)

It can be easily observed that $\mathcal{D}$ annihilates constants, and hence
the zeros of the reaction term, that are

$\tilde{p}_{0}=0,\qquad\tilde{p}_{1}=1,\qquad\tilde{p}_{2}=\dfrac{1}{2},$ (17)

are the constant stationary solutions of (14)-(15)-(16).
We remark that $\tilde{p}_{0}=0$ corresponds to a stationary state in which
all the agents have opinion $-1$, while $\tilde{p}_{1}=1$ corresponds to a
stationary state in which all the agents have opinion $+1$, i.e.
$\tilde{p}_{0,1}$ depict a polarization scenario. The third stationary state
$\tilde{p}_{2}=\dfrac{1}{2}$, implying $\tilde{q}_{2}=\dfrac{1}{2}$, depicts a
non-polarized scenario in which half of the population has opinion $+1$ and
the other half $-1$, thus implying coexistence of the two opposite opinions.
According to this notation we have that

$h_{c}(w,t)=\tilde{p}(c,t)\delta(w-1)+\tilde{q}(c,t)\delta(w+1),$

and then we can link $\tilde{p}$ to the average opinion $m_{w}(c,t)$ for a
given connectivity $c$, that is defined as

$m_{w}(c,t)\coloneqq\int_{\\{{-1,1\\}}}wh_{c}(w,t)\,dw=\tilde{p}(c,t)-\tilde{q}(c,t)=g(c,t)\left(p(c,t)-q(c,t)\right),$

and remark that the total average opinion over the network is

$M_{w}(t)=\int_{\mathbb{R}_{+}}m_{w}(c,t)g(c,t)\,dc=\int_{\mathbb{R}_{+}}\left(p(c,t)-q(c,t)\right)\,dc.$
(18)

Then, if $\tilde{p}^{\infty}$ is a constant stationary state, the actual
stationary state is $c$-dependent as given by
$p^{\infty}(c)=\tilde{p}^{\infty}g^{\infty}(c)$ (whose existence is to be
discussed later and that is simply $g(c)$ when the connectivity is fixed).
Therefore:

$m_{w}^{\infty}(c)=(\tilde{p}^{\infty}-\tilde{q}^{\infty})=(2\tilde{p}^{\infty}-1),$

which means that the normalization function $g$ and the constant
$\tilde{p}_{\infty}$ determine the average opinion. The latter, together with
(18), also implies that

$M_{w}^{\infty}=(2\tilde{p}^{\infty}-1).$ (19)

We remark that (19), that holds for every $\omega$, implies that the polarized
scenarios described by $M_{w}^{\infty}=\pm 1$ correspond to
$\tilde{p}^{\infty}=\tilde{p}_{0,1}$ (the polarized configurations). On the
other hand, the coexistence of the two different opinions, described by
$M^{\infty}_{w}=0$, corresponds to $\tilde{p}^{\infty}=\tilde{p}_{2}$. In the
specific choice considered in [11], it is shown that $\tilde{p}_{0,1}$ are
stable, while the coexistence configuration $\tilde{p}_{2}$ is unstable. Our
aim, now, is to analyse the stability of the stationary states
$\tilde{p}_{i},\,i=0,1,2$ for a generic $\omega$, in order to see if the non-
polarized scenario is always unstable and if the polarized scenarios are
always stable.

###  2.3 Linear stability analysis of the Allen-Cahn model

From now on we shall drop the $\tilde{}$ in the notation, bearing, then, in
mind the fact that $p(c,t)$ is the fraction of agents having connectivity $c$
at time $t$ and it may be constant. Let us now consider a perturbation of the
stationary states, i.e.

$p_{i}+k_{i}(c,t),\qquad i=0,1,2.$

Plugging the latter in (14), it can be easily verified that

$\mathcal{D}(p_{i}+k_{i})=\mathcal{D}(p_{i})+\mathcal{D}^{\prime}(p_{i})k_{i}+\mathcal{O}(k_{i}^{2}),$

where

$\mathcal{D}^{\prime}(p)k=\int_{\mathbb{R}_{+}^{2}}\tilde{\omega}(c,c_{\ast},c_{\ast\ast})\left[\big{(}p(c_{\ast},t)k(c_{\ast\ast},t)+p(c_{\ast\ast},t)k(c_{\ast},t)-2p(c,t)k(c,t)\big{)}(1-p(c,t))\right.\\\
\left.-k(c,t)\big{(}p(c_{\ast},t)p(c_{\ast\ast},t)-p(c,t)^{2}\big{)}-k(c,t)\big{(}(1-p(c_{\ast},t))(1-p(c_{\ast\ast},t))-(1-p(c,t))^{2}\big{)}\right.\\\
\left.+p(c,t)\big{(}k(c_{\ast},t)(1-p(c_{\ast\ast},t))+(1-p(c_{\ast},t))k(c_{\ast\ast},t)-2(1-p(c,t))k(c,t)\big{)}\right]\,dc_{\ast}dc_{\ast\ast},$
(20)

and, therefore

$\displaystyle p_{0}=0:$
$\displaystyle\qquad\partial_{t}k_{0}=\mathcal{D}^{\prime}(p_{0})k_{0}-\Gamma
k_{0};$ $\displaystyle p_{1}=1:$
$\displaystyle\qquad\partial_{t}k_{1}=\mathcal{D}^{\prime}(p_{1})k_{1}-\Gamma
k_{1};$ $\displaystyle p_{2}=\dfrac{1}{2}:$
$\displaystyle\qquad\partial_{t}k_{2}=\mathcal{D}^{\prime}(p_{2})k_{2}+\dfrac{\Gamma}{2}k_{2}.$

We remark that for $i=0,1$ $\mathcal{D}(p_{i})=\mathcal{D}^{\prime}(p_{i})=0$:
this implies stability of the fully polarized configurations. On the other
hand, for the stationary state $p_{2}$ we get

$\partial_{t}k_{2}=\int_{\mathbb{R}_{+}^{2}}\dfrac{\omega(c,c_{\ast},c_{\ast\ast})}{2}g(c_{\ast})g(c_{\ast\ast})\big{(}k_{2}(c_{\ast},t)+k_{2}(c_{\ast\ast},t)-2k_{2}(c,t)\big{)}\,dc_{\ast}\,dc_{\ast\ast}+\dfrac{\Gamma(c)}{2}k_{2}.$
(21)

If we now consider $k_{2}=e^{\lambda t}k(c)$, the latter becomes

$\lambda
k(c)=\int_{\mathbb{R}_{+}^{2}}\dfrac{\omega(c,c_{\ast},c_{\ast\ast})}{2}g(c_{\ast})g(c_{\ast\ast})[k(c_{\ast})+k(c_{\ast\ast})]\,dc_{\ast}\,dc_{\ast\ast}-\Gamma(c)k(c)+\dfrac{\Gamma(c)}{2}k(c).$
(22)

Hence, we consider the following

######  Assumption 1.

The kernel $\omega>0$ is symmetric with respect to the connectivity of the
influencing individuals, i.e.

$\omega(c,c_{\ast},c_{\ast\ast})=\omega(c,c_{\ast\ast},c_{\ast}).$

The latter means that there is, reasonably, symmetry with respect to the two
influencing agents. As a consequence (22) becomes

$\lambda
k(c)=\int_{\mathbb{R}_{+}^{2}}\omega(c,c_{\ast},c_{\ast\ast})g(c_{\ast})g(c_{\ast\ast})k(c_{\ast})\,dc_{\ast}\,dc_{\ast\ast}-\dfrac{\Gamma(c)}{2}k(c),$
(23)

that reduces to

$k(c)=\dfrac{\int_{\mathbb{R}_{+}^{2}}\omega(c,c_{\ast},c_{\ast\ast})g(c_{\ast})g(c_{\ast\ast})k(c_{\ast})\,dc_{\ast}\,dc_{\ast\ast}}{\lambda+\Gamma(c)/2}.$
(24)

As our guess is that the coexistence configuration $\tilde{p}_{2}$ is always
unstable, we want to show the existence of a positive eigenvalue $\lambda$ by
studying (24). We shall do this by exploiting an indirect argument. First of
all, we introduce the measure space

$L^{1}(\mathbb{R}_{+},\mu),\qquad d\mu(c)=g(c)dc,$

that is a finite space measure as $g$ is a probability density function on
$\mathbb{R}_{+}$ (see (4)) and we denote it $L^{1}_{\mu}$. We consider the
space $L^{1}(\mathbb{R}_{+},\mu)$ to be endowed with its natural $L^{1}$ norm

$\|\phi\|_{L^{1}(\mathbb{R}_{+},\mu)}\coloneqq\int_{\mathbb{R}_{+}}\phi(c)g(c)\,dc.$
(25)

We remark that, as $p$ defined in (1) is in $L^{1}(\mathbb{R}_{+})$, then
$\tilde{p}$ defined in (11) naturally belongs to $L^{1}(\mathbb{R}_{+},\mu)$,
and the same holds for the perturbation $k$, that is actually a perturbation
of $\tilde{p}$. Specifically we have that

$\|\tilde{p}\|_{L^{1}(\mathbb{R}_{+},\mu)}=\|p\|_{L^{1}(\mathbb{R}_{+})}.$

We also make the further, reasonable, assumption on $\omega$, namely

######  Assumption 2.

The kernel $\omega>0$ and the distribution of the connectivities $g$ are
defined in such a way to satisfy

$\omega\in
L^{\infty}(\mathbb{R}_{+}\times\mathbb{R}_{+}\times\mathbb{R}_{+},\mu\times\mu\times\mu).$
(26)

We now introduce

$G(c,c_{\ast}):=\int_{\mathbb{R}_{+}}\omega(c,c_{\ast},c_{\ast\ast}),\,d\mu(c_{\ast\ast})$
(27)

and we remark that (26) implies that

$G\in L^{\infty}(\mathbb{R}_{+}\times\mathbb{R}_{+},\mu\times\mu),$ (28)

and also

$\Gamma\in L^{\infty}(\mathbb{R}_{+},\mu).$ (29)

Moreover, as $\omega>0$, both $G$ and $\Gamma$ are positive.

Next, we introduce the operator

$\displaystyle A_{\lambda}:L^{1}(\mathbb{R}_{+},\mu)$
$\displaystyle\longmapsto L^{1}(\mathbb{R}_{+},\mu)$ (30) $\displaystyle k$
$\displaystyle\longmapsto
A_{\lambda}k(c):=\int_{\mathbb{R}_{+}}G_{\lambda}(c,c_{\ast})k(c_{\ast})\,d\mu(c_{\ast}),$
(31)

where

$G_{\lambda}(c,c_{\ast})=\dfrac{G(c,c_{\ast})}{\lambda+\Gamma(c)/2}.$ (32)

We remark that, as $\Gamma>0$, then for each $\lambda$ we have that
$\lambda+\Gamma/2>\lambda$, and therefore $G_{\lambda}<\dfrac{G}{\lambda}$, so
that

$G_{\lambda}\in L^{\infty}(\mathbb{R}_{+}\times\mathbb{R}_{+},\mu\times\mu),$
(33)

as (28) holds. Then, for the Holder inequality, $A_{\lambda}k(c)\in
L^{1}(\mathbb{R}_{+},\mu)$ and the operator $A_{\lambda}$ is well defined.
Specifically, because of (33), $A_{\lambda}$ is a bounded operator and for the
embedding of finite measure spaces, then we also have that

$G_{\lambda}\in L^{2}(\mathbb{R}_{+}\times\mathbb{R}_{+},\mu\times\mu),$ (34)

then $A_{\lambda}$ is compact [9].

Now, we study for fixed $\lambda$ the eigenvalue problem

$A_{\lambda}k(c)=\delta k(c)$

and we want to show that there exists $\lambda>0$ for the eigenvalue
$\delta=1$ with a positive eigenfunction. This corresponds to prove that there
exists $\lambda>0$ for (24). We call $\mathcal{K}\subset
L^{1}(\mathbb{R}_{+},\mu)$ the cone of nonnegative functions, that is
reproducing. For $\lambda>0$ $A_{\lambda}$ is a strongly positive operator on
$\mathcal{K}$. Moreover, as a consequence of (33), that is granted by
Assumption 2, we have that

$\dfrac{\Gamma(c)}{\lambda+\dfrac{\Gamma(c)}{2}}\in
L^{\infty}(\mathbb{R}_{+},\mu).$ (35)

Therefore, we can apply the Krein-Rutman theorem and, then, for each
$\lambda>0$, we have that there exists $\delta=\delta(\lambda)>0$, that is
also the spectral radius of $A_{\lambda}$, associated to a positive
eigenfunction that is the only positive eigenfunction. We can remark that for
$k\equiv 1$ , $\delta(\lambda)=1$ is an eigenvalue-eigenfunction couple if

$\int_{\mathbb{R}_{+}}\dfrac{\Gamma(c)}{\lambda+\dfrac{\Gamma(c)}{2}}g(c)\,dc=1.$
(36)

The latter is not verified if $\lambda<0$. In fact, let us suppose that there
exists $\lambda<0$ satisfying (36). Then $\lambda+\Gamma(c)/2<\Gamma(c)/2$.
Therefore $\dfrac{\Gamma(c)/2}{\lambda+\frac{\Gamma(c)}{2}}>1$, and then
$\int_{\mathbb{R}_{+}}\dfrac{\frac{\Gamma(c)}{2}g(c)}{\lambda+\frac{\Gamma(c)}{2}}\,dc>\int_{\mathbb{R}_{+}}g(c)\,dc=1$,
which is a contradiction. We also remark that, if $\omega$ does not depend on
$c$, we find

$1=\dfrac{\Gamma}{\lambda+\Gamma/2},$

implying

$\lambda=\dfrac{\Gamma}{2}.$

Specifically, when $\omega(c,c_{\ast},c_{\ast\ast})=c_{\ast}c_{\ast\ast}$ as
in [11], then $\Gamma=m_{C}^{2}$, so that

$\lambda=m_{C}^{2}/2>0.$

In conclusion, we have found a positive $\lambda$ and, then,
$p_{2}=\frac{1}{2}$ is unstable.

We can summarize these observations in one theorem.

######  Theorem 1.

Let $\omega$ satisfy Assumptions 1-2. Then the operator
$\mathcal{D}^{\prime}\left(\frac{1}{2}\right)+\frac{\Gamma}{2}Id$ has a
positive eigenvalue and this means that the stationary state
$\tilde{p}_{2}=\frac{1}{2}$ of equation (14) is linearly unstable.

This allows to conclude that the stationary kinetic distribution of the
opinions is the Dirac delta centered in $1$ or $-1$, i.e.
$h^{\infty}(w)=\delta(w\pm 1)$.

######  Remark.

We remark that Assumption 2 that concerns the regularity of $\omega$ in the
finite measure space $L_{1}(\mathbb{R}_{+}^{3},\mu^{3})$, that is defined by
$g$, depends on the interplay of $\omega$ and $g$. Specifically, we remark
that kernels in the form
$\omega(c,c_{\ast},c_{\ast\ast})=cc_{\ast}c_{\ast\ast}$ or
$\omega(c,c_{\ast},c_{\ast\ast})=c_{\ast}c_{\ast\ast}$ are positive as
$c,c_{\ast},c_{\ast\ast}>0$. Moreover, they may be seen as a particular case
of kernels in the form
$\omega(c,c_{\ast},c_{\ast\ast})=\omega_{1}(c)\omega_{2}(c_{\ast})\omega_{3}(c_{\ast\ast})$
with $\omega_{i}$ functions with certain growth (and for the symmetry
assumption 1 most probably $\omega_{2}=\omega_{3}$), then:

$\displaystyle
k(c)=\omega_{1}(c)\frac{\left(\int_{\mathbb{R}_{+}}\omega_{2}(c_{\ast})k(c_{\ast})\,d\mu(c_{\ast})\right)\left(\int_{\mathbb{R}_{+}}\omega_{3}(c_{\ast\ast})\,d\mu(c_{\ast\ast})\right)}{\lambda+\frac{1}{2}\omega_{1}(c)\left(\int_{\mathbb{R}_{+}}\omega_{2}(c_{\ast})\,d\mu(c_{\ast})\right)\left(\int_{\mathbb{R}_{+}}\omega_{3}(c_{\ast\ast})\,d\mu(c_{\ast\ast})\right)}.$

If $\lambda<0$, then $\lambda+\Gamma(c)/2<\Gamma(c)$ and hence:

$k(c)>\int_{\mathbb{R}_{+}}\frac{\omega_{2}(c_{\ast})}{\int_{\mathbb{R}_{+}}\omega_{2}(c_{\ast})\,d\mu(c_{\ast})}k(c_{\ast})\,d\mu(c_{\ast})$

and we can introduce the probability measure
$\nu\coloneqq\frac{\omega_{2}(c_{\ast})}{\int_{\mathbb{R}_{+}}\omega_{2}(c_{\ast})\,d\mu(c_{\ast})}\mu$,
which is absolutely continuous with respect to $\mu$, then
$k(c)>\mathbb{E}_{\nu}[k]$ for every $c\in\mathbb{R}_{+}$, where
$\mathbb{E}_{\nu}$ is the expected value with respect to $\nu$. But this not
possible.

##  3 Linear problems

In this section we apply the methodology illustrated in the previous section
to some kinetic models in opinion dynamics implementing binary interactions.
As the starting point is a binary interaction, the kinetic model will not be
transformed into an Allen-Cahn equation, but into a linear scattering
equation. By analysing a suitable eigenvalue-eigenfunction problem we shall
study the asymptotic states.

###  3.1 The Ochrombel-type simplified model

With the same notations of the previous section, we present a further model
exposed in [11], that is the Ochrombel simplified model, which involves binary
interactions. In this model, two identical opinions are not necessary to
convince an individual, but the probability that the first individual is
convinced by the opinion of the second one is proportional to the connectivity
of the latter and the duration of the interaction. In terms of Boltzmann-type
equations, these assumptions read:

$\frac{d}{dt}\int_{\mathbb{R}_{+}}\int_{\\{-1,1\\}}\phi(w,c)f(w,c,t)\,dw\,dc\\\
=\frac{1}{2}\int_{\mathbb{R}_{+}^{2}}\int_{\\{-1,1\\}^{2}}B(c,c_{\ast})(\phi(w,c_{\ast})-\phi(w_{\ast},c_{\ast}))f(w,c,t)f(w_{\ast},c_{\ast},t)\,dw\,dw_{\ast}\,dc\,dc_{\ast}.$
(37)

In [11], the choice $B(c,c_{\ast})=\eta c$ is considered. Again, by deriving a
closed form for the evolution equation for the total average, the authors in
[11] show that the stationary average opinion may be any value in $[-1,1]$. We
now want to see if the same result holds for a generic $B$. Then, again
choosing $\phi(w,c)=\varphi(w)\psi(c)$ and performing the change of variables
$(c,w)\mapsto(c_{\ast},w_{\ast})$ at the right hand side, we have the
$c$-strong form for (37):

$\frac{d}{dt}\int_{\\{-1,1\\}}\varphi(w)f(w,c,t)\,dw\\\
=\frac{1}{2}\int_{\mathbb{R}_{+}}\int_{\\{-1,1\\}^{2}}B(c,c_{\ast})(\varphi(w_{\ast})-\varphi(w))f(w,c,t)f(w_{\ast},c_{\ast},t)\,dw\,dw_{\ast}\,dc_{\ast}.$

If $\varphi(w)=\frac{w+1}{2}$, the evolution equation for
$\tilde{p},\tilde{q}$ defined in (11) is

$\partial_{t}\tilde{p}(c,t)=\frac{1}{2}\int_{\mathbb{R}_{+}}B(c,c_{\ast})(\tilde{p}(c_{\ast},t)\tilde{q}(c,t)-\tilde{q}(c_{\ast},t)\tilde{p}(c,t))\,dc_{\ast}.$

Repeating the same argument as before we obtain

$\displaystyle\partial_{t}\tilde{p}(c,t)$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}_{+}}B(c,c_{\ast})(\tilde{p}(c_{\ast},t)(1-\tilde{p}(c,t))-(1-\tilde{p}(c_{\ast},t))\tilde{p}(c,t))g(c_{\ast})\,dc_{\ast}$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}_{+}}B(c,c_{\ast})(\tilde{p}(c_{\ast},t)-\tilde{p}(c,t))g(c_{\ast})\,dc_{\ast},$

and, if we define

$\Gamma_{B}(c)\coloneqq\int_{\mathbb{R}_{+}}B(c,c_{\ast})g(c_{\ast})\,dc_{\ast},$
(38)

we get

$\partial_{t}\tilde{p}(c,t)=\frac{1}{2}\left[\int_{\mathbb{R}_{+}}B(c,c_{\ast})\tilde{p}(c_{\ast},t)g(c_{\ast})\,dc_{\ast}-\Gamma_{B}(c)\tilde{p}(c,t)\right],$
(39)

that is a linear scattering equation (or linear Boltzmann equation). We make
the following reasonable assumption.

######  Assumption 3.

The kernel $B>0$ and the distribution $g$ are defined in such a way to satisfy

$B\in L^{\infty}(\mathbb{R}_{+}\times\mathbb{R}_{+},\mu\times\mu).$ (40)

As a consequence, $\Gamma_{B}\in L^{\infty}(\mathbb{R}_{+},\mu)$ and
$\Gamma_{B}>0$.

Typically, we introduce

$\displaystyle\mathcal{T}_{B}:L^{1}(\mathbb{R}_{+},\mu)$
$\displaystyle\longmapsto L^{1}(\mathbb{R}_{+},\mu)$ (41) $\displaystyle k$
$\displaystyle\longmapsto\mathcal{T}_{B}k(c):=\int_{\mathbb{R}_{+}}B(c,c_{\ast})k(c_{\ast})\,d\mu(c_{\ast}),$
(42)

that is well defined, bounded and compact. The operator $\mathcal{T}_{B}$
allows to rewrite (39) as

$\partial_{t}\tilde{p}=\mathcal{T}_{B}\tilde{p}-\Gamma_{B}\tilde{p}.$ (43)

We shall denote by $\mathcal{T}_{B}^{*}$ the adjoint. As we want to determine
the stationary state of (43), we introduce $n$ and $m$ the eigenfunctions of
the stationary equation and of its adjoint, i.e. respectively

$\Gamma_{B}(c)n(c)=\mathcal{T}_{B}n(c),$ (44)
$\Gamma_{B}(c)m(c)=\mathcal{T}^{*}_{B}m(c).$ (45)

In the present case, because of the definition of $\Gamma_{B}$, we have that
the adjoint $\mathcal{T}^{*}_{B}$ satisfies micro-reversibility, i.e.

$B(c,c_{\ast})g(c_{\ast})m(c_{\ast})=B(c_{\ast},c)g(c)m(c),$

and it is then conservative. On the other hand, the eigenfunction of the
operator $\mathcal{T}_{B}$ is

$n=1,$

up to multiplicative constants (any constant is an eigenfunction).
Specifically, we may argue that the operator

$\mathcal{L}_{B}=\dfrac{1}{\Gamma_{B}}\mathcal{T}_{B}$

is linear and it is bounded. As $T_{B}\in
L^{\infty}(\mathbb{R}_{+}\times\mathbb{R}_{+},\mu\times\mu)\subset
L^{2}(\mathbb{R}_{+}\times\mathbb{R}_{+},\mu\times\mu)$, then the operator
$\mathcal{L}_{B}$ is also compact. Hence we can apply the Krein-Rutman
theorem, and, therefore, there exists a positive eigenvalue (that is the
spectral radius) with a positive eigenfunction that is the unique positive one
(up to a multiplicative constant). We remark that $n\equiv 1$, $\delta=1$ is
an eigenvalue-eigenfunction couple and, in particular, all the functions
belonging to $\langle 1\rangle\subset L^{1}(\mathbb{R}_{+},\mu)$ are
eigenfunctions relative to the unitary eigenvalue. The same argument can be
applied to $\mathcal{T}_{B}^{*}$ and $m$ is a positive eigenfunction unique up
to multiplicative constants.

Then $\tilde{p}^{\infty}\in\langle 1\rangle$, and thus it is a constant. In
order to detect the exact stationary state and the decay to equilibrium, we
use classical arguments of General Relative Entropy [20]. Firstly, in order to
uniquely define the eigenfunctions, we normalize them by setting

$\int_{\mathbb{R}_{+}}n(c)g(c)\,dc=1,\qquad\int_{\mathbb{R}_{+}}n(c)m(c)g(c)\,dc=1.$
(46)

Specifically, as $g$ satisfies (4), the latter implies

$n=1.$

We can state the following general result

######  Lemma 1.

Let us consider Assumption 3, the normalization conditions (46) and solutions
$h(c,t)$ to (39), and let $H$ be a convex functional. Then we have that

$\dfrac{d}{dt}\int_{\mathbb{R}_{+}}m(c)n(c)H\left(\dfrac{h(c,t)}{n(c)}\right)\,g(c)\,dc\\\
=\int_{\mathbb{R}_{+}^{2}}B(c,c_{\ast})m(c)n(c_{\ast})\left[H\left(\dfrac{h(c,t)}{n(c)}\right)-H\left(\dfrac{h(c_{\ast},t)}{n(c_{\ast})}\right)\right.\\\
\left.+H^{\prime}\left(\dfrac{h(c_{\ast},t)}{n(c_{\ast})}\right)\left(\dfrac{h(c_{\ast},t)}{n(c_{\ast})}-\dfrac{h(c,t)}{n(c)}\right)\right]\,g(c_{\ast})dc_{\ast}g(c)dc\leq
0.$ (47)

In particular, we have that

$\rho:=\int_{\mathbb{R}_{+}}m(c)h(c,t)\,g(c)dc=\int_{\mathbb{R}_{+}}m(c)h(c,0)\,g(c)dc$
(48)

is constant in time. Moreover, there exists a constant $\nu_{3}>0$ such that

$\int_{\mathbb{R}_{+}}m(c)n(c)\left(\dfrac{h(c,t)-\rho
n(c)}{n(c)}\right)^{2}\,g(c)dc\leq\int_{\mathbb{R}_{+}}m(c)n(c)\left(\dfrac{h(c,0)-\rho
n(c)}{n(c)}\right)^{2}\,g(c)dc\exp^{-\nu_{3}t}$ (49)

###### Proof.

The proof of this Lemma relies on classical results of General Relative
Entropy as illustrated in [15], that we here only adapt to the case of
operators defined on the finite measure space $L^{1}(\mathbb{R}_{+},\mu)$. We
write it for completeness in the Appendix 7. ∎

Thanks to Lemma 1, it is possible to prove that

######  Theorem 2.

Let $B$ satisfy Assumption 3, then

$\tilde{p}^{\infty}=\rho,\qquad\rho=\int_{\mathbb{R}_{+}}m(c)p(c,0)\,dc$ (50)

is an asymptotic stable equilibrium of (39).

The proof relies on applying Lemma 1 to $h(c,t)=\tilde{p}(c,t)$ whose
evolution is ruled by (39). The constant $\rho$ is determined by (48) by
remembering that $\tilde{p}(c,0)g(c)=p(c,0)$. The fact that a priori any
constant $\tilde{p}^{\infty}$ can be the stationary state, is analogous to the
specific case considered in [11], where in the simplified model, differently
with respect to the Two-against-one model, there may be final average opinions
that can be intermediate values in $[-1,1]$. The specific intermediate value
is here determined by $\rho$, that is fixed by the initial condition $p(c,0)$,
and by the connectivity distribution $g$ and by the kernel $B$ that determine
together the operator $\mathcal{T}_{B}$ and, thus, $m$. Of course, in general
$\tilde{p}$ is not conserved (the conserved quantity is $\hat{p}+\hat{q}$),
unless $\mathcal{T}_{B}$ is self-adjoint. In fact, we have that
$\mathcal{T}_{B}^{*}$ satisfies micro-reversibility, and therefore
$\int_{\mathbb{R}_{+}}T_{B}(c,c_{\ast})\,d\mu(c_{\ast})=1$, and it is, thus,
conservative. Then, if $\mathcal{T}_{B}=\mathcal{T}_{B}^{*}$, we have that
also $m=1$ and $\rho=\|\tilde{p}\|_{L^{1}_{\mu}}=\|p\|_{L^{1}}$.

In conclusion, the kinetic asymptotic stable equilibrium of (37) is

$f^{\infty}(w,c)=p^{\infty}(c)\delta(w-1)+q^{\infty}(c)\delta(w+1),$

where

$p^{\infty}(c)=\rho g(c)\qquad q^{\infty}(c)=(1-\rho)g(c),$

$\rho$ defined in (48).

###  3.2 Continuous opinion model

This method of analysis can be applied in further cases. Let us for example
consider a multi-agent system where now the opinion is a continuous variable
$w\in[-1,1]$ evolving as a consequence of the binary interaction rules [25]

$w^{\prime}=w-\nu K(c,c_{\ast})(w-w_{\ast})+\xi D(w,c),$ (51)

where $\nu\in(0,1/2)$ and $\xi$ is a white noise s.t.
$\xi\geq\dfrac{-1+\gamma}{D}$. The latter model extends the popular
interaction rules for opinion formation introduced in [24]. In [25], the
authors consider a multi-agent system in which the agent is characterized by
the opinion $w\in[-1,1]$ and by the connectivity $c\in\mathbb{R}_{+}$ and
modify the rule by adding the dependence on $c$ in $K$ and in $D$. The
evolution equation for
$f(w,c,t):[-1,1]\times\mathbb{R}_{+}\times\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$,
in weak form, is

$\frac{d}{dt}\int_{[-1,1]}\phi(w)f(w,c,t)\,dw\\\
=\left\langle\int_{\mathbb{R}_{+}}\int_{[-1,1]^{2}}B(c,c_{\ast})(\phi(w^{\prime})-\phi(w))f(w,c,t)f(w_{\ast},c_{\ast},t)\,dw\,dw_{\ast}\,dc_{\ast}\right\rangle.$
(52)

In general, $f$ may be written as in (6), where now $w\in[-1,1]$ and we may
define the average opinion given a fixed connectivity as

$m_{w}(c,t):=\int_{[-1,1]}wh_{c}(w,t)\,dw.$

In [25], the authors show that when there is statistical independence, i.e.
(7) holds true, then $m_{w}$ does not depend on $c$, and the total average
$M_{w}$ defined as

$M_{w}(t):=\int_{\mathbb{R}_{+}}\int_{[-1,1]}wh_{c}(w,t)\,dwg(c)\,dc$ (53)

is conserved as $g$ is time independent. Conversely, when there is not
statistical independence, then the stationary conditional density function
$h_{c}^{\infty}$ can be determined by means of the quasi-invariant limit
procedure. It is shown that $h_{c}^{\infty}$ has a constant (independent of
$c$) average $m_{w}^{\infty}$, that is actually equal to $M_{w}^{\infty}$.
Then, they remark that if the ansatz ”$m_{w}(c,t)$ independent of $c$” holds
true for each $t>0$, then $m_{w}$ and $M_{w}$ are conserved in time, and,
then, it is possible to determine $M^{\infty}_{w}$. Else, its value can be
known only if $h_{c}$ is known at each time or assumptions can be made on
$D(w,c)$ in order to obtain an eventually nonlinear equation for $M_{w}$ that,
however, may have a non-unique solution.

Now, setting $\phi(w)=w$ in (52), we may find the evolution equation of
$m_{w}(c,t)$ that is

$\dfrac{\partial}{\partial
t}m_{w}(c,t)=\int_{\mathbb{R}_{+}}B(c,c_{\ast})K(c,c_{\ast})\left(m_{w}(c_{\ast},t)-m_{w}(c,t)\right)g(c_{\ast})\,dc_{\ast},$
(54)

that is again a linear scattering equation of the same type of (39) and one
can repeat the same analysis, having in this case $m_{w}$ instead of
$\tilde{p}$.

We now define

$BK(c,c_{\ast}):=B(c,c_{\ast})K(c,c_{\ast}),$

and assume the following.

######  Assumption 4.

The kernels $B>0$ and $K>0$ and the distribution $g$ are defined in such a way
that

$BK\in L^{\infty}(\mathbb{R}_{+}\times\mathbb{R}_{+},\mu\times\mu).$ (55)

Then, we can define

$\Gamma_{BK}(c):=\int_{\mathbb{R}_{+}}BK(c,c_{\ast})\,g(c_{\ast})dc_{\ast},$
(56)

satisfying $0<\Gamma_{BK}\in L^{\infty}(\mathbb{R}_{+},\mu),$ and the operator

$\displaystyle\mathcal{T}_{BK}:L^{1}(\mathbb{R}_{+},\mu)$
$\displaystyle\longmapsto L^{1}(\mathbb{R}_{+},\mu)$ (57) $\displaystyle k$
$\displaystyle\longmapsto\mathcal{T}_{BK}k(c):=\int_{\mathbb{R}_{+}}BK(c,c_{\ast})k(c_{\ast})\,d\mu(c_{\ast}),$
(58)

that is well defined, bounded and compact. We shall denote by
$\mathcal{T}_{BK}^{*}$ its adjoint. We denote by $n^{BK}$ and $m^{BK}$ the
eigenfunctions of the stationary equation and of its adjoint, i.e.
respectively

$\Gamma_{BK}(c)n^{BK}(c)=\mathcal{T}_{BK}n^{BK}(c),$ (59)
$\Gamma_{BK}(c)m^{BK}(c)=\mathcal{T}^{*}_{BK}m^{BK}(c).$ (60)

Again, because of the definition of $\Gamma_{BK}$, we have that the adjoint
$\mathcal{T}^{*}_{BK}$ satisfies micro-reversibility, i.e.

$BK(c,c_{\ast})g(c_{\ast})m^{BK}(c_{\ast})=BK(c_{\ast},c)g(c)m^{BK}(c),$

while the eigenfunction of the operator $\mathcal{T}_{BK}$ is

$n^{BK}=1$

Again, the operator $\mathcal{T}_{BK}$ is compact and the Krein-Rutman theorem
applies. Following the same procedure as before and Lemma 1, we can state the
following result.

######  Theorem 3.

Let $B,K$ satisfy Assumption 4. Then

$m_{w}^{\infty}=\rho_{w},\,\qquad\rho_{w}=\int_{\mathbb{R}_{+}}m^{BK}(c)m_{w}(c,0)g(c)\,dc.$
(61)

is an asymptotic stable equilibrium.

Again, a priori any possible constant may be the stationary state of (54). The
specific value is $\rho_{w}$ that is fixed by the initial condition
$m_{w}(c,0)$, by the connectivity distribution $g$ and by the kernel $BK$ that
determines $m^{BK}$. With respect to the previous section, where there is a
conserved quantity that is (3), in this case a priori there is no conserved
quantity. If there is conservation, then $M_{w}$ is simply equal to the
initial condition. In particular, here it is possible to see that $M_{w}$ is
conserved in time if and only if $m_{w}$ is a constant, i.e. it does not
depend on $c$, and this is a consequence of Theorem 3. Moreover, we have that
$m_{w}$ is conserved if and only if $\mathcal{T}_{BK}=\mathcal{T}_{BK}^{*}$.
This also implies that $M_{w}$ is conserved, being both $M_{w}$ and $m_{w}$
constant and equal. If there is no conservation, anyway we know the stationary
asymptotic constant value $\rho_{w}$.

Then, with respect to [25], we have shown that the independence of $m_{w}$
with respect to $c$ is a sufficient and necessary condition for conservation.
The latter is granted if $\mathcal{T}_{BK}$ is self-adjoint, as $m=n=1$, and
this defines a condition on $B$ and $K$. Else, when there is no conservation,
we can determine the, anyway stable, stationary state of (54) by using (61),
that, differently from [25], does not require the knowledge of $h_{c}$.

##  4 Co-evolving networks

We now consider, in the same spirit as [2], a co-evolving network. As a
consequence, we assume that the connectivity of the agent may vary in time
according to a given microscopic interaction that is independent of $w$, e.g.
via a process of the form

$dc=I(c)\leavevmode\nobreak\ dt+D(c)\leavevmode\nobreak\ dW,$

where $I(c)$ is a deterministic contribution, while $D(c)$ is a diffusion and
$W$ is a Wiener process. For the opinion evolution we consider the Ochrombel
type simplified model and the Two-against-one model with a kinetic version of
this process. We remark that, as $c$ varies, then $g=g(c,t)$ will depend on
time.

###  4.1 Ochrombel simplified model

The weak kinetic equation for $f$ implementing the Ochrombel type simplified
dynamics for the opinion and an independent microscopic dynamics for the
connectivity is

$\frac{d}{dt}\int_{\mathbb{R}_{+}}\int_{\\{-1,1\\}}\phi(w,c)f(w,c,t)\,dw\,dc\\\
=\frac{1}{2}\int_{\mathbb{R}_{+}^{2}}\int_{\\{-1,1\\}^{2}}B(c,c_{\ast})(\phi(w,c_{\ast})-\phi(w_{\ast},c_{\ast}))f(w,c,t)f(w_{\ast},c_{\ast},t)\,dw\,dw_{\ast}\,dc\,dc_{\ast}\\\
+\chi\left\langle\int_{\mathbb{R}_{+}}\int_{\\{-1,1\\}}(\phi(w,c^{\prime})-\phi(w,c))f(w,c,t)\,dw\,dc\right\rangle,$
(62)

with

$c^{\prime}=I(c)+D(c)\xi,$ (63)

being $\xi$ a Gaussian random variable (or drawn from some other noise model
with mean zero), $\langle\cdot\rangle$ denotes its expectation, and $\chi$ the
frequency of update of the connectivity. Equation (62) can be rewritten as

$\frac{d}{dt}\int_{\mathbb{R}_{+}}\int_{\\{-1,1\\}}\phi(w,c)f(w,c,t)\,dw\,dc\\\
=\frac{1}{2}\int_{\mathbb{R}_{+}^{2}}\int_{\\{-1,1\\}^{2}}B(c,c_{\ast})(\phi(w_{\ast},c)-\phi(w,c))f(w,c,t)f(w_{\ast},c_{\ast},t)\,dw\,dw_{\ast}\,dc\,dc_{\ast}\\\
+\chi\left\langle\int_{\mathbb{R}_{+}}\int_{\\{-1,1\\}}(\phi(w,c^{\prime})-\phi(w,c))f(w,c,t)\,dw\,dc\right\rangle,$
(64)

with (63). Concerning the evolution of the connectivity, the term related to
the variation of $c$ (second term in the right-hand side in (64)) may be
rewritten as [12]

$\frac{d}{dt}\int_{\mathbb{R}_{+}}\int_{\\{-1,1\\}}\phi(w,c)f(w,c,t)\,dw\,dc\\\
=\frac{1}{2}\int_{\mathbb{R}_{+}^{2}}\int_{\\{-1,1\\}^{2}}B(c,c_{\ast})(\phi(w_{\ast},c)-\phi(w,c))f(w,c,t)f(w_{\ast},c_{\ast},t)\,dw\,dw_{\ast}\,dc\,dc_{\ast}\\\
+\chi\int_{\mathbb{R}_{+}}\int_{\\{-1,1\\}}\phi(w,c)\left(\int_{\mathbb{R}_{+}}P(c|c^{\prime})f(t,w,c^{\prime})dc^{\prime}-f(w,c,t)\right)\,dw\,dc$
(65)

where $P(c^{\prime}|c)$ is a (conditional) probability density function of
changing connectivity into $c^{\prime}$ given the prior connectivity $c$, and
satisfying

$\int_{\mathbb{R}_{+}}P(c^{\prime}|c)dc^{\prime}=1,\quad\int_{\mathbb{R}_{+}}P(c^{\prime}|c)c^{\prime}dc^{\prime}=I(c),\quad\int_{\mathbb{R}_{+}}P(c^{\prime}|c)(c^{\prime}-I(c))^{2}dc^{\prime}=D^{2}(c),$

i.e. it has average $I(c)$ and variance $D^{2}(c)$. The existence of such a
probability distribution for (63) is an implicit assumption on $I$ and $D$. In
strong form, (65) may be rewritten as

$\frac{d}{dt}\int_{\\{-1,1\\}}\varphi(w)f(w,c,t)\,dw\ \\\
=\frac{1}{2}\int_{\mathbb{R}_{+}}\int_{\\{-1,1\\}^{2}}B(c,c_{\ast})(\varphi(w_{\ast})-\varphi(w))f(w,c,t)f(w_{\ast},c_{\ast},t)\,dw\,dw_{\ast}\,dc_{\ast}\,\\\
+\chi\int_{\\{-1,1\\}}\varphi(w)\left(\int_{\mathbb{R}_{+}}P(c|c^{\prime})f(t,w,c^{\prime})dc^{\prime}-f(w,c,t)\right)\,dw.$
(66)

Therefore, setting $\varphi(w)=\dfrac{w+1}{2}$ and dividing by $g(c,t)$,
multiplying the second term on the right-hand side by
$g(c^{\prime},t)/g(c^{\prime},t)$ we get

$\partial_{t}\tilde{p}(c,t)=\frac{1}{2}\int_{\mathbb{R}_{+}}B(c,c_{\ast})\tilde{p}(c_{\ast},t)g(c_{\ast},t)\,dc_{\ast}-\frac{1}{2}\,\Gamma_{B}(c)\tilde{p}(c,t)\\\
+\chi\left(\int_{\mathbb{R}_{+}}P(c|c^{\prime})\dfrac{g(c^{\prime},t)}{g(c,t)}\tilde{p}(t,c^{\prime})dc^{\prime}-\tilde{p}(c,t)\right)-\tilde{p}(c,t)\dfrac{\partial_{t}g(c,t)}{g(c,t)}.$
(67)

Now, setting $\varphi(w)=1$ in (66), we have that

$\partial_{t}g(c,t)=\chi\left(\int_{\mathbb{R}_{+}}P(c|c^{\prime})g(c^{\prime},t)dc^{\prime}-g(c,t)\right),$
(68)

and therefore (67) becomes

$\partial_{t}\tilde{p}(c,t)=\frac{1}{2}\int_{\mathbb{R}_{+}}B(c,c_{\ast})\tilde{p}(c_{\ast})g(c_{\ast},t)\,dc_{\ast}-\frac{1}{2}\,\Gamma_{B}(c)\tilde{p}(c,t)\\\
+\chi\left(\int_{\mathbb{R}_{+}}P_{g}(c,c^{\prime})\tilde{p}(t,c^{\prime})g(c^{\prime},t)\,dc^{\prime}-\tilde{p}(c,t)\right)-\tilde{p}(c,t)\chi\left(\int_{\mathbb{R}_{+}}P_{g}(c,c^{\prime})g(c^{\prime},t)\,dc^{\prime}-1\right),$
(69)

where

$P_{g}(c,c^{\prime})=P(c|c^{\prime})\dfrac{1}{g(c,t)}.$ (70)

Equation (67) may be rewritten in compact form as

$\partial_{t}\tilde{p}(c,t)=\int_{\mathbb{R}_{+}}\left[\frac{1}{2}B(c,c_{\ast})+\chi
P_{g}(c,c_{\ast})\right]\tilde{p}(c_{\ast},t)g(c_{\ast},t)\,dc_{\ast}-\left[\frac{1}{2}\,\Gamma_{B}(c)+\chi\Gamma_{P}(c)\right]\tilde{p}(c,t),$
(71)

where

$\Gamma_{P}(c)=\int_{\mathbb{R}_{+}}P_{g}(c,c_{\ast})g(c_{\ast},t)\,dc_{\ast}.$
(72)

Again, the equation for $\tilde{p}$ is a linear scattering equation. However
now, as we are dealing with a time-varying $g$, but we want to determine the
stationary states of (71), firstly we need to identify the asymptotic limit
$g^{\infty}$. In the present notation, we report here the result given in [6].

######  Theorem 4.

Let the mapping $c_{\ast}\mapsto P(\cdot\,|\,c_{\ast})$ be Lipschitz
continuous with respect to the $1$-Wasserstein metric $W_{1}$ in the space of
the probability measures on $\mathbb{R}_{+}$ , i.e. let a constant
$\textrm{Lip}{(P)}>0$ exist such that

$W_{1}\bigl{(}P(\cdot\,|\,c_{\ast}),\,P(\cdot\,|\,d_{\ast})\bigr{)}\leq\textrm{Lip}{(P)}\left\lvert
d_{\ast}-c_{\ast}\right\rvert,\qquad\forall\,\,c_{\ast},\,d_{\ast}\in\mathbb{R}_{+}.$
(73)

If $\chi\textrm{Lip}{(P)}<\frac{1}{2}$ then (68) admits a unique equilibrium
distribution $g_{\infty}$, which is a probability measure on $\mathbb{R}_{+}$
and which is also globally attractive, i.e.

$\lim_{t\to+\infty}W_{1}(g(\cdot,\,t),\,g^{\infty})=0$

for every solution $g$ to (68).

Then, we introduce the finite measure space related to the stationary
probability density function $g^{\infty}$

$L^{1}(\mathbb{R}_{+},\mu^{\infty}),\qquad d\mu^{\infty}=g^{\infty}(c)dc.$

We can remark that the stationary solutions $\tilde{p}^{\infty}$ of (71) are
given by

$\left[\frac{1}{2}\,\Gamma_{B}^{\infty}(c)+\chi\Gamma_{P}^{\infty}(c)\right]\tilde{p}^{\infty}(c)=\mathcal{T}_{BP}\tilde{p}^{\infty}(c)$

where

$\displaystyle\mathcal{T}_{BP}:L^{1}(\mathbb{R}_{+},\mu^{\infty})$
$\displaystyle\longmapsto L^{1}(\mathbb{R}_{+},\mu^{\infty})$ (74)
$\displaystyle k$
$\displaystyle\longmapsto\mathcal{T}_{BP}k(c):=\mathcal{T}_{B}^{\infty}k(c)+\mathcal{T}_{P}k(c)$
(75)

and

$\displaystyle\mathcal{T}_{P}:L^{1}(\mathbb{R}_{+},\mu^{\infty})$
$\displaystyle\longmapsto L^{1}(\mathbb{R}_{+},\mu^{\infty})$ (76)
$\displaystyle k$
$\displaystyle\longmapsto\mathcal{T}_{P}k(c):=\int_{\mathbb{R}_{+}}P_{g}(c,c_{\ast})k(c_{\ast})\,g^{\infty}(c_{\ast})dc_{\ast}$
(77)

and being $P^{\infty}_{g},\Gamma_{B}^{\infty},\Gamma_{P}^{\infty}$ defined by
(70),(38),(72) with $g=g^{\infty}$. We consider again Assumption 3, and,
similarly, also the following.

######  Assumption 5.

We assume that $P$ and $g$ are defined in such a way that

$P_{g}^{\infty}\in
L^{\infty}(\mathbb{R}_{+}\times\mathbb{R}_{+},\mu\times\mu),\quad
P_{g}^{\infty}>0\,.$ (78)

We again introduce the eigenfunctions $n^{BP},m^{BP}$ of the operators
$\mathcal{T}_{BP}$ and $\mathcal{T}^{*}_{BP}$, and we have again that
$n^{BP}=1$ and $m^{BP}$ is the unique positive eigenfunction determined by the
normalization condition
$\int_{\mathbb{R}_{+}}n^{BP}(c)g^{\infty}(c)\,dc=\int_{\mathbb{R}_{+}}m^{BP}(c)g^{\infty}(c)\,dc=1$.

Following the same steps as in Section 3, again we can state the following
result.

######  Theorem 5.

Let $P$ satisfy (73) with $\chi\textrm{Lip}(P)<\frac{1}{2}$, and let
$B,P_{g}^{\infty}$ satisfy Assumptions 3,5. Then the asymptotic stationary
equilibrium of (71) is

$\tilde{p}^{\infty}=\rho_{g},\qquad\rho_{g}=\int_{\mathbb{R}_{+}}m^{BP}(c)p(c,0)\,dc.$

Remembering that the constant steady state is actually $\tilde{p}^{\infty}$
defined in (11), going back to the original definition of $p$ (1), we then
have that the stationary distribution of the connectivity of the agents having
opinion $w=1$ is

$p^{\infty}(c)=g^{\infty}(c)\rho_{g}.$ (79)

Again, $\tilde{p}^{\infty}$ is conserved if $\mathcal{T}_{BP}$ is self-
adjoint. However, we know the stationary solution $\rho_{g}$ if $m^{BP}$ is
determined and the initial condition $p(c,0)$ is given.

###  4.2 Two-against-one model

We now consider the Two-against-one model on a co-evolving network. Exploiting
the previous computations, the $c$-strong kinetic equation for $f$
implementing the Two-against-one dynamics for the opinion and the same
dynamics for the connectivity is

$\frac{d}{dt}\int_{\\{-1,\,1\\}}\varphi(w)f(w,c,t)\,dw=\frac{\eta}{3}\int_{\mathbb{R}_{+}^{2}}\int_{\\{-1,\,1\\}^{3}}\omega(c,c_{\ast},c_{\ast\ast})\bigl{(}\varphi(w_{\ast\ast})-\varphi(w)\bigr{)}\\\
\times
f(w,c,t)f(w_{\ast\ast},c_{\ast},t)f(w_{\ast\ast},c_{\ast\ast},t)\,dw\,dw_{\ast}\,dw_{\ast\ast}\,dc_{\ast}\,dc_{\ast\ast}\\\
+\chi\int_{\\{-1,1\\}}\varphi(w)\left(\int_{\mathbb{R}_{+}}P(c|c^{\prime})f(t,w,c^{\prime})dc^{\prime}-f(w,c,t)\right)\,dw.$
(80)

Then, following the same procedure as in Sec. 2.2 and 4.1, we find the
evolution equation for the quantity (11), that, dropping
the$\leavevmode\nobreak\ \tilde{}$ is

$\partial_{t}p(c,t)=\mathcal{D}(p)+\Gamma(c)\left[p(1-p)(2p-1)\right]+\int_{\mathbb{R}_{+}}\chi
P_{g}(c,c_{\ast})p(c_{\ast},t)g(c_{\ast},t)\,dc_{\ast}-\chi\Gamma_{P}(c)p(c,t)$
(81)

where $\mathcal{D}$ is defined as in (15). The stationary states $p_{i}$ of
(81) are the solutions of

$\mathcal{D}(p_{i})+\Gamma(c)\left[p_{i}(1-p_{i})(2p_{i}-1)\right]+\int_{\mathbb{R}_{+}}\chi
P_{g}^{\infty}(c,c_{\ast})p_{i}(c_{\ast},t)g(c_{\ast},t)\,dc_{\ast}-\chi\Gamma^{\infty}_{P}(c)p_{i}(c,t)=0.$
(82)

We remark that, under the assumption (78), following Sec. 2.2, we find that
$p_{0}=0,p_{1}=1,p_{2}=\frac{1}{2}$ are constant solutions of (82). Repeating
the same linear stability analysis as in Sec. 2.3, we find that
$p_{0}=0,p_{1}=1$ are stable stationary states. On the other hand, for the
stationary state $p_{2}$, we have that the evolution equation of the
perturbation $k_{2}$ is

$\partial_{t}k_{2}=\int_{\mathbb{R}_{+}^{2}}\dfrac{\omega(c,c_{\ast},c_{\ast\ast})}{2}g(c_{\ast})g(c_{\ast\ast})\big{(}k_{2}(c_{\ast},t)+k_{2}(c_{\ast\ast},t)-2k_{2}(c,t)\big{)}\,dc_{\ast}\,dc_{\ast\ast}+\dfrac{\Gamma(c)}{2}k_{2}\\\
+\int_{\mathbb{R}_{+}}\chi
P_{g}(c,c_{\ast})k_{2}(c_{\ast},t)g(c_{\ast},t)\,dc_{\ast}-\chi\Gamma_{P}(c)k_{2}(c,t).$

If we now consider $k_{2}=e^{\lambda t}k(c)$, the latter becomes

$\lambda
k(c)=\int_{\mathbb{R}_{+}^{2}}\omega(c,c_{\ast},c_{\ast\ast})g(c_{\ast})g(c_{\ast\ast})k(c_{\ast})\,dc_{\ast}\,dc_{\ast\ast}-\dfrac{\Gamma(c)}{2}k(c)\\\
+\int_{\mathbb{R}_{+}}\chi
P_{g}(c,c_{\ast})k(c_{\ast})g(c_{\ast},t)\,dc_{\ast}-\chi\Gamma_{P}(c)k(c),$
(83)

where we have used Assumption 1. We now introduce

$\displaystyle\mathcal{B}_{\lambda}:L_{1}(\mathbb{R}_{+},\mu^{\infty})\longrightarrow
L_{1}(\mathbb{R}_{+},\mu^{\infty})$ (84)
$\displaystyle\mathcal{B}_{\lambda}k(c):=\dfrac{\displaystyle\int_{\mathbb{R}_{+}}G^{\infty}(c,c_{\ast})g^{\infty}(c_{\ast})k(c_{\ast})\,dc_{\ast}+\chi\int_{\mathbb{R}_{+}}P_{g}^{\infty}(c,c_{\ast})g^{\infty}(c_{\ast})k(c_{\ast})\,dc_{\ast}}{\lambda+\dfrac{\Gamma^{\infty}(c)}{2}+\chi\Gamma_{P}^{\infty}(c)},$

where the apex ∞ indicates the corresponding quantity defined with
$g=g^{\infty}$. As a consequence of Assumptions 2,5, we have that

$\dfrac{G^{\infty}(c,c_{\ast})+\chi
P_{g}^{\infty}(c,c_{\ast})}{\lambda+\dfrac{\Gamma^{\infty}(c)}{2}+\chi\Gamma_{P}^{\infty}(c)}\in
L^{\infty}(\mathbb{R}_{+}\times\mathbb{R}_{+},\mu^{\infty}\times\mu^{\infty}).$

Following the same arguments as in Section 2.3, we can apply the Krein-Rutman
Theorem and we remark that for $k\equiv 1$, $\delta(\lambda)=1$ is an
eigenvalue-eigenfunction couple if

$\int_{\mathbb{R}_{+}}\dfrac{\Gamma^{\infty}(c)+\chi\Gamma_{P}^{\infty}(c)}{\lambda+\Gamma^{\infty}(c)/2+\chi\Gamma_{P}^{\infty}(c)}\,g^{\infty}(c)dc=1.$

Arguing like in Section 2.3, we can see that the latter is not verified if
$\lambda<0$.

We can summarize these observations in one theorem.

######  Theorem 6.

Let $\omega$ satisfy Assumptions 1-2, $B$ satisfy assumption 3, $P$ satisfy
(73) and $P_{g}^{\infty}$ satisfy Assumption 5. Then the operator
$\mathcal{D}^{\prime}\left(\frac{1}{2}\right)+\frac{\Gamma}{2}Id+\chi\mathcal{T}_{P}-\chi
Id$, where $\mathcal{T}_{P}$ is defined in (76), has a positive eigenvalue and
this means that the stationary state $\tilde{p}_{2}=\frac{1}{2}$ of equation
(81) is linearly unstable.

##  5 Numerical tests

We now present some numerical tests, in order to illustrate the asymptotic
behavior of the discussed models, both in the case of Tow-against-one and
Ochrombel simplified microscopic interactions. Moreover, we shall consider the
case of the continuous opinion dynamics.

###  5.1 Numerical scheme

In order to integrate numerically Eq. (14) we consider its equivalent
formulation

$\partial_{t}p(c)=\int_{\mathbb{R}_{+}^{2}}\omega(c,c_{\ast},c_{\ast\ast})g(c_{\ast})g(c_{\ast\ast})\left[(1-p(c))p(c_{\ast})p(c_{\ast\ast})-p(c)(1-p(c_{\ast}))(1-p(c_{\ast\ast}))\right]\,dc_{\ast}\,dc_{\ast\ast}$

where we have dropped the dependence of $p$ on $t$. Let us consider a time
discretization $\\{t^{n}\\}_{n}$ where $dt:=t^{n+1}-t^{n}$ is the time step.
Hence, denoting in a semi-continuous notation $p^{n}(c)$ an approximation of
$p(c,t^{n})$, we can consider the following explicit-implicit scheme

$\dfrac{p^{n+1}(c)-p^{n}(c)}{dt}=(1-p^{n+1}(c))A^{n}-p^{n+1}(c)B^{n}$

where

$A^{n}:=\int_{\mathbb{R}_{+}^{2}}\omega(c,c_{\ast},c_{\ast\ast})g(c_{\ast})g(c_{\ast\ast})p^{n}(c_{\ast})p^{n}(c_{\ast\ast})\,dc_{\ast}\,dc_{\ast\ast}$

and

$B^{n}:=\int_{\mathbb{R}_{+}^{2}}\omega(c,c_{\ast},c_{\ast\ast})g(c_{\ast})g(c_{\ast\ast})(1-p^{n}(c_{\ast}))(1-p^{n}(c_{\ast\ast}))\,dc_{\ast}\,dc_{\ast\ast}.$

These means that

$p^{n+1}(c)=\frac{p^{n}(c)+dt\,A^{n}}{1+dt(A^{n}+B^{n})}.$

One can proceed likewise with equation (37).

###  5.2 Two-against one model

In this subsection, we illustrate the asymptotic trend of the Allen-Cahn
equation (14) for $\tilde{p}$. In Fig. 1, we consider two different pairs of
connectivity distributions $g$ and interaction kernels $\omega$. In both cases
we consider a constant initial $\tilde{p}(c,0)=$ and, specifically, we
selected different $\tilde{p}(c,0)$ in a neighbourhood of $0.5$, that is the
coexistence unstable asymptotic equilibrium. Then, in $(a)$:
$g(c)=\frac{c}{2}\mathbbm{1}\\{0\leq c\leq 2\\}$ and
$\omega(c,c_{\ast},c_{\ast\ast})=cc_{\ast}c_{\ast\ast}$, while in $(b)$
$g(c)=0.5\,\mathbbm{1}\\{0\leq c\leq 2\\}$ and
$\omega(c,c_{\ast},c_{\ast\ast})=\mathbb{1}\\{|c_{\ast}-c_{\ast\ast}|\leq
1\\}$. In both cases we remark that Assumption 2 holds as $g$ has compact
support. In both cases, we remark that the unstable constant
$\tilde{p}_{2}=\frac{1}{2}$ is the stationary state only if it is the initial
condition, i.e. $\tilde{p}(c,0)=\frac{1}{2}$. Else, we remark that the
stationary state is either $+1$ or $-1$ according to the value of the initial
constant condition. In Fig. 2, the same choice of the connectivity
distribution $g(c)=0.1e^{-0.1c}$ and interaction kernel
$\omega(c,c_{\ast},c_{\ast\ast})=c\mathbb{1}\\{|c_{\ast}-c_{\ast\ast}|$ is
made. Then, in $(a)$ the initial datum is a non rescaled truncated Gaussian
distribution centered in $1.5$ with unitary variance, whereas in $(b)$ the
same initial distribution is normalised. Again, we remark that convergence is
towards one of the two polarized configurations.

Figure 1: Time evolution of the numerical simulations of the solution
$\tilde{p}(c,t)$ to the model given by (14) with $\tilde{p}(c,0)=$constant. In
both cases, we selected different $\tilde{p}(c,0)$ in a neighbourhood of
$0.5$. In $(a)$: $g(c)=\frac{c}{2}\mathbbm{1}\\{0\leq c\leq 2\\}$ and
$\omega(c,c_{\ast},c_{\ast\ast})=cc_{\ast}c_{\ast\ast}$, while in $(b)$
$g(c)=0.5\,\mathbbm{1}\\{0\leq c\leq 2\\}$ and
$\omega(c,c_{\ast},c_{\ast\ast})=\mathbb{1}\\{|c_{\ast}-c_{\ast\ast}|\leq
1\\}$.

Figure 2: Time evolution of the numerical simulations of the solution
$\tilde{p}(c,t)$ to the model given by (14) with, $g(c)=0.1e^{-0.1c}$ and
$\omega(c,c_{\ast},c_{\ast\ast})=c\mathbb{1}\\{|c_{\ast}-c_{\ast\ast}|\leq
1\\}$. In the first case the initial datum is a non rescaled truncated
Gaussian distribution centered in 1.5 with unitary variance, whereas in the
second figure the same distribution is normalised.

###  5.3 Ochrombel simplified model

In this subsection we consider the evolution equation (39), that corresponds
to the Ochrombel simplified model (37). The results are reported in Figure 3.
As distribution of the connectivity, we consider $g(c)=e^{-c}$, while the
interaction kernel is proportional to the interactions of both agents:
$B(c,c_{\ast})=cc_{\ast}$. We remark that for the properties of $g$,
Assumption 3 holds. Two different initial data are considered: in $(a)$
$\tilde{p}(c,0)=e^{-c}$, while in $(b)$ $\tilde{p}(c,0)$ is a truncated (on
the positive axis) Gaussian centered in $2$ with unitary variance in the
second one, i.e. $\tilde{p}(c,0)=e^{-(c-2)^{2}}$. In both cases the asymptotic
value (50) is recovered: while in $(a)$
$\rho=\int_{\mathbb{R}_{+}}e^{-2c}\,dc=\frac{1}{2}$, in $(b)$ we have that
$\rho=\frac{1}{\int_{\mathbb{R}_{+}}e^{-\frac{1}{2}(c-2)^{2}}\,dc}\int_{\mathbb{R}_{+}}e^{-\frac{1}{2}(c-2)^{2}-c}\,dc\simeq
0.19$.

Figure 3: Time evolution of the numerical simulations of the solution
$\tilde{p}(c,t)$ to the model given by (39) with $g(c)=e^{-c}$,
$B(c,c_{\ast})=cc_{\ast}$, $\tilde{p}(c,0)=e^{-c}$ in the first figure and
$\tilde{p}(c,0)$ truncated (on the positive axis) Gaussian centered in $2$
with unitary variance in the second one. One can recover the respective
asymptotic values $\rho=\int_{\mathbb{R}_{+}}e^{-2c}\,dc=\frac{1}{2}$ and
$\rho=\frac{1}{\int_{\mathbb{R}_{+}}e^{-\frac{1}{2}(x-2)^{2}}\,dc}\int_{\mathbb{R}_{+}}e^{-\frac{1}{2}(c-2)^{2}-c}\,dc\simeq
0.19$.

###  5.4 Continuous opinion model

In this subsection, we present the numerical solution of (54) and its
stationary solution. The results are shown in Figure 4. Here $B=1$,
$K(c,c_{\ast})=\frac{c_{\ast}}{c+c_{\ast}}$ initial datum $m_{w}(c,0)=e^{-c}$
as in [25]. The choice of $K$ and $B$ corresponds to a left eigenfunction
$m^{BK}=c$. In $(a)$ the distribution of the connectivity is $g(c)=e^{-c}$,
that has thin tails, while in $(b)$ a fat-tailed distribution is chosen:
$g(c)=\frac{e^{-\frac{1}{c}}}{c^{2}}$ (the same as in [25]). We show that we
recover the respective asymptotic values $\rho_{w}$ defined by (61), that are
$\rho_{w}=\int_{\mathbb{R}_{+}}ce^{-2c}\,dc=0.25$ in $(a)$, and
$\rho_{w}=\int_{\mathbb{R}_{+}}\dfrac{e^{-\frac{1+c^{2}}{c}}}{c^{2}}\,dc\simeq
0.28$ in $(b)$.

Figure 4: Time evolution of the numerical simulations of the solution
$m_{w}(c,t)$ to equation (54) with $B=1$,
$K(c,c_{\ast})=\frac{c_{\ast}}{c+c_{\ast}}$, initial datum
$m_{w}(c,0)=e^{-c}$. In $(a)$ $g(c)=e^{-c}$, while in $(b)$
$g(c)=\frac{e^{-\frac{1}{c}}}{c^{2}}$. One can recover the respective
asymptotic values $\rho_{w}=\int_{\mathbb{R}_{+}}ce^{-2c}\,dc=0.25$ and
$\rho_{w}=\int_{\mathbb{R}_{+}}\dfrac{e^{-\frac{1+c^{2}}{c}}}{c^{2}}\,dc\simeq
0.28$.

##  6 Conclusion

In this paper, we have studied the asymptotic stationary equilibria of
Boltzmann-type equations for opinion formation on a social network. The
microscopic dynamics prescribes for each agent a degree of connectivity and a
binary value of the opinion evolving according to majority-like rules.
Specifically, we have considered a Boltzmann-type kinetic model for the Two-
against-one model that involves ternary interactions. In this case we have
derived an Allen-Cahn equation for the time-evolution of the fraction of
individuals with a given connectivity holding one of the two opposite
opinions. The linear stability analysis of the Allen-Cahn equation is
performed by solving an eigenfunction-eigenvalue problem related to an
integral operator defined by the interaction kernel on a suitable finite
measure space defined by the distribution of the connectivity. The linear
stability analysis of the Allen-Cahn equation has allowed us to conclude that
the polarised configurations are always stable, while the coexistence
configuration is always unstable. This is true for any choice of the
interaction kernel that gives the rate at which individuals interact according
to their degree of connectivity.

We have then considered the Ochrombel-simplified model, that relies on binary
interactions. In this case, the evolution equation for the fraction of
individuals with a given connectivity holding one of the two opinions, is a
linear scattering equation. Also in this case, the study of the stationary
equilibria amounts to the analysis of an eigenvalue-eigenfunction problem for
a suitable integral operator related to the interaction kernel and set on a
finite measure space defined by the distribution of the connectivity. In this
case, the possible stationary solutions correspond to an average opinion that
may be any value in $[-1,1]$, even though the possible opinion states are $\pm
1$. In this case, then, coexistence is possible for any interaction kernel.
Moreover, thanks to General Relative Entropy results, the exact asymptotic
stationary stable equilibrium could be determined.

We have then considered considered a model for opinion formation on a social
network when the microscopic opinion value is a continuous one that ranges in
$[-1,1]$ as done in [25]. The application of the same methodology as in the
Ochrombel case has allowed us to determine the value of the asymptotic
stationary stable average opinion for a fixed degree of connectivity, by only
considering regularity assumptions on the interaction kernels and not on the
dependence of the average opinion on the degree of connectivity. Specifically,
this methodology allows to determine the asymptotic average opinion also in
the case where it is not constant with respect to $c$ and without the need of
determining it from the conditioned kinetic distribution of the opinion
$h_{c}(w)$.

Eventually, we have applied the methodology to the case of a co-evolving
network, by considering both the case of the Two-against-one model and the
Ochrombel simplified model. In all cases, the regularity assumptions on the
interaction kernels on the finite measure space defined by the distribution of
the connectivity are minimal ($L^{\infty}$) and depend on the interplay
between the interaction kernels ($\omega,B,K$) and $g$.

The interest of the present manuscript relies on the possibility of
determining rigorously the stationary equilibria and their stability of an
opinion formation problem that is formerly described by a kinetic
Boltzmann–type model, which can be derived from appropriate microscopic
stochastic processes that can model accurately the interactions of the agents
on a social network, which, in the present case, is described statistically.

Possible future works include the study of the continuous opinion model on co-
evolving networks and the study of the equilibria of an opinion formation
process (both in the case of a discrete and a continuous opinion) on a finite
graph, a model in the same spirit as the one proposed in [14, 13] in the case
of the diffusion of a virus.

## Acknowledgements

MB acknowledges support from DESY (Hamburg, Germany), a member of the
Helmholtz Association HGF. NL is member of INdAM-GNFM. NL acknowledges support
from GNFM (Gruppo Nazionale per la Fisica Matematica) of IN- dAM (Istituto
Nazionale di Alta Matematica) through a “Progetto Giovani 2020” grant. This
work was also partially supported by the Italian Ministry of University and
Research (MUR) through the “Dipartimenti di Ec- cellenza” Programme
(2018-2022), Department of Mathematical Sciences “G. L. Lagrange”, Politecnico
di Torino (CUP: E11G18000350001). This work is also part of the activities of
the PRIN 2020 project (No. 2020JLWP23) “Integrated Mathematical Approaches to
Socio–Epidemiological Dynamics”.

##  7 Appendix

###### Proof.

It is straightforward to verify that if $h(c,t)$ is a solution of (39), then

$\partial_{t}\left(m(c)n(c)H\left(\dfrac{h(c,t)}{n(c)}\right)\right)\\\
+\int_{\mathbb{R}_{+}}B(c,c_{\ast})\left[m(c_{\ast})n(c)H\left(\dfrac{h(c,t)}{n(c)}\right)-m(c)n(c_{\ast})H\left(\dfrac{h(c_{\ast},t)}{n(c_{\ast})}\right)\right]\,g(c_{\ast})dc_{\ast}\\\
=\int_{\mathbb{R}_{+}}B(c,c_{\ast})m(c)n(c_{\ast})\left[H\left(\dfrac{h(c,t)}{n(c)}\right)-H\left(\dfrac{h(c_{\ast},t)}{n(c_{\ast})}\right)+H^{\prime}\left(\dfrac{h(c_{\ast},t)}{n(c_{\ast})}\right)\left(\dfrac{h(c_{\ast},t)}{n(c_{\ast})}-\dfrac{h(c,t)}{n(c)}\right)\right]\,g(c_{\ast})dc_{\ast}$
(85)

and, after integration on $g(c)dc$, (47) holds.

Choosing $H(u)=u$ (that is convex) allows to see that

$\dfrac{d}{dt}\int_{\mathbb{R}_{+}}m(c)h(c,t)\,g(c)\,dc=0,$

and therefore (48) holds.

Then, we choose $H(u)=u^{2}$ and we work on $h(c,t)=p(c,t)-\rho n(c)$. We
remark that $\int_{\mathbb{R}_{+}}m(c)h(c)\,g(c)dc=0$. We have that

$\dfrac{d}{dt}\int_{\mathbb{R}_{+}}m(c)n(c)H\left(\dfrac{p(c,t)}{n(c)}\right)\,g(c)\,dc=-\int_{\mathbb{R}_{+}^{2}}B(c_{\ast},c)m(c)n(c)\left[\dfrac{h(c)}{n(c)}-\dfrac{h(c_{\ast})}{n(c_{\ast})}\right]^{2}\,g(c_{\ast})dc_{\ast}g(c)dc$

It is then possible to prove a Poincaré inequality

$\nu_{3}\int_{\mathbb{R}_{+}}m(c)n(c)\left(\dfrac{h}{n}\right)^{2}g(c)dc\leq\int_{\mathbb{R}_{+}^{2}}B(c_{\ast},c)m(c)n(c)\left[\dfrac{h(c)}{n(c)}-\dfrac{h(c_{\ast})}{n(c_{\ast})}\right]^{2}\,g(c_{\ast})dc_{\ast}g(c)dc$

that holds for any $\psi>0$ as long as it is possible to define

$\nu_{1}=\int_{\mathbb{R}_{+}}n(c)m^{2}(c)/\psi(c)\,g(c)dc<\infty\qquad\nu_{2}\psi(c_{\ast})n(c)\leq
B(c,c_{\ast}),\qquad\nu_{3}=(\nu_{1}\nu_{2})^{-1}.$ (86)

In fact, whenever $\int_{\mathbb{R}_{+}}m(c)h(c,t)g(c)\,dc=0$, then

$\int_{\mathbb{R}_{+}}m(c)n(c)\left(\dfrac{h}{n}\right)^{2}g(c)dc\leq\\\
\nu_{1}\int_{\mathbb{R}_{+}}\int_{\mathbb{R}_{+}}m(c)n(c)\left[\dfrac{h(c,t)}{n(c)}-\dfrac{h(c_{\ast},t)}{n(c_{\ast})}\right]^{2}\psi(c_{\ast})n(c_{\ast})g(c_{\ast})\,dc_{\ast}g(c)\,dc\\\
\leq\nu_{1}\nu_{2}\int_{\mathbb{R}_{+}}\int_{\mathbb{R}_{+}}m(c)n(c_{\ast})B(c,c_{\ast})\left[\dfrac{h(c,t)}{n(c)}-\dfrac{h(c_{\ast},t)}{n(c_{\ast})}\right]^{2}g(c)g(c_{\ast})\,dc_{\ast}dc$

where the first inequality follows from the Holder inequality on the finite
measure space $L^{2}(\mathbb{R}_{+},\mu),L^{1}(\mathbb{R}_{+},\mu)$ and from
the first in (86), while the second inequality follows from the second in
(86). Specifically, we can choose

$\psi(c_{\ast})=\displaystyle\min_{c}B(c,c_{\ast}),\quad\nu_{2}\in(0,1),$

remembering that $B>0$. Then,

$\dfrac{d}{dt}\int_{\mathbb{R}_{+}}m(c)n(c)H\left(\dfrac{p(c,t)}{n(c)}\right)\,g(c)\,dc\leq-\nu_{3}\int_{\mathbb{R}_{+}}m(c)n(c)\left(\dfrac{h}{n}\right)^{2}g(c)dc$

and (49) follows by a simple use of Gronwall lemma. ∎

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