2.1. Deterministic Particle Approximation (DPA)

We begin this section with the rigorous formulation of the particle evolution already sketched in the Introduction. We consider in $\mathbb{R}^d$ a network of $M$ nodes and we locate an agent $a^{\mathcal{i}}\in \mathbb{R}^d$ with $\mathcal{i}\in \mathcal{M}=\{1,\ldots,M\}$ in each node. Assume that each agent may have opinion ranging on a compact set $\Omega \subset \mathbb{R}$ , without loss of generality we consider $\Omega = [{-}1,1]$ . To each agent we associate a finite opinion strength $\sigma ^{\mathcal{i}}$ and an initial opinion density $\bar{\rho }^{\mathcal{i}}(x)\in L^1(\Omega )$ such that

\begin{equation*} \sigma ^{\mathcal {i}} = \int _\Omega \bar {\rho }^{\mathcal {i}} (y)\,dy\,, \qquad \forall \mathcal {i}\in \mathcal {M}\,. \end{equation*}

Given $N\in \mathbb{N}$ , we consider the strength fractions $\sigma ^{\mathcal{i}}_N = \sigma ^{\mathcal{i}}/ N$ , and we introduce for each $\mathcal{i} \in \mathcal{M}$ the $\{\bar{x}_k^{\mathcal{i}}\}$ partition of $\Omega$ with $k\in \mathcal{N}=\{0,\ldots,N\}$ given by

(2.1) \begin{equation} \begin{aligned} &\bar{x}_0^{\mathcal{i}} = -1 \,,\\ &\bar{x}_k^{\mathcal{i}} = \inf \left \{ x \in \Omega \, : \, \int _{\bar{x}_{k-1}}^x \bar{\rho }^{\mathcal{i}} (y)\,dy = \sigma ^{\mathcal{i}}_N \right \} \,, \qquad k\in \mathcal{N}\setminus \left \{0,N\right \}\\ &\bar{x}_N^{\mathcal{i}} = 1 \,. \end{aligned} \end{equation}

Note that $\bar{x}_k^{\mathcal{i}}\lt \bar{x}_{k+1}^{\mathcal{i}}$ , for any $\mathcal{i}\in \mathcal{M}$ and $k\in \mathcal{N}\setminus \left \{N\right \}$ . This procedure allows to associate with each agent a finite number of time-evolving opinions $x^{\mathcal{i}}_k(t)$ . Assume that initially all the nodes $\bar{a}^{\mathcal{i}} = a^{\mathcal{i}}(t=0)$ are located in a certain smooth and bounded domain $\Lambda \in \mathbb{R}^d$ . We then let the nodes evolve in time depending on the distances $a^{\mathcal{ij}}$ between the agents $a^{\mathcal{i}}$ and $a^{\mathcal{j}}$ and the mean opinion of the agents.

We define the discrete opinion densities for the $\mathcal{i}$ -th agent as

(2.2) \begin{equation} \rho ^{\mathcal{i}}_k(t) = \frac{\sigma ^{\mathcal{i}}_N}{|I_k^{\mathcal{i}}(t)|},\quad \textit{ with }\quad I_k^{\mathcal{i}}(t)=[x^{\mathcal{i}}_k(t),x^{\mathcal{i}}_{k+1}(t)) \end{equation}

with $k\in \mathcal{N}\setminus \left \{N\right \}$ , and the discrete mean opinions by

(2.3) \begin{equation} \mu _{x^{\mathcal{i}}}^{\mathcal{N}}(t) = \frac{1}{N+1} \sum _{k\in \mathcal{N}} x^{\mathcal{i}}_k(t)\,, \end{equation}

where $a^{\mathcal{ij}}_{\mathcal{N}}$ is the Euclidean distance between the agents after the opinion discretisation. In the following, we may denote with $\textbf{x}^{\mathcal{i},\mathcal{N}}(t)\,:\!=\left (x_0^{\mathcal{i}}(t),\ldots,x_N^{\mathcal{i}}(t)\right )$ , for all $\mathcal{i}\in \mathcal{M}$ .

Thus, we consider the following system of ODEs

(2.4a) \begin{align} &\dot{x}^{\mathcal{i}}_k(t) = \frac{\beta ^{\mathcal{i}}_k}{\sigma _N^{\mathcal{i}}} \left (\Phi ^{\mathcal{i}}(\rho ^{\mathcal{i}}_{k-1}) - \Phi ^{\mathcal{i}}(\rho ^{\mathcal{i}}_{k}) \right ) + \theta ^{\mathcal{i}}_k \end{align}

(2.4b) \begin{align} &\dot{a}_{\mathcal{N}}^{\mathcal{i}} (t) = \sum _{\mathcal{j}\in \mathcal{M}}\mathbf{V}(\mu _{x^{\mathcal{i}}}^{\mathcal{N}}, \mu _{x^{\mathcal{j}}}^{\mathcal{N}}, a_{\mathcal{N}}^{\mathcal{ij}}), \end{align}

for $k\in \mathcal{N}\setminus \left \{0,N\right \}$ and $\mathcal{i} \in \mathcal{M}$ , endowed with the boundary conditions

(2.5) \begin{equation} \dot{x}^{\mathcal{i}}_0(t) = 0\quad, \quad \dot{x}^{\mathcal{i}}_{N}(t) = 0 \qquad \text{for all }\mathcal{i} \in \mathcal{M}, \end{equation}

and initial conditions

(2.6) \begin{equation} x^{\mathcal{i}}_k(0) = \bar{x}^{\mathcal{i}}_k\quad, \quad a_{\mathcal{N}}^{\mathcal{i}}(0) = \bar{a}^{\mathcal{i}} \qquad \text{for all }\mathcal{i} \in \mathcal{M},\,k\in \mathcal{N}. \end{equation}

In (2.4), we have denoted with $\beta ^{\mathcal{i}}_k(t)$ the discrete diffusion mobilities

(2.7) \begin{equation} \beta ^{\mathcal{i}}_k(t) = \sum _{\mathcal{j}\in \mathcal{M}} \sum _{l\in \mathcal{N}} \sigma ^{\mathcal{j}}_N\mathbf{A}^{\mathcal{ij}}(x^{\mathcal{i}}_k,x^{\mathcal{j}}_l,a_{\mathcal{N}}^{\mathcal{ij}})\,, \end{equation}

and with $\Phi ^{\mathcal{i}}$ the nonlinear diffusion for the agent $\mathcal{i}$ , see assumption (Dif) below. The contribution of the diffusion at the particle level can be interpreted assuming that opinions evolve with a speed equal to the osmotic velocity associated with the diffusion process, see [Reference Fagioli and Radici25, Reference Fagioli and Radici26, Reference Russo46].

Functions $\theta ^{\mathcal{i}}_k(t)$ describe the discrete transports and are given by

(2.8) \begin{equation} \theta ^{\mathcal{i}}_k(t) = \sum _{\mathcal{j}\in \mathcal{M}} \sum _{l\in \mathcal{N}} \sigma ^{\mathcal{j}}_N\mathbf{K}^{\mathcal{ij}}(x^{\mathcal{i}}_k,x^{\mathcal{j}}_l,a_{\mathcal{N}}^{\mathcal{ij}})\,. \end{equation}

We briefly comment on the boundary conditions in (2.5). As mentioned earlier, in (1.1) we did not introduce any boundary conditions. However, we are interested in the case where individual opinions do not exit the considered interval. Condition (2.5) enforces zero velocity for extreme opinions, which, along with the results of Lemma 3.2 on the preservation of the order of opinions, implies zero-flux boundary conditions for the opinion densities, as specified in equation (2.11) below.

2.2. Preliminaries and assumptions

We now present some tools from optimal transport that will be useful in the sequel. The Wasserstein distance is the right notion of distance for the opinions since it allows to measure the distances between measures (densities) with same mass. For a fixed mass $\sigma \gt 0$ , we consider the space

\begin{equation*} \mathfrak{M}_\sigma = \bigl \{\mu \text{ Radon measure on } \mathbb{R} \colon \mu \ge 0 \text { and }\mu ({\mathbb {R}})=\sigma \bigr \}. \end{equation*}

Given $\mu \in \mathfrak{M}_\sigma$ , we introduce the pseudo-inverse function $X_\mu \in L^1([0,\sigma ]\,{;}\,{\mathbb{R}})$ as

(2.9) \begin{equation} X_\mu (z) = \inf \bigl \{ x \in{\mathbb{R}} \colon \mu (({-}\infty,x]) \gt z \bigr \}. \end{equation}

In particular, if $\sigma =1$ , then $\mathfrak{M}_1$ is the set of non-negative probability densities on $\mathbb{R}$ , and it is possible to consider the one-dimensional $1$ -Wasserstein distance between each pair of densities $\rho _1,\rho _2\in \mathfrak{M}_1$ . As shown in [Reference Carrillo and Toscani16], in the one-dimensional setting the $p$ -Wasserstein distance can be equivalently defined in terms of the $L^1$ -distance between the respective pseudo-inverse mappings as

\begin{equation*} d_{W^p}(\rho _1,\rho _2) = \|X_{\rho _1}-X_{\rho _2}\|_{L^p([0,1];\,{\mathbb {R}})}. \end{equation*}

For generic $\sigma \gt 0$ , we recall the definition for the scaled $1$ -Wasserstein distance between $\rho _1,\rho _2\in \mathfrak{M}_\sigma$ as

(2.10) \begin{equation} d_{W^1_\sigma }(\rho _1,\rho _2)= \|X_{\rho _1}-X_{\rho _2}\|_{L^1([0,\sigma ];\,{\mathbb{R}})}, \end{equation}

We refer to [Reference Ambrosio, Gigli and Savaré4, Reference Santambrogio47, Reference Villani55] for a complete presentation of the subject.

We assume that the initial densities are under the following assumptions:

1. (In1) $\bar{\rho }^{\mathcal{i}} \in BV(\Omega \,;\,{\mathbb{R}}^+)$ with $\|\bar{\rho }^{\mathcal{i}}\|_{L^1(\Omega )} = \sigma ^{\mathcal{i}}$ , for some $\sigma ^{\mathcal{i}}\gt 0$ ,

2. (In2) there exists $m^{\mathcal{i}},M^{\mathcal{i}} \gt 0$ such that $m^{\mathcal{i}} \leq \bar{\rho }^{\mathcal{i}}(x) \leq M^{\mathcal{i}}$ for every $x \in \Omega$ .

Due to technical constraints, initial data with a vacuum region cannot be considered, as evident in the proof of Proposition 3.6. The lower bound is employed to control the time derivatives of extreme opinion densities $\dot{\rho }_0^{\mathcal{i}}$ and $\dot{\rho }_N^{\mathcal{i}}$ . The essential point is the need for either a uniform control over these quantities or an estimation at an appropriate rate of $N$ . We believe that this technical issue can be resolved, especially in the case of nonlinear diffusion, where finite speed of propagation is known and not utilised in the numerical Section. However, addressing this matter is currently beyond our capabilities. We now introduce the assumptions for the diffusive and transport operators.

1. (A) We assume that ${\mathbf{A}}^{\mathcal{ij}} \,{:}\, \Omega \times \Omega \times{\mathbb{R}}\to{\mathbb{R}}$ is a $C^1$ non-negative function w.r.t. the first variable and for all pairs $(\mathcal{i},\mathcal{j})$ it exists $c_{\mathbf{A}} \gt 0$ such that

   \begin{equation*} |{\mathbf {A}}^{\mathcal {ij}}(x^i_{k^*}, x^j_{s^*}, a^{\mathcal {ij}}) - {\mathbf {A}}^{\mathcal {ij}}(x^i_k, x^j_s,a^{\mathcal {ij}})| \le c_{\mathbf {A}} \left ( |x^i_{k^*} - x^i_k| + |x^j_{s^*} - x^j_s| \right )\,, \end{equation*}
   and it exists $c_{1,{\mathbf{A}}} \gt 0$ such that
   \begin{equation*} |\partial _1{\mathbf {A}}^{\mathcal {ij}}(x^i_{k^*}, x^j_{s^*}, a^{\mathcal {ij}})-\partial _1{\mathbf {A}}^{\mathcal {ij}}(x^i_k, x^j_s,a^{\mathcal {ij}})|\leq c_{1,{\mathbf {A}}}|x^j_{s^*} - x^j_s|, \end{equation*}
   for all $(k,k^*)$ and $(s,s^*)$ pairs of indexes in $\mathcal{N}\times \mathcal{N}$ , where $\partial _1{\mathbf{A}}^{\mathcal{ij}}$ denotes the derivatives with respect to the first entrance.

2. (K) We assume that $\mathbf{K}^{\mathcal{ij}} \,{:}\, \Omega \times \Omega \times{\mathbb{R}}\to{\mathbb{R}}$ is bounded, continuous, and for all pairs $(\mathcal{i},\mathcal{j})$ , it exists $c_{\mathbf{K}} \gt 0$ such that

   \begin{equation*} |\mathbf {K}^{\mathcal {ij}}(x^i_{k^*}, x^j_{s^*},a^{\mathcal {ij}}) - \mathbf {K}^{\mathcal {ij}}(x^i_k, x^j_s,a^{\mathcal {ij}})| \le c_{\mathbf {K}} \left ( |x^i_{k^*} - x^i_k| + |x^j_{s^*} - x^j_s|\right )\,, \end{equation*}
   for all $(k,k^*)$ and $(s,s^*)$ pairs of indexes in $\mathcal{N}\times \mathcal{N}$ . We further assume that
   \begin{equation*}\mathbf {K^{\mathcal {i}\mathcal {i}}}(x,x,a^{\mathcal {i}\mathcal {i}})=0.\end{equation*}

3. (Dif) $\Phi ^{\mathcal{i}}\,{:}\, [0,\infty ) \to{\mathbb{R}}$ is a nondecreasing Lipschitz function, with $\Phi ^{\mathcal{i}}(0)=0$ .

4. (V) The network velocity $\mathbf{V}$ is a $C^1$ bounded function on $\Omega \times \Omega \times{\mathbb{R}}^{+}$ .

2.3. Continuous reconstruction and main result

Given the preliminary assumptions, we give the definition of weak solutions to equation (1.1) together with the statement of the main result.

By setting $\Omega _T = [0,T]\times \Omega$ and $\partial \Omega _T=[0,T]\times \left \{-1,1\right \}$ , and considering $\bar{\rho }^{\mathcal{i}}\in L^1\cap L^\infty (\Omega )$ and $\bar{a}^{\mathcal{i}}\in \Lambda \subset{\mathbb{R}}^d$ , we are going to deal with the following PDE-ODE system

(2.11) \begin{equation} \begin{cases} \partial _t \rho ^{\mathcal{i}} = \partial _x\! \left ( \beta ^{\mathcal{i}}(\boldsymbol{\rho },\mathbf{a};\,x) \partial _x \Phi ^{\mathcal{i}}(\rho ^{\mathcal{i}} )\right ) - \partial _x \left ( \rho ^{\mathcal{i}}\theta ^{\mathcal{i}}(\boldsymbol{\rho },\mathbf{a};\,x)\right ),& (t,x)\in \Omega _T,\\ \partial _t a^{\mathcal{i}}(t) = \mathop{\sum}\limits _{\mathcal{j}\in \mathcal{M}}\mathbf{V}(\mu _{\rho ^{\mathcal{i}}}(t), \mu _{\rho ^{\mathcal{j}}}(t), a^{\mathcal{ij}}),& t\in [0,T],\\ \beta ^{\mathcal{i}}(\boldsymbol{\rho },\mathbf{a};\,x) \partial _x \Phi ^{\mathcal{i}}(\rho ^{\mathcal{i}}(t,x) )-\rho ^{\mathcal{i}}(t,x)\theta ^{\mathcal{i}}(\boldsymbol{\rho },\mathbf{a};\,x)=0,& (t,x)\in \partial \Omega _T,\\ \rho ^{\mathcal{i}}(0,\cdot ) = \bar{\rho }^{\mathcal{i}}, & x\in \Omega,\\ a^{\mathcal{i}}(0)=\bar{a}^{\mathcal{i}}, \end{cases}\, \end{equation}

for all $\mathcal{i}\in \mathcal{M}$ , where the bold notation refers to the vectors $\boldsymbol{\rho } = (\rho ^1,\ldots,\rho ^M)$ and the set $\mathbf{a}=(a^1,\ldots,a^M)$ .

We state the notion of weak solution for the system (2.11) as follows

Definition 2.1 (Weak solution). We say that the couple $\left (\boldsymbol{\rho },\mathbf{a}\right )$ is a weak solution of (1.1) in the formulation (2.11) if

• • $\rho ^{\mathcal{i}} \in L^\infty \cap BV(\Omega _T)$ , with $\rho ^{\mathcal{i}}(0,\cdot )= \bar{\rho }^{\mathcal{i}}$ , for all $\mathcal{i}\in \mathcal{M}$

• • $a^{\mathcal{i}}\in C^2\left (\left [0,T\right ];\,{\mathbb{R}}^d\right )$ ,

and taken $\zeta \in C_0^{\infty }(\Omega _T)$ , for all $\mathcal{i}\in \mathcal{M}$ it satisfies

(2.12) \begin{equation} \begin{aligned} \int _{\Omega _T} \rho ^{\mathcal{i}}(t,x) &\partial _t\zeta (t,x)+\rho ^{\mathcal{i}}(t,x)\theta ^{\mathcal{i}}(\boldsymbol{\rho },\mathbf{a};\,x) \partial _x \zeta (x)\, dx\,dt \\ +\int _{\Omega _T}&\Phi ^{\mathcal{i}}\left (\rho ^{\mathcal{i}}(t,x)\right ) \left ( \partial _x\beta ^{\mathcal{i}}(\boldsymbol{\rho },\mathbf{a};\,x) \partial _x \zeta (t,x)+\beta ^{\mathcal{i}}(\boldsymbol{\rho },\mathbf{a};\,x) \partial _{xx} \zeta (t,x)\right )\, dx=0\,\end{aligned} \end{equation}

and

(2.13) \begin{equation} a^{\mathcal{i}}(t) = \bar{a}^{\mathcal{i}}+\sum _{\mathcal{j}\in \mathcal{M} }\int _0^t \mathbf{V}\left (\mu _{\rho ^{\mathcal{i}}}(\tau ), \mu _{\rho ^{\mathcal{j}}}(\tau ), a^{\mathcal{ij}}(\tau )\right )\,d\tau \end{equation}

for all $t\in [0,T]$ .

Given the discrete opinion densities defined in (2.2), we consider the following piecewise constant density reconstructions

(2.14) \begin{equation} \rho ^{\mathcal{i},\mathcal{N}}(t,x) = \sum _{k\in \mathcal{N}\setminus \left \{N\right \}}\rho ^{\mathcal{i}}_k(t)\, \chi _{I_k^{\mathcal{i}}(t)}(x)\,. \end{equation}

The main result of the paper reads as follows

Note that non-negativeness of the limit functions $\rho$ easily follows by construction, see also (3.4) below, as well as the mass preservation. To avoid lack of notation, we highlight that from now on while considering the limit $N\to \infty$ we refer to the limit to infinity of the cardinality of the set of indexes $\mathcal{N}=\{0,\ldots,N\}$ .

3.1. Basic estimates

We start providing some fundamental estimates that allow to deduce the well-posedness of (2.4) and of the discrete densities (2.2). Let us recall that in (2.2), we introduced the intervals

(3.1) \begin{equation} I_k^{\mathcal{i}}(t) = \left [{x}^{\mathcal{i}}_{k}(t),{x}^{\mathcal{i}}_{k+1}(t)\right ),\quad |I_k^{\mathcal{i}}|(t)=|{x}^{\mathcal{i}}_{k+1} -{x}^{\mathcal{i}}_{k}|, \end{equation}

for $\mathcal{i}\in \mathcal{M}$ and $k\in \mathcal{N}\setminus \left \{N\right \}$ . The first step is to prove that such intervals are well-defined. We start with the following auxiliary lemma, that follows directly from Assumption $(\mathbf{K})$

Lemma 3.1. With the setting of the Main Theorem 2.2, and referring to the previous definitions, given $\mathbf{K^{\mathcal{ij}}}$ under Assumption $(\mathbf{K})$ , then it exists $C\gt 0$ such that the following inequality holds

(3.2) \begin{equation} |\theta _{k+1}^i(t) - \theta _k^i(t)| \le C |x^i_{k+1}(t) - x^i_{k}(t)| \quad \forall \mathcal{i} \in \mathcal{M}, \, \forall t \in [0,T]\,, \end{equation}

with $C= c_{\mathbf{K}} \sigma ^{\mathcal{M}}$ , where $\sigma ^{\mathcal{M}} = \sum _{j\in \mathcal{M}} \sigma ^{\mathcal{j}}$ .

Lemma 3.2 (Ordering preservation). Assume ${\mathbf{A}}^{\mathcal{ij}}$ and $\mathbf{K}^{\mathcal{ij}}$ under assumptions $({\mathbf{A}})$ and $(\mathbf{K})$ , respectively, for all $\mathcal{i},\mathcal{j}\in \mathcal{M}$ . Let us consider the DPA system described by (2.4) with initial conditions $\bar{x}^{\mathcal{i}}$ constructed in (2.1), for $\mathcal{i}\in \mathcal{M}$ , and a finite time $T\gt 0$ . Then, for all $t\in [0,T)$ there is a positive constant $\mu$ independent from $\mathcal{N}$ – and so from $N$ too – such that the distance between two adjacent opinions $x^{\mathcal{i}}_k, x^{\mathcal{i}}_{k+1} \in C[0,T]$ is bounded from below by

\begin{equation*} \left (x^{\mathcal {i}}_{k+1} - x^{\mathcal {i}}_k\right )(t) \ge \underset {\mathcal {i},k}{\min }(\bar {x}^{\mathcal {i}}_{k+1} - \bar {x}^{\mathcal {i}}_k) \, e^{-\mu T}, \end{equation*}

for all $k\in \mathcal{N}\setminus \left \{N\right \}$ and $\mathcal{i} \in \mathcal{M}$ .

Proof. Given $T\gt 0$ and $\mathcal{i}\in \mathcal{M}$ , we define $\tau _1$ as

\begin{equation*} \tau _1 = \inf \big \{ s \in (0,T) \,:\, \exists \, k \in \mathcal {N}\backslash \{N\} \text { s.t. } (x^{\mathcal {i}}_{k+1} - x^{\mathcal {i}}_{k} )(s) = (\bar {x}^{\mathcal {i}}_{k+1} - \bar {x}^{\mathcal {i}}_{k})\, e^{-\mu s} \big \}\,, \end{equation*}

then the same index $k$ corresponds also to the one of the maximum $\rho ^{\mathcal{i}}_k$ at time $\tau _1$ because $I^i_k$ is the minimum interval of the $\mathcal{N}$ partition of $\Omega$ for the $i$ -th agent at time $\tau _1$ .

At this point, let assume that exists $\tau _2 \in (\tau _1, T)$ such that

\begin{equation*}\left (x^{\mathcal {i}}_{k+1} - x^{\mathcal {i}}_k\right )(s) \lt (\bar {x}^{\mathcal {i}}_{k+1} - \bar {x}^{\mathcal {i}}_k) e^{-\mu s} \quad \forall s \in (\tau _1,\tau _2)\,.\end{equation*}

We show that the existence of $\tau _2$ would bring to a contradiction. Let us consider the evolution of the interval $I^i_k$

\begin{align*} \frac{d}{dt} \left [ e^{\mu t} \left (x^{\mathcal{i}}_{k+1} - x^{\mathcal{i}}_k\right )(t)\right ]_{|t=\tau _1} &= e^{\mu \tau _1 } \frac{\beta ^{\mathcal{i}}_{k+1}(\tau _1) }{\sigma _N^{\mathcal{i}}}\left [ \Phi ^{\mathcal{i}}(\rho ^{\mathcal{i}}_{k}(\tau _1) ) - \Phi ^{\mathcal{i}}(\rho ^{\mathcal{i}}_{k+1}(\tau _1) ) \right ] + e^{\mu \tau _1} \theta ^{\mathcal{i}}_{k+1}(\tau _1) \\ &\quad - e^{\mu \tau _1} \frac{\beta ^{\mathcal{i}}_{k}(\tau _1) }{\sigma _N^{\mathcal{i}}}\left [ \Phi ^{\mathcal{i}}(\rho ^{\mathcal{i}}_{k-1}(\tau _1) ) - \Phi ^{\mathcal{i}}(\rho ^{\mathcal{i}}_{k}(\tau _1) )\right ] - e^{\mu \tau _1} \theta ^{\mathcal{i}}_{k}(\tau _1) \\ &\quad + \mu e^{\mu \tau _1} (x^{\mathcal{i}}_{k+1}(\tau _1) - x^{\mathcal{i}}_k(\tau _1) )\,. \end{align*}

At time $t=\tau _1$ , as highlighted before, by construction of the discrete densities in (2.2) we have $\rho ^{\mathcal{i}}_k \ge \rho ^{\mathcal{i}}_l$ for all $l\ne k$ .

Then, the monotonicity of $\Phi ^{\mathcal{i}}$ gives

\begin{align*} \frac{d}{dt} \left [ e^{\mu t} \left (x^i_{k+1} - x^i_k\right )(t)\right ]_{|t=\tau _1} &\ge e^{\mu \tau _1}\left [ \mu \, \left (x^{\mathcal{i}}_{k+1}(\tau _1) - x^{\mathcal{i}}_k(\tau _1) \right )+\left (\theta ^{\mathcal{i}}_{k+1}(\tau _1) - \theta ^{\mathcal{i}}_{k}(\tau _1)\right ) \right ]. \end{align*}

Thanks to (3.2) we get

\begin{align*} \frac{d}{dt} \left [ e^{\mu t} \left (x^{\mathcal{i}}_{k+1} - x^{\mathcal{i}}_k\right )(t)\right ]_{|t=\tau _1} &\ge e^{\mu \tau _1}\, (\mu - C) (x^{\mathcal{i}}_{k+1}(\tau _1) - x^{\mathcal{i}}_{k}(\tau _1)) \ge 0\,, \end{align*}

which holds while choosing $\mu \ge C$ . At this point, we fix $t^*\in (\tau _1,\tau _1+\delta \lt \tau _2)$ with $\delta$ as small as we want, and we get

\begin{equation*} e^{\mu t^*} \left (x^{\mathcal {i}}_{k+1} - x^{\mathcal {i}}_k\right )(t^*) = e^{\mu \tau _1} \left (x^{\mathcal {i}}_{k+1} - x^{\mathcal {i}}_k\right )(\tau _1) + \int _{\tau _1}^{t^*} \frac {d}{ds} \left [ e^{\mu t} \left (x^{\mathcal {i}}_{k+1} - x^{\mathcal {i}}_k\right )(s)\right ] \, ds \,, \end{equation*}

due to the positiveness of the last term we get the wished result which show the absurd,

\begin{equation*} \left (x^{\mathcal {i}}_{k+1} - x^{\mathcal {i}}_k\right )(t^*) \ge e^{- \mu t^*} \left (\bar {x}^{\mathcal {i}}_{k+1} - \bar {x}^{\mathcal {i}}_k\right )\,, \end{equation*}

this proves that $I^i_k(t)$ cannot decrease faster than $I^i_k(0) e^{- \mu t}$ . Nevertheless, this does not deny the existence of an index $\tilde{k}$ such that the interval $I^i_{\tilde{k}}(t)$ , satisfying $I^i_{\tilde{k}} (\tau _1) \gt I^i_{\tilde{k}}(0)$ , decreases faster than $I^i_k(t)$ . This means that could exists $\tilde{\tau }_1 \gt \tau _1$ for which $I^i_{\tilde{k}}(\tilde{\tau }_1) = I^i_{\tilde{k}}(0)$ , with $\tilde{\tau }_1 \lt T$ . At this point, we should prove that there exists $\tilde{\mu }$ such that $I^i_{\tilde{k}}(t) \ge e^{-\tilde{\mu }t} I^i_{\tilde{k}}(0)$ for all $t \in (\tilde{\tau }_1, T)$ . To prove it, we repeat the same procedure explained before but defining $\tilde{\tau }_1$ considering the set of indexes $\mathcal{N}/ \{k\}$ . The final exponential rate will be the largest $\mu$ among those considered.

Let us also consider the case with $k$ not unique, i.e. for $\tau _1$ there are several intervals satisfying the definition of $\tau _1$ , the set of these indexes is denoted by $\mathcal{J}=\{ k_j \}$ . If there is at least one $k_{j^*}$ not adjacent to other indexes of $\{k_j\}$ , then we take that index and we follow the proof above. In the event that there are three indexes of $\{k_j\}$ in a row, we take without distinction that one with the fastest decrease in time of the interval $I^i_{k_j}$ . At this point, we are back to the steps shown above. This concludes the proof.

Remark 3.3. A similar procedure of the one in Lemma 3.2 allows to produce the upper bound on discrete opinions

(3.3) \begin{equation} \left (x^{\mathcal{i}}_{k+1} - x^{\mathcal{i}}_k\right )(t) \le \underset{k}{\max }(\bar{x}^{\mathcal{i}}_{k+1} - \bar{x}^{\mathcal{i}}_k) \, e^{CT} \qquad \forall t\in [0,T]\,. \end{equation}

This estimate, together with the one in Lemma 3.2, allows to deduce the following bounds on the discrete densities in (2.2)

(3.4) \begin{equation} m^{\mathcal{i}} e^{-CT}\leq \rho _{k}^{\mathcal{i}}\leq M^{\mathcal{i}} e^{\mu T}\,, \textit{ for all }\,\mathcal{i}\in \mathcal{M},\,\,k\in \mathcal{N}\setminus \left \{N\right \}, \end{equation}

with $m^{\mathcal{i}}$ and $M^{\mathcal{i}}$ in assumption (In2).

Lemma 3.4 (Velocity boundedness). Fix $T\gt 0$ and assume ${\mathbf{A}}^{\mathcal{ij}}$ and $\mathbf{K}^{\mathcal{ij}}$ , respectively, under assumptions $({\mathbf{A}})$ and $(\mathbf{K})$ for all $(\mathcal{i},\mathcal{j})\in \mathcal{M}\times \mathcal{M}$ . Then, solutions to system (2.4) satisfy

\begin{equation*} \sup _{t\in \left [0,T\right ]}\|\dot {\mathbf {x}}^{\mathcal {i},\mathcal {N}}(t)\|_{\infty }\lt +\infty, \,\textit { for all }\, \mathcal {i}\in \mathcal {M}. \end{equation*}

Proof. Using the equation for the evolution of the partitioning, we get

\begin{align*} \frac{1}{2}\frac{d}{dt} \left (\dot{x}^{\mathcal{i}}_k(t)\right ) ^2 &\le x_k^{\mathcal{i}} \left ( \frac{\beta ^{\mathcal{i}}_{k}}{\sigma _N^{\mathcal{i}}} \left (\Phi ^{\mathcal{i}}(\rho ^{\mathcal{i}}_{k-1}) - \Phi ^{\mathcal{i}}(\rho ^{\mathcal{i}}_{k}) \right ) -\theta ^{\mathcal{i}}_k\right )\, \\ &\le |x_k^{\mathcal{i}}|\left ( \frac{1}{\sigma _N^{\mathcal{i}}}\|\beta ^{\mathcal{i}}_{k}\|_{\infty } Lip\left (\Phi ^{\mathcal{i}}\right )\Big | \rho ^{\mathcal{i}}_{k} - \rho ^{\mathcal{i}}_{k-1}\Big | + \Big |\theta ^{\mathcal{i}}_k\Big |\right )\\ \leq &c_1|x_k^{\mathcal{i}}|^2+c_2\Big | \rho ^{\mathcal{i}}_{k} - \rho ^{\mathcal{i}}_{k-1}\Big |^2+c_3\Big |\theta ^{\mathcal{i}}_k\Big |^2, \end{align*}

for some constants $c_1,c_2,c_3\gt 0$ . Then, the thesis follows from equation (3.4), assumption $(\mathbf{K})$ and the fact that $x_k^{\mathcal{i}}\in \Omega$ because (2.5) and Lemma 3.2.

We now have all the tools needed to prove the convergence in some strong sense for the piecewise constant densities of equation (2.14). We based our strategy on the ones proposed in the context of DPA, see for instance the proofs in [Reference Fagioli and Radici25, Reference Fagioli and Tse27], that are using the generalised Aubin-Lions lemma version in [Reference Rossi and Savaré45], that we report here in a simplified version adapted to our setting.

The result reads as follows

Proof. The proof reduces to the application of Theorem 3.5, in order to show that we can apply that result we first prove that

(3.5) \begin{equation} \sup _N \int _0^T TV[\rho ^{\mathcal{i},\mathcal{N}}(t,\cdot )]\,dt \le \infty \,. \end{equation}

To show this result, we look for a Grönwall type inequality. We start performing the following preliminary computation

\begin{align*} \dot{\rho }_{k}^{\mathcal{i}} & = -\frac{\rho _{k}^{\mathcal{i}}}{|I_k^{\mathcal{i}}|} \left(\dot{x}_{k+1}^{\mathcal{i}}(t)-\dot{x}_{k}^{\mathcal{i}}(t)\right )\\ & = -\frac{\rho _{k}^{\mathcal{i}}}{|I_k^{\mathcal{i}}|} \Big ( \underbrace{\frac{\beta ^{\mathcal{i}}_{k+1}}{\sigma _N^{\mathcal{i}}} \left(\Phi ^{\mathcal{i}}\left (\rho _k^{\mathcal{i}}\right ) - \Phi ^{\mathcal{i}}\left (\rho _{k+1}^{\mathcal{i}}\right )\right) - \frac{\beta ^{\mathcal{i}}_{k}}{\sigma _N^{\mathcal{i}}} \left(\Phi ^{\mathcal{i}}\left (\rho _{k-1}^{\mathcal{i}}\right ) - \Phi ^{\mathcal{i}}\left (\rho _{k}^{\mathcal{i}}\right )\right)}_{:=B_k^{\mathcal{i}}} + \underbrace{(\theta _{k+1}^{\mathcal{i}} - \theta _{k}^{\mathcal{i}})}_{:=|I_k^{\mathcal{i}}|\Theta _k^{\mathcal{i}}} \Big ) \\&= -\frac{{\rho }_{k}^{\mathcal{i}}}{|I_k^{\mathcal{i}}|}B_k^{\mathcal{i}}+ \rho _{k}^{\mathcal{i}} \Theta _k^{\mathcal{i}}. \end{align*}

The first step consists of proving the Lipschitz continuity in time of $t \to TV[\rho _t^{\mathcal{i},\mathcal{N}}(t,\cdot )]$ . The total variation can be explicitly computed as

\begin{align*} TV[\rho ^{\mathcal{i},\mathcal{N}}(t,\cdot )] &= \rho _0^{\mathcal{i}}(t) + \sum _{k\in \mathcal{N}\setminus \left \{0,N\right \}} |\rho _k^{\mathcal{i}}(t) - \rho _{k-1}^{\mathcal{i}}(t)| + \rho _{N-1}^{\mathcal{i}}(t). \end{align*}

From the boundedness in Lemmas 3.2 and 3.4, for all $s,t\in (0,T)$ , we estimate

\begin{align*} |\rho _k^{\mathcal{i}}(t)-\rho _k^{\mathcal{i}}(s)| \leq &\left |\int _s^t \dot{\rho }_k^{\mathcal{i}} (\tau )\,d\tau \right |\leq \int _s^t \frac{\rho _{k}^{\mathcal{i}}}{|I_k^{\mathcal{i}}|} \left |\dot{x}_{k+1}^{\mathcal{i}}(t)-\dot{x}_{k}^{\mathcal{i}}(t)\right |\,d\tau \\ &\le 2\frac{M^{\mathcal{i}}}{\underset{k,\mathcal{i}}{\min } |\bar{x}^{\mathcal{i}}_{k+1} - \bar{x}^{\mathcal{i}}_{k}| } \sup _{\tau \in \left [0,T\right ]}\|\dot{\mathbf{x}}^{\mathcal{i},\mathcal{N}}(\tau )\|_{\infty } |t-s|. \end{align*}

Then, it follows

\begin{align*} \left |TV[\rho ^{\mathcal{i},\mathcal{N}}(t,\cdot )]-TV[\rho ^{\mathcal{i},\mathcal{N}}(s,\cdot )]\right | \leq &|\rho _0^{\mathcal{i}}(t)-\rho _0^{\mathcal{i}}(s)| + |\rho _{N-1}^{\mathcal{i}}(t)-\rho _{N-1}^{\mathcal{i}}(s)| \\ &+ \sum _{k\in \mathcal{N}\setminus \left \{0,N\right \}}\left ( |\rho _k^{\mathcal{i}}(t) - \rho _{k}^{\mathcal{i}}(s)|+ |\rho _{k-1}^{\mathcal{i}}(t) - \rho _{k-1}^{\mathcal{i}}(s)|\right )\\ &\le 2\max _{k,\mathcal{i}} \frac{M^{\mathcal{i}}}{\underset{k,\mathcal{i}}{\min } |\bar{x}^{\mathcal{i}}_{k+1} - \bar{x}^{\mathcal{i}}_{k}| } \sup _{\tau \in \left [0,T\right ]}\|\dot{\mathbf{x}}^{\mathcal{i},\mathcal{N}}(\tau )\|_{\infty } |t-s|, \end{align*}

that proves the asserted Lipschitz continuity.

We now consider the time derivative

\begin{equation*} \frac {d}{dt}TV[\rho ^{\mathcal {i},\mathcal {N}}(t,\cdot )] = \dot {\rho }_0^{\mathcal {i}}(t) + \sum _{k\in \mathcal {N}\setminus \left \{0,N\right \}}\textrm {sign}\big ({\rho }_k^{\mathcal {i}}(t) - {\rho }_{k-1}^{\mathcal {i}}(t)\big ) \, \left (\dot {\rho }_k^{\mathcal {i}}(t) - \dot {\rho }_{k-1}^{\mathcal {i}}(t)\right ) + \dot {\rho }_{N-1}^{\mathcal {i}}(t) \,. \end{equation*}

Rearranging the sum and defining the operator

\begin{equation*} {\textrm {s}}_{k} = \begin {cases} 1- \textrm {sign}\big ({\rho }_1^{\mathcal {i}}(t) - {\rho }_{0}^{\mathcal {i}}(t)\big ) & k=0, \\ \textrm {sign}\big ({\rho }_k^{\mathcal {i}}(t) - {\rho }_{k-1}^{\mathcal {i}}(t)\big ) - \textrm {sign}\big ({\rho }_{k+1}^{\mathcal {i}}(t) - {\rho }_{k}^{\mathcal {i}}(t)\big ),&k=1,\ldots,N-2,\\ 1+\textrm {sign}\big ({\rho }_{N-1}^{\mathcal {i}}(t) - {\rho }_{N-2}^{\mathcal {i}}(t)\big ), & k=N-1, \end {cases} \end{equation*}

with we can rewrite the previous equation as

(3.6) \begin{equation} \frac{d}{dt}TV[\rho _t^{\mathcal{i},\mathcal{N}}(t,\cdot )] ={\textrm{s}}_{0}\dot{\rho }_0^{\mathcal{i}}(t) + \sum _{k=1}^{N-2}{\textrm{s}}_{k} \dot{\rho }_k^{\mathcal{i}}(t) +{\textrm{s}}_{N-1}\dot{\rho }_{N-1}^{\mathcal{i}}(t) \,. \end{equation}

At this point, we show that the terms involving the diffusion are always negative, i.e.

\begin{equation*} -{\textrm {s}}_k\frac {{\rho }_{k}^{\mathcal {i}}}{|I_k^{\mathcal {i}}|}B_k^{\mathcal {i}}\leq 0\quad k=2,\ldots,N-2. \end{equation*}

In order to prove this statement, we should distinguish different cases. First, we observe that ${\textrm{s}}_k$ is always zero if $\rho _{k-1}^{\mathcal{i}}\lt \rho _{k}^{\mathcal{i}}\lt \rho _{k+1}^{\mathcal{i}}$ or $\rho _{k+1}^{\mathcal{i}}\lt \rho _{k}^{\mathcal{i}}\lt \rho _{k-1}^{\mathcal{i}}$ . In the other two cases, namely $\rho _{k}^{\mathcal{i}}\lt \rho _{k-1}^{\mathcal{i}}$ and $\rho _{k}^{\mathcal{i}}\lt \rho _{k+1}^{\mathcal{i}}$ , or $\rho _{k-1}^{\mathcal{i}},\,\rho _{k+1}^{\mathcal{i}}\lt \rho _{k}^{\mathcal{i}}$ and $\rho _{k-1}^{\mathcal{i}}\lt \rho _{k}^{\mathcal{i}}$ , the monotonicity of $\Phi ^{\mathcal{i}}$ implies, respectively, ${\textrm{s}}_k\leq 0$ and $B_k^{\mathcal{i}}\leq 0$ , and ${\textrm{s}}_k\geq 0$ and $B_k^{\mathcal{i}}\geq 0$ , and hence, the negativity of the diffusion contribution is proved.

Concerning the term involving $\Theta _k^{\mathcal{i}}$ , we can rearrange the sum as follows

\begin{align*} \sum _{k=1}^{N-2}{\textrm{s}}_k\rho _k^{\mathcal{i}}\Theta _k^{\mathcal{i}} = &\, \textrm{sign}(\rho _1^{\mathcal{i}}-\rho _0^{\mathcal{i}})\rho _1^{\mathcal{i}}\Theta _1^{\mathcal{i}}-\textrm{sign}(\rho _{N-1}^{\mathcal{i}}-\rho _{N-2}^{\mathcal{i}})\rho _{N-2}^{\mathcal{i}}\Theta _{N-2}^{\mathcal{i}}\\ &+ \sum _{k=2}^{N-2}\textrm{sign}(\rho _{k}^{\mathcal{i}}-\rho _{k-1}^{\mathcal{i}})(\rho _{k}^{\mathcal{i}}-\rho _{k-1}^{\mathcal{i}})\Theta _{k}^{\mathcal{i}}\\ &+ \sum _{k=2}^{N-2}\textrm{sign}(\rho _{k}^{\mathcal{i}}-\rho _{k-1}^{\mathcal{i}})(\Theta _{k}^{\mathcal{i}}-\Theta _{k-1}^{\mathcal{i}})\rho _{k-1}^{\mathcal{i}}. \end{align*}

Observing that $|\Theta _k^{\mathcal{i}}|\leq C$ because of equation (3.2) and that

\begin{align*} \left |\Theta _{k}^{\mathcal{i}}-\Theta _{k-1}^{\mathcal{i}}\right |\leq &\left |\frac{1}{|I_k^{\mathcal{i}}|}\left [\left (\theta _{k+1}^{\mathcal{i}}-\theta _{k}^{\mathcal{i}}\right )-\left (\theta _{k}^{\mathcal{i}}-\theta _{k-1}^{\mathcal{i}}\right )\right ]\right |\\&+\left |\left (\frac{1}{|I_k^{\mathcal{i}}|}-\frac{1}{|I_{k-1}^{\mathcal{i}}|}\right )\left (\theta _{k}^{\mathcal{i}}-\theta _{k-1}^{\mathcal{i}}\right )\right |\\ \leq & C\frac{\rho _k^{\mathcal{i}}}{\sigma _N^{\mathcal{i}}}\left (|I_k^{\mathcal{i}}|^2+|I_{k-1}^{\mathcal{i}}|^2+\left ||I_{k}^{\mathcal{i}}|-|I_{k-1}^{\mathcal{i}}|\right |\right )+\frac{C}{\sigma _N^{\mathcal{i}}}\left |\rho _{k}^{\mathcal{i}}-\rho _{k-1}^{\mathcal{i}}\right ||I_{k-1}^{\mathcal{i}}|, \end{align*}

we can bound

\begin{align*} \left |\sum _{k=1}^{N-2}{\textrm{s}}_k\rho _k^{\mathcal{i}}\theta _k^{\mathcal{i}}\right | \leq & 2M^{\mathcal{i}} C\left (1+|\Omega |\right )+ 3C\, TV[\rho ^{\mathcal{i},\mathcal{N}}(t)]. \end{align*}

We can finally estimate

\begin{align*} \frac{d}{dt}TV[\rho _t^{\mathcal{i},\mathcal{N}}(t,\cdot )] \leq & 2C\frac{M^{\mathcal{i}}}{m^{\mathcal{i}}}\left (\rho _0^{\mathcal{i}}(t) +\rho _{N-1}^{\mathcal{i}}(t)\right )+ 2M^{\mathcal{i}} C\left (1+|\Omega |\right )+ 3C\, TV[\rho ^{\mathcal{i},\mathcal{N}}(t)] \,, \end{align*}

and thus, equation (3.5) follows by Grönwall type argument.

We now prove that the second requirement of Theorem 3.5 holds, namely there exists a positive constant $C$ such that

(3.7) \begin{equation} d_{W^1}\big (\rho ^{\mathcal{i},\mathcal{N}}(t,\cdot ),\rho ^{\mathcal{i},\mathcal{N}}(s,\cdot )\big ) \le C |t-s| \qquad \forall \, s,t\in [0,T]\,. \end{equation}

In order to do this, we use the isometry of equation (2.10), where the pseudo-inverse function for $\rho ^{\mathcal{i},\mathcal{N}}$ is given by

\begin{equation*} X_{\rho ^{\mathcal {i},\mathcal {N}}}(m,t) = \sum _{k\in \mathcal {N}\setminus \left \{N\right \}}\left ( x^{\mathcal {i}}_k(t) + \frac {m - k \sigma ^{\mathcal {i}}_N }{\rho ^{\mathcal {i}}_k(t)} \right ) \chi _{\big [k \sigma ^{\mathcal {i}}_N, (k+1)\sigma ^{\mathcal {i}}_N\big ]}(m)\,. \end{equation*}

Then, for any $t\gt s$ , we have

\begin{align*} d_{W^1}\big (\rho ^{\mathcal{i},\mathcal{N}}(t,\cdot ),\rho ^{\mathcal{i},\mathcal{N}}(s,\cdot )\big ) &\le \sum _{k\in \mathcal{N}\setminus \left \{N\right \}}\int _{k\sigma ^{\mathcal{i}}_N}^{(k+1)\sigma ^{\mathcal{i}}_N} \Big | x^{\mathcal{i}}_k(t) - x^{\mathcal{i}}_k(s) \Big |\, dm\\ &\quad + \sum _{k\in \mathcal{N}\setminus \left \{N\right \}}\int _{k\sigma ^{\mathcal{i}}_N}^{(k+1)\sigma ^{\mathcal{i}}_N} \Bigg | (m - k \sigma ^{\mathcal{i}}_N) \bigg ( \frac{1}{\rho ^{\mathcal{i}}_k(t)} - \frac{1}{\rho ^{\mathcal{i}}_k(s)} \bigg ) \Bigg | \, dm \\ &\le \sum _{k\in \mathcal{N}\setminus \left \{N\right \}}\sigma ^{\mathcal{i}}_N \int _s^t \Big | \dot{x}^{\mathcal{i}}_k(\tau ) \Big |\, d\tau +\frac{(\sigma ^{\mathcal{i}}_N)^2}{2} \int _s^t \Bigg | \frac{d}{d\tau } \frac{1}{\rho ^{\mathcal{i}}_k(\tau )} \Bigg | \,d\tau \\ &\le 3 \sigma ^{\mathcal{i}}_N \sum _{k\in \mathcal{N}\setminus \left \{N\right \}} \int _s^t \Big | \dot{x}^{\mathcal{i}}_k(\tau ) \Big |\, d\tau \\ &\leq C|t-s|, \end{align*}

in view of Lemma 3.4.

Once proved the bounds described by equations (3.5) and (3.7), we can apply Theorem 3.5, which concludes the proof.

Lemma 3.7 (Convergence of momenta). Given $\mu _{\rho ^{\mathcal{i}}}(t)$ and $\mu _{x^{\mathcal{i}}}^{\mathcal{N}}(t)$ , respectively,

\begin{equation*} \mu _{\rho ^{\mathcal {i}}}(t) = \frac {1}{\sigma ^{\mathcal {i}}|\Omega |}\int _\Omega \rho ^{\mathcal {i}}(y,t) y\,dy, \,\qquad \mu _{x^{\mathcal {i}}}^{\mathcal {N}}(t) = \frac {1}{N+1} \sum _{k=0}^{N} x^{\mathcal {i}}_k(t), \end{equation*}

we have that

\begin{equation*} \lim _{N\to \infty } \big (\mu _{\rho ^{\mathcal {i}}}(t) - \mu _{x^{\mathcal {i}}}^{\mathcal {N}}(t)\big ) = 0 \end{equation*}

for all $t\gt 0$ .

Proof. We recall that $x^i_0 + x^i_N = 0$ , and $\Omega = [{-}1,1]$ , from the definitions we have

\begin{align*} \mu _{x^{\mathcal{i}}}^{\mathcal{N}}(t) & = \frac{1}{2(N+1)} \sum _{k\in \mathcal{N}\setminus \left \{N\right \}}( x^{\mathcal{i}}_{k+1} + x^{\mathcal{i}}_k)= \frac{1}{2(N+1)} \sum _{k\in \mathcal{N}\setminus \left \{N\right \}}\frac{(x^{\mathcal{i}}_{k+1} + x^{\mathcal{i}}_k) ( x^{\mathcal{i}}_{k+1} - x^{\mathcal{i}}_k)}{ x^{\mathcal{i}}_{k+1} - x^{\mathcal{i}}_k}\\ &= \frac{N}{(N+1)} \sum _{k\in \mathcal{N}\setminus \left \{N\right \}}\frac{\rho ^{\mathcal{i}}_k }{\sigma ^{\mathcal{i}}} \frac{(x^{\mathcal{i}}_{k+1})^2 - (x^{\mathcal{i}}_k)^2}{2}= \frac{N}{(N+1)} \sum _{k\in \mathcal{N}\setminus \left \{N\right \}} \frac{\rho ^{\mathcal{i}}_k(t)}{\sigma ^{\mathcal{i}}}\, \int _{-\infty }^{\infty } x\, \chi _{I^{\mathcal{i}}_k} \,dx \\ &= \frac{N}{(N+1)}\, \mu _{\rho ^{\mathcal{i},N}}(t) \,, \end{align*}

where we used equation (2.14). We conclude that there exists a constant depending only on the domain $\Omega$ such that

\begin{align*} |\mu _{\rho ^{\mathcal{i}}}(t) - \mu _{x^{\mathcal{i}}}^{\mathcal{N}}(t) | &\le \int _\Omega |\rho ^{\mathcal{i}} - \frac{N}{N+1}\rho ^{\mathcal{i},N}|\, |x|\, dx\\ &\le C(|\Omega |) \| \rho ^{\mathcal{i}} - \rho ^{\mathcal{i},N} \|_{L^1} + o\left (\frac{C}{N}\right )\,, \end{align*}

which concludes the proof.

The following Proposition concerns the convergence of the approximated nodes $a_{\mathcal{N}}^{\mathcal{i}}$ .

Proof. We first notice that from (2.4b) and the boundedness of $\mathbf{V}$ , we have the uniform bound

\begin{equation*} |a_{\mathcal {N}}^{\mathcal {i}}(t)|\leq |\bar {a}^{\mathcal {i}}|+T\|\mathbf {V}\|_{\infty },\quad \textit{ with }t\in [0,T], \end{equation*}

for all $\mathcal{i}\in \mathcal{M}$ uniformly in $N$ . Thus, there exist $a^{\mathcal{i}}$ such that $a_{\mathcal{N}}^{\mathcal{i}}(t)$ pointwise converges (up to subsequences) to $a^{\mathcal{i}}(t)$ as $N\to \infty$ . Consider now $N_1,N_2\in \mathbb{N}$ , then

\begin{align*} \sum _{\mathcal{i}\in \mathcal{M}}|a_{\mathcal{N}_1}^{\mathcal{i}}(t)-a_{\mathcal{N}_2}^{\mathcal{i}}(t)|\leq & \sum _{\mathcal{i},\mathcal{j}\in \mathcal{M}} \int _0^t\left | \mathbf{V}(\mu _{x^{\mathcal{i}}}^{\mathcal{N}_1}, \mu _{x^{\mathcal{j}}}^{\mathcal{N}_1}, a_{\mathcal{N}_1}^{\mathcal{ij}})-\mathbf{V}(\mu _{x^{\mathcal{i}}}^{\mathcal{N}_2}, \mu _{x^{\mathcal{j}}}^{\mathcal{N}_2}, a_{\mathcal{N}_2}^{\mathcal{ij}})\right |d\tau \\ \leq & C\sum _{\mathcal{i},\mathcal{j}\in \mathcal{M}}\int _0^t\left | \mu _{x^{\mathcal{i}}}^{\mathcal{N}_1}(\tau )-\mu _{x^{\mathcal{i}}}^{\mathcal{N}_2}(\tau )\right |+\left | \mu _{x^{\mathcal{j}}}^{\mathcal{N}_1}(\tau )-\mu _{x^{\mathcal{j}}}^{\mathcal{N}_2}(\tau )\right |\,d\tau \\ & + C\sum _{\mathcal{j}\in \mathcal{M}}\int _0^t\left | a_{\mathcal{N}_1}^{\mathcal{ij}}(\tau )-a_{\mathcal{N}_2}^{\mathcal{ij}}(\tau )\right |\,d\tau \\ \leq & 2C\sum _{\mathcal{i}\in \mathcal{M}}\sum _{l=1,2}\| \rho ^{\mathcal{i}}-\rho ^{\mathcal{i},\mathcal{N}_l}\|_{L^1(\Omega _T)} + 2C\int _0^t\sum _{\mathcal{i}\in \mathcal{M}}|a_{\mathcal{N}_1}^{\mathcal{i}}(t)-a_{\mathcal{N}_2}^{\mathcal{i}}(t)|\,d\tau, \end{align*}

where we used the estimate from the proof of Lemma 3.7 and the fact that by straightforward manipulations we have

\begin{align*} \sum _{\mathcal{i},\mathcal{j}\in \mathcal{M}} \left | a_{\mathcal{N}_1}^{\mathcal{ij}}(\tau )-a_{\mathcal{N}_2}^{\mathcal{ij}}(\tau )\right |=& \sum _{\mathcal{i},\mathcal{j}\in \mathcal{M}}\left | \|a_{\mathcal{N}_1}^{\mathcal{i}}(\tau )-a_{\mathcal{N}_1}^{\mathcal{j}}(\tau )\|-\|a_{\mathcal{N}_2}^{\mathcal{i}}(\tau )-a_{\mathcal{N}_2}^{\mathcal{j}}(\tau )\|\right |\\ & =\sum _{\mathcal{i},\mathcal{j}\in \mathcal{M}}\left | \|a_{\mathcal{N}_1}^{\mathcal{i}}(\tau )\pm a_{\mathcal{N}_2}^{\mathcal{i}}(\tau )\pm a_{\mathcal{N}_2}^{\mathcal{j}}(\tau )-a_{\mathcal{N}_1}^{\mathcal{j}}(\tau )\|-\|a_{\mathcal{N}_2}^{\mathcal{i}}(\tau )-a_{\mathcal{N}_2}^{\mathcal{j}}(\tau )\|\right |\\ & \leq \sum _{\mathcal{i},\mathcal{j}\in \mathcal{M}}\left | \|a_{\mathcal{N}_1}^{\mathcal{i}}(\tau )-a_{\mathcal{N}_2}^{\mathcal{i}}(\tau )\|+\|a_{\mathcal{N}_2}^{\mathcal{i}}-a_{\mathcal{N}_2}^{\mathcal{j}}(\tau )\|\right .\\ &-\left .\|a_{\mathcal{N}_2}^{\mathcal{i}}(\tau )-a_{\mathcal{N}_2}^{\mathcal{j}}(\tau )\|+\|a_{\mathcal{N}_2}^{\mathcal{j}}(\tau )-a_{\mathcal{N}_1}^{\mathcal{j}}(\tau )\|\right |\\ & \leq \sum _{\mathcal{i},\mathcal{j}\in \mathcal{M}} \left |a_{\mathcal{N}_1}^{\mathcal{i}}(\tau )-a_{\mathcal{N}_2}^{\mathcal{i}}(\tau )\right |+\left |a_{\mathcal{N}_2}^{\mathcal{j}}(\tau )-a_{\mathcal{N}_1}^{\mathcal{j}}(\tau )\right |\\ &\leq 2\sum _{\mathcal{i}\in \mathcal{M}}|a_{\mathcal{N}_1}^{\mathcal{i}}(t)-a_{\mathcal{N}_2}^{\mathcal{i}}(t)|\,. \end{align*}

Thus, by using the integral version of Gronwall’s inequality we can deduce

\begin{align*} \sup _{t\in [0,T]}\sum _{\mathcal{i}\in \mathcal{M}}|a_{\mathcal{N}_1}^{\mathcal{i}}(t)-a_{\mathcal{N}_2}^{\mathcal{i}}(t)|\leq & 2C\sum _{\mathcal{i}\in \mathcal{M}}\sum _{l=1,2}\| \rho ^{\mathcal{i}}-\rho ^{\mathcal{i},\mathcal{N}_l}\|_{L^1(\Omega _T)} e^{2CT}. \end{align*}

and then

\begin{align*} \sup _{t\in [0,T]}|a_{\mathcal{N}_1}^{\mathcal{i}}(t)-a_{\mathcal{N}_2}^{\mathcal{i}}(t)|\leq & 2C\sum _{\mathcal{i}\in \mathcal{M}}\sum _{l=1,2}\| \rho ^{\mathcal{i}}-\rho ^{\mathcal{i},\mathcal{N}_l}\|_{L^1(\Omega _T)} e^{2CT}, \end{align*}

that ensure the uniform convergences of $a_{\mathcal{N}_1}^{\mathcal{i}}$ to $a^{\mathcal{i}},$ for all $\mathcal{i} \in \mathcal{M}$ .

In order to show that $a^{\mathcal{i}}$ satisfies (2.13), it is enough to observe that we can invoke the dominated convergence theorem since by the continuity of $\mathbf{V}$ and the uniform converges proved we have that

\begin{equation*} \mathbf {V}(\mu _{x^{\mathcal {i}}}^{\mathcal {N}}, \mu _{x^{\mathcal {j}}}^{\mathcal {N}}, a_{\mathcal {N}}^{\mathcal {ij}})\to \mathbf {V}(\mu _{\rho ^{\mathcal {i}}}, \mu _{\rho ^{\mathcal {j}}}, a^{\mathcal {ij}})\quad \text{a.e. in} \quad t\in [0,T], \end{equation*}

and $\mathbf{V}(\mu _{x^{\mathcal{i}}}^{\mathcal{N}}, \mu _{x^{\mathcal{j}}}^{\mathcal{N}}, a_{\mathcal{N}}^{\mathcal{ij}})$ is uniformly bounded w.r.t. $N$ . Thus,

\begin{equation*} \int _0^t\mathbf {V}(\mu _{x^{\mathcal {i}}}^{\mathcal {N}}, \mu _{x^{\mathcal {j}}}^{\mathcal {N}}, a_{\mathcal {N}}^{\mathcal {ij}})\,d\tau \to \int _0^t \mathbf {V}(\mu _{\rho ^{\mathcal {i}}}, \mu _{\rho ^{\mathcal {j}}}, a^{\mathcal {ij}})\,d\tau, \end{equation*}

for all $t\in [0,T]$ .

We now prove that the empirical measures associated with the solution of equation (2.4a) and the piecewise constant densities in equation (2.14) share the same limit with respect to a suitable topology.

Lemma 3.9. For any $T\gt 0$ , the empirical measures associated with the solution of equation (2.4a), defined by

(3.8) \begin{equation} \tilde{\rho }^{\mathcal{i},\mathcal{N}}(t,x)=\sigma _N^{\mathcal{i}}\sum _{k\in \mathcal{N}}\delta _{x_k^{\mathcal{i}}(t)}(x), \quad \mathcal{i}\in \mathcal{M}, \end{equation}

satisfy

\begin{equation*} d_{W^1}\left (\tilde {\rho }^{\mathcal {i},\mathcal {N}}(t,\cdot ),\rho ^{\mathcal {i}}(t,\cdot )\right )\to 0,\,\textit { as }\, N\to \infty, \end{equation*}

for all $t\in [0,T]$ , where $\rho ^{\mathcal{i}}$ is the limit obtained in Proposition 3.6.

Proof. We take again advantage of the isometry between the $1-$ Wasserstain space for probability measures and the $L^1$ space in the space pseudo-inverse functions by noticing that the pseudo-inverse of an empirical measure is piecewise constant and then

\begin{align*} d_{W^1}\big (\tilde{\rho }^{\mathcal{i},\mathcal{N}}(t,\cdot ),\rho ^{\mathcal{i},\mathcal{N}}(t,\cdot )\big ) = & \|X_{\tilde{\rho }^{\mathcal{i},\mathcal{N}}}(t,\cdot )-X_{\rho ^{\mathcal{i},\mathcal{N}}}(t,\cdot )\|_{L^1([0,\sigma ^{\mathcal{i}}])}\\ \leq & \sum _{k\in \mathcal{N}\setminus \left \{N\right \}}\int _{k\sigma ^{\mathcal{i}}_N}^{(k+1)\sigma ^{\mathcal{i}}_N} \Bigg | (m - k \sigma ^{\mathcal{i}}_N)\frac{1}{\rho ^{\mathcal{i}}_k(t)} \Bigg | \, dm\\ = & \frac{\sigma _N^{\mathcal{i}}}{2}|\Omega |. \end{align*}

The statement then follows from a triangulation argument.

3.2. Convergence to weak solutions

With $\mathbf{a}^{\mathcal{N}}$ we denoted the vector $(a^{\mathcal{i}}_{\mathcal{N}})_{\mathcal{i}\in \mathcal{M}}$ , of agents with piecewise constant opinion distribution $\rho ^{\mathcal{i},\mathcal{N}}$ , while $\mathbf{a}$ is related to the vector of continuous distributions $\boldsymbol{\rho }$ . This distinction is not stressed in the rest of the paper where the context does not allow misunderstanding.

Remark 3.10. We notice that evaluating the operator $\beta ^{\mathcal{i}}$ in (3.8) we have

(3.9) \begin{equation} \begin{aligned} \beta ^{\mathcal{i}}(\boldsymbol{\tilde{\rho }}^N,\boldsymbol{a}^N;\,x_{k}^{\mathcal{i}}) =& \sum _{\mathcal{j} \in \mathcal{M}} \int _\Omega \mathbf{A}^{\mathcal{ij}}(x_k^{\mathcal{i}},y,a_N^{\mathcal{ij}})\tilde{\rho }^{\mathcal{j},N}(y,t)\,dy\\ =& \sum _{\mathcal{j} \in \mathcal{M}} \sum _{l\in \mathcal{N}}\sigma _N^{\mathcal{j}}\mathbf{A}^{\mathcal{ij}}(x_k^{\mathcal{i}},x_l^{\mathcal{j}},a_N^{\mathcal{ij}})\\ =&\beta _k^{\mathcal{i}}. \end{aligned} \end{equation}

Moreover,

(3.10) \begin{equation} \partial _x\beta ^{\mathcal{i}}\left (\tilde{\boldsymbol{\rho }}^{N},\boldsymbol{a}^N;\,x\right )=\sum _{\mathcal{j}\in \mathcal{M}}\int _{\Omega }\partial _1{\mathbf{A}}^{\mathcal{ij}}(x,y;\,a_N^{\mathcal{ij}})\tilde{\rho }^{\mathcal{j},N}(t,y)\,dy, \end{equation}

Lemma 3.11. Let $T\gt 0$ , and consider the kernels ${\mathbf{A}}^{\mathcal{ij}}$ , $\mathbf{K}^{\mathcal{ij}}$ under assumptions $({\mathbf{A}})$ and $(\mathbf{K})$ , respectively. Let $\rho ^{\mathcal{i},\mathcal{N}}$ and $\tilde{\rho }^{\mathcal{i},\mathcal{N}}$ be the sequences defined in equations (2.14) and (3.8), respectively, and their limits $\rho ^{\mathcal{i}}$ given by Proposition 3.6 and Lemma 3.9 , for all $\mathcal{i} \in \mathcal{M}$ . Then, for every $\zeta \in C_{0}^\infty \left (\Omega _T\right )$ we have

(3.11) \begin{equation} \int _{\Omega _T}\rho ^{\mathcal{i},\mathcal{N}}\partial _t\zeta \,dx\,dt\to \int _{\Omega _T}\rho ^{\mathcal{i}}\partial _t\zeta \,dx\,dt \end{equation}

(3.12) \begin{equation} \int _{\Omega _T}\Phi ^{\mathcal{i}}\left (\rho ^{\mathcal{i},\mathcal{N}}\right )\partial _x\beta ^{\mathcal{i}}\left (\tilde{\boldsymbol{\rho }}^{\mathcal{N}},\mathbf{a}^{\mathcal{N}};\,x\right )\partial _x\zeta \,dx\,dt\to \int _{\Omega _T}\Phi ^{\mathcal{i}}\left (\rho ^{\mathcal{i}}\right )\partial _x\beta ^{\mathcal{i}}\left (\boldsymbol{\rho },\mathbf{a};\,x\right )\partial _x\zeta \,dx\,dt \end{equation}

(3.13) \begin{equation} \int _{\Omega _T}\Phi ^{\mathcal{i}}\left (\rho ^{\mathcal{i},\mathcal{N}}\right )\beta ^{\mathcal{i}}\left (\tilde{\boldsymbol{\rho }}^{\mathcal{N}},\mathbf{a}^{\mathcal{N}};\,x\right )\partial _{xx}\zeta \,dx\,dt\to \int _{\Omega _T}\Phi ^{\mathcal{i}}\left (\rho ^{\mathcal{i}}\right )\beta ^{\mathcal{i}}\left (\boldsymbol{\rho },\mathbf{a};\,x\right )\partial _{xx}\zeta \,dx\,dt \end{equation}

(3.14) \begin{equation} \int _{\Omega _T}\rho ^{\mathcal{i},\mathcal{N}}\theta ^{\mathcal{i}}\left (\tilde{\boldsymbol{\rho }}^{\mathcal{N}},\mathbf{a}^{\mathcal{N}};\,x\right )\partial _x\zeta \,dx\,dt\to \int _{\Omega _T}\rho ^{\mathcal{i}}\theta ^{\mathcal{i}}\left (\boldsymbol{\rho },\mathbf{a};\,x\right )\partial _x\zeta \,dx\,dt \end{equation}

as $N\to \infty$ .

Proof. We only prove equation (3.12), since equations (3.13) and (3.14) follow from similar argument, and equation (3.11) is a direct consequence of the $L^1$ strong compactness proved in Proposition 3.6. We first split the terms as following

\begin{align*} &\left |\int _{\Omega _T}\left (\Phi ^{\mathcal{i}}\left (\rho ^{\mathcal{i},\mathcal{N}}\right )\partial _x\beta ^{\mathcal{i}}\left (\tilde{\boldsymbol{\rho }}^{\mathcal{N}},\mathbf{a}^{\mathcal{N}};\,x\right )-\Phi ^{\mathcal{i}}\left (\rho ^{\mathcal{i}}\right )\partial _x\beta ^{\mathcal{i}}\left (\boldsymbol{\rho },\mathbf{a};\,x\right )\right )\partial _x\zeta (t,x)dx\,dt\right |\\ &\leq \left |\int _{\Omega _T}\left (\Phi ^{\mathcal{i}}\left (\rho ^{\mathcal{i},\mathcal{N}}\right )-\Phi ^{\mathcal{i}}\left (\rho ^{\mathcal{i}}\right )\right )\partial _x\beta ^{\mathcal{i}}\left (\tilde{\boldsymbol{\rho }}^{\mathcal{N}},\mathbf{a}^{\mathcal{N}};\,x\right )\partial _x\zeta (t,x)\,dx\,dt \right |\\&\quad +\left |\int _{\Omega _T}\Phi ^{\mathcal{i}}\left (\rho ^{\mathcal{i}}\right )\left (\partial _x\beta ^{\mathcal{i}}\left (\tilde{\boldsymbol{\rho }}^{\mathcal{N}},\mathbf{a}^{\mathcal{N}};\,x\right )-\partial _x\beta ^{\mathcal{i}}\left (\boldsymbol{\rho },\mathbf{a};\,x\right )\right )\partial _x\zeta (t,x)dx\,dt\right |\\ & =|I|+|II|. \end{align*}

We now treat the two terms separately. Assumption $({\mathbf{A}})$ and equation (3.10) ensure the following bound

\begin{align*} |I|\leq & \sum _{\mathcal{j}\in \mathcal{M}} \int _{\Omega _T}\left |\Phi ^{\mathcal{i}}\left (\rho ^{\mathcal{i},\mathcal{N}}\right )-\Phi ^{\mathcal{i}}\left (\rho ^{\mathcal{i}}\right )\right |\int _{\Omega }\left |\partial _1{\mathbf{A}}^{\mathcal{ij}}(x,y;\,a^{\mathcal{ij}})\tilde{\rho }^{\mathcal{j},\mathcal{N}}(t,y)\,dy\right |\left |\partial _x\zeta (t,x)\right |\,dx\,dt\\ \leq & C\|\rho ^{\mathcal{i},\mathcal{N}}-\rho ^{\mathcal{i}}\|_{L^1(\Omega _T)}. \end{align*}

where $C$ is a constant depending on $\left \|\partial _1{\mathbf{A}}^{\mathcal{ij}}\right \|_{\infty }$ , $\left \|\partial _x\zeta \right \|_{\infty }$ , $Lip\left (\Phi ^{\mathcal{i}}\right )$ and $\sigma ^{\mathcal{M}}$ . In order to bound the second integral, let us introduce $\Pi ^{\mathcal{i},\mathcal{N}}$ an optimal transport plan between $\tilde{\rho }^{\mathcal{i},\mathcal{N}}$ and $\rho ^{\mathcal{i}}$ . Then, we have

\begin{align*} |II|\leq & \sum _{\mathcal{j}\in \mathcal{M}} \int _{\Omega _T}\Phi ^{\mathcal{i}}\left (\rho ^{\mathcal{i}}\right )\int _{\Omega ^2}\left |\partial _1{\mathbf{A}}^{\mathcal{ij}}(x,y;\,a_{\mathcal{N}}^{\mathcal{ij}})-\partial _1{\mathbf{A}}^{\mathcal{ij}}(x,z;\,a^{\mathcal{ij}})\right |\,d\Pi ^{\mathcal{i},\mathcal{N}}(y,z)|\partial _x\zeta |\,dx\,dt\\ \leq & C\sum _{\mathcal{j}\in \mathcal{M}} \int _{\Omega _T}\int _{\Omega \times \Omega }\left |y-z\right |\,d\Pi ^{\mathcal{i},\mathcal{N}}(y,z)\,dx\,dt\\ \leq & C M|\Omega |\int _0^T d_{W^1}(\tilde{\rho }^{\mathcal{i},\mathcal{N}}(t,\cdot ),\rho ^{\mathcal{i}}(t,\cdot ))\,dt, \end{align*}

where in this case $C$ is a constant depending on $\|\rho ^{\mathcal{i}}\|_\infty$ , $\left \|\partial _x\zeta \right \|_{\infty }$ , and the constant $c_{1,A}$ from Assumption $({\mathbf{A}})$ . The convergences in Proposition 3.6 and Lemma 3.9 ensure that equation (3.12) holds.

We are now in the position of proving that the limit densities and nodes satisfy the weak formulation in the sense of Definition 2.1. More precisely, we are going to show that for $N\to +\infty$ we have

(3.15) \begin{equation} \begin{aligned} \int _{\Omega _T}&\rho ^{\mathcal{i},\mathcal{N}}\partial _t\zeta +\rho ^{\mathcal{i},\mathcal{N}}\theta ^{\mathcal{i}}(\boldsymbol{\tilde{\rho }}^{\mathcal{N}},\mathbf{a}^{\mathcal{N}};\,x) \partial _x \zeta \, dx\,dt\\ &+\int _{\Omega _T}\Phi ^{\mathcal{i}}\left (\rho ^{\mathcal{i},\mathcal{N}}\right ) \left ( \partial _x\beta ^{\mathcal{i}}(\boldsymbol{\tilde{\rho }}^{\mathcal{N}},\mathbf{a}^{\mathcal{N}};\,x) \partial _x \zeta +\beta ^{\mathcal{i}}(\boldsymbol{\tilde{\rho }}^{\mathcal{N}},\mathbf{a}^{\mathcal{N}};\,x) \partial _{xx} \zeta \right )\, dx\,dt\to 0 \end{aligned} \end{equation}

that combined with the convergences in Lemma 3.11 gives the assertion. We state the following

Proof. We start considering the term involving the time derivative. By definition of $\rho ^{\mathcal{i},\mathcal{N}}$ in equation (2.14), a discrete integration by parts and Fundamental Theorem of Calculus give

\begin{align*} \int _{\Omega _T}\rho ^{\mathcal{i},\mathcal{N}}\partial _t\zeta =&\sum _{k\in \mathcal{N}\setminus \left \{N\right \}}\int _0^T\rho _k^{\mathcal{i}}(t)\int _{I_k^{\mathcal{i}}(t)}\partial _t\zeta (t,x)\,dx\,dt\\ =&\sum _{k\in \mathcal{N}\setminus \left \{N\right \}}\int _0^T\rho _k^{\mathcal{i}}(t)\dot{x}_{k+1}^{\mathcal{i}}(t)\left (\int _{I_k^{\mathcal{i}}(t)}\zeta (t,x)\,dx-\zeta (t,x_{k+1}^{\mathcal{i}}(t))\right )\,dt\\ &-\sum _{k\in \mathcal{N}\setminus \left \{N\right \}}\int _0^T\rho _k^{\mathcal{i}}(t)\dot{x}_{k}^{\mathcal{i}}(t)\left ( \int _{I_k^{\mathcal{i}}(t)}\zeta (t,x)\,dx-\zeta (t,x_{k}^{\mathcal{i}}(t))\right )\,dt. \end{align*}

A second-order expansion of $\zeta$ around $x_{k+1}^{\mathcal{i}}(t)$ in the first average integral and around $x_{k}^{\mathcal{i}}(t)$ in the second average integral produces

\begin{align*} \int _{\Omega _T}\rho ^{\mathcal{i},\mathcal{N}}\partial _t\zeta =&-\frac{\sigma _N^{\mathcal{i}}}{2}\sum _{k\in \mathcal{N}\setminus \left \{N\right \}}\int _0^T\dot{x}_{k+1}^{\mathcal{i}}(t)\partial _x\zeta (t,x_{k+1}^{\mathcal{i}})\,dt\\ &+\frac{1}{2}\sum _{k\in \mathcal{N}\setminus \left \{N\right \}}\int _0^T\rho _k^{\mathcal{i}}(t)\dot{x}_{k+1}^{\mathcal{i}}(t)\int _{I_k^{\mathcal{i}}(t)}\partial _{xx}\zeta (t,\hat{x}_{k+1}^{\mathcal{i}})\left (x-x_{k+1}^{\mathcal{i}}(t))\right )^2\,dx\,dt\\ &-\frac{\sigma _N^{\mathcal{i}}}{2}\sum _{k\in \mathcal{N}\setminus \left \{N\right \}}\int _0^T\dot{x}_{k}^{\mathcal{i}}(t)\partial _x\zeta (t,x_{k}^{\mathcal{i}})\,dt\\ &-\frac{1}{2}\sum _{k\in \mathcal{N}\setminus \left \{N\right \}}\int _0^T\rho _k^{\mathcal{i}}(t)\dot{x}_{k}^{\mathcal{i}}(t)\int _{I_k^{\mathcal{i}}(t)}\partial _{xx}\zeta (t,\hat{x}_{k}^{\mathcal{i}})\left (x-x_{k}^{\mathcal{i}}(t))\right )^2\,dx\,dt, \end{align*}

where $\hat{x}_{k}^{\mathcal{i}}$ and $\hat{x}_{k+1}^{\mathcal{i}}$ are points in $\left [x_{k}^{\mathcal{i}},x\right ]$ and equation $\left [x,x_{k+1}^{\mathcal{i}}\right ]$ , respectively. We now combine the first and third terms on the r.h.s. above and use (2.4) in order to obtain

\begin{equation*} -\frac {\sigma _N^{\mathcal {i}}}{2}\sum _{k\in \mathcal {N}\setminus \left \{N\right \}}\int _0^T\left (\dot {x}_{k}^{\mathcal {i}}(t)\partial _x\zeta (t,x_{k}^{\mathcal {i}})+\dot {x}_{k+1}^{\mathcal {i}}(t)\partial _x\zeta (t,x_{k+1}^{\mathcal {i}})\right )\,dt = A_1+A_2, \end{equation*}

where

\begin{align*} A_1 = - \frac{1}{2}\sum _{k\in \mathcal{N}\setminus \left \{N\right \}}\int _0^T&\left (\beta ^{\mathcal{i}}_k\left (\Phi ^{\mathcal{i}}(\rho ^{\mathcal{i}}_{k-1}) - \Phi ^{\mathcal{i}}(\rho ^{\mathcal{i}}_{k}) \right ) \partial _x\zeta (t,x_{k}^{\mathcal{i}})\right .\\ &+\left .\beta ^{\mathcal{i}}_{k+1} \left (\Phi ^{\mathcal{i}}(\rho ^{\mathcal{i}}_{k}) - \Phi ^{\mathcal{i}}(\rho ^{\mathcal{i}}_{k+1}) \right ) \partial _x\zeta (t,x_{k+1}^{\mathcal{i}})\right )\,dt, \end{align*}

and

\begin{equation*} A_2= - \frac {\sigma _N^{\mathcal {i}}}{2}\sum _{k\in \mathcal {N}\setminus \left \{N\right \}}\int _0^T\left (\theta _{k}^{\mathcal {i}}(t)\partial _x\zeta (t,x_{k}^{\mathcal {i}})+\theta _{k+1}^{\mathcal {i}}(t)\partial _x\zeta (t,x_{k+1}^{\mathcal {i}})\right )\,dt. \end{equation*}

We now combine the integral $A_1$ with the two terms involving the diffusion in equation (3.15) in order to show that

\begin{equation*} A_1 + \int _{\Omega _T}\Phi ^{\mathcal {i}}\left (\rho ^{\mathcal {i},\mathcal {N}}\right )\partial _x \left ( \beta ^{\mathcal {i}}(\boldsymbol {\tilde {\rho }}^{\mathcal {N}},\textbf {a}^{\mathcal {N}};\,x) \partial _x \zeta (t,x)\right )\, dx\,dt=0. \end{equation*}

Invoking equation (3.9) and the fact that $\partial _x\zeta (t,x_{0}^{\mathcal{i}})=\partial _x\zeta (t,x_{N}^{\mathcal{i}})=0$ , we can compute

\begin{align*} &\int _{\Omega _T}\Phi ^{\mathcal{i}}\left (\rho ^{\mathcal{i},\mathcal{N}}\right )\partial _x \left ( \beta ^{\mathcal{i}}(\boldsymbol{\tilde{\rho }}^{\mathcal{N}},\mathbf{a}^{\mathcal{N}};\,x) \partial _x \zeta (t,x)\right )\, dx\,dt\\ =&\sum _{k\in \mathcal{N}\setminus \left \{N\right \}}\int _0^T\Phi ^{\mathcal{i}}\left (\rho _k^{\mathcal{i}}\right ) \int _{I_k^{\mathcal{i}}(t)}\partial _x \left ( \beta ^{\mathcal{i}}(\boldsymbol{\tilde{\rho }}^{\mathcal{N}},\mathbf{a}^{\mathcal{N}};\,x) \partial _x \zeta (t,x)\right )\, dx\,dt\\ =&\sum _{k\in \mathcal{N}\setminus \left \{N\right \}}\int _0^T\Phi ^{\mathcal{i}}\left (\rho _k^{\mathcal{i}}\right ) \left ( \beta ^{\mathcal{i}}(\boldsymbol{\tilde{\rho }}^{\mathcal{N}},\mathbf{a}^{\mathcal{N}};\,x_{k+1}^{\mathcal{i}}) \partial _x \zeta (t,x_{k+1}^{\mathcal{i}})-\beta ^{\mathcal{i}}(\boldsymbol{\tilde{\rho }}^{\mathcal{N}},\mathbf{a}^{\mathcal{N}};\,x_{k}^{\mathcal{i}}) \partial _x \zeta (t,x_{k}^{\mathcal{i}})\right )\,dt\\ =&\sum _{k\in \mathcal{N}\setminus \left \{N\right \}}\int _0^T\Phi ^{\mathcal{i}}\left (\rho _k^{\mathcal{i}}\right ) \left ( \beta _{k+1}^{\mathcal{i}}\partial _x \zeta (t,x_{k+1}^{\mathcal{i}})-\beta _k^{\mathcal{i}} \partial _x \zeta (t,x_{k}^{\mathcal{i}})\right )\,dt\\ =&\sum _{k\in \mathcal{N}\setminus \left \{0,N\right \}}\int _0^T\beta ^{\mathcal{i}}_k \left (\Phi ^{\mathcal{i}}(\rho ^{\mathcal{i}}_{k-1}) - \Phi ^{\mathcal{i}}(\rho ^{\mathcal{i}}_{k}) \right ) \partial _x\zeta (t,x_{k}^{\mathcal{i}})\,dt, \end{align*}

which can be combined with $A_1$ by shifting the indexes and using again the fact that the test function vanishes at the boundary.

Recalling the definition of $\theta _k^{\mathcal{i}}$ in equation (2.8), rearranging the indexes in $A_2$ , using the fact that $\partial _x\zeta$ vanishes at the boundary of $\Omega$ and summing with the second term in equation (3.15) we obtain

\begin{align*} &\sum _{k\in \mathcal{N}\setminus \left \{0,N\right \}}\sum _{\mathcal{i}\in \mathcal{M}}\sum _{l\in \mathcal{N}\setminus \left \{N\right \}}\sigma _N^{\mathcal{i}}\sigma _N^{\mathcal{j}}\int _0^T\mathbf{K}^{\mathcal{ij}}(x_k^{\mathcal{i}},x_l^{\mathcal{j}};\,a_N^{\mathcal{ij}})\partial _x\zeta (t,x_{k}^{\mathcal{i}})\,dt\\&+ \sum _{k\in \mathcal{N}\setminus \left \{N\right \}}\sum _{\mathcal{i}\in \mathcal{M}}\sum _{l\in \mathcal{N}\setminus \left \{N\right \}}\sigma _N^{\mathcal{j}}\int _0^T\rho _k^{\mathcal{i}}(t) \int _{I_{k}^{\mathcal{i}}} \mathbf{K}^{\mathcal{ij}}(x,x_l^{\mathcal{j}};\,a_N^{\mathcal{ij}}) \partial _x \zeta (t,x)\, dx\,dt. \end{align*}

A first-order expansion on $\partial _x\zeta$ around $x_k^{\mathcal{i}}$ for $\hat{\hat{x}}_k^{\mathcal{i}}\in \left [x_k^{\mathcal{i}},x\right ]$ , together with the definition of $\rho _k^{\mathcal{i}}$ and assumption $(\mathbf{K})$ , yields

\begin{align*} &\left |\sum _{k\in \mathcal{N}\setminus \left \{0,N\right \}}\sum _{\mathcal{i}\in \mathcal{M}}\sum _{l\in \mathcal{N}\setminus \left \{N\right \}}\sigma _N^{\mathcal{j}}\int _0^T\rho _k^{\mathcal{i}}\partial _x\zeta (t,x_{k}^{\mathcal{i}})\int _{I_{k}^{\mathcal{i}}} \mathbf{K}^{\mathcal{ij}}(x_k^{\mathcal{i}},x_l^{\mathcal{j}};\,a_N^{\mathcal{ij}})-\mathbf{K}^{\mathcal{ij}}(x,x_l^{\mathcal{j}};\,a_N^{\mathcal{ij}})\,dx\,dt\right .\\&\left .+ \sum _{k\in \mathcal{N}\setminus \left \{N\right \}}\sum _{\mathcal{i}\in \mathcal{M}}\sum _{l\in \mathcal{N}\setminus \left \{N\right \}}\sigma _N^{\mathcal{j}}\int _0^T\rho _k^{\mathcal{i}}(t) \int _{I_{k}^{\mathcal{i}}} \mathbf{K}^{\mathcal{ij}}(x,x_l^{\mathcal{j}};\,a_N^{\mathcal{ij}}) (x-x_k^{\mathcal{i}})\partial _{xx} \zeta (t,\hat{\hat{x}}_k^{\mathcal{i}})\, dx\,dt\right |\\ &\leq \sum _{k\in \mathcal{N}\setminus \left \{0,N\right \}}\sum _{\mathcal{i}\in \mathcal{M}}\sum _{l\in \mathcal{N}\setminus \left \{N\right \}}\sigma _N^{\mathcal{j}} \left (c_{\mathbf{K}}\|\partial _x\zeta \|_\infty +\|\mathbf{K}^{\mathcal{ij}}\|_\infty \|\partial _{xx}\zeta \|_\infty \right )\int _0^T\rho _k^{\mathcal{i}}\int _{I_{k}^{\mathcal{i}}}|x_k^{\mathcal{i}}-x|\,dx\,dt\\ &=\sum _{k\in \mathcal{N}\setminus \left \{0,N\right \}}\sum _{\mathcal{i}\in \mathcal{M}}\frac{\sigma ^{\mathcal{j}}}{2} \left (c_{\mathbf{K}}\|\partial _x\zeta \|_\infty +\|\mathbf{K}^{\mathcal{ij}}\|_\infty \|\partial _{xx}\zeta \|_\infty \right )\int _0^T\sigma _N^{\mathcal{i}}|I_k^{\mathcal{i}}|(t)\,dt\\ &\leq \sigma _N^{\mathcal{i}}\sum _{\mathcal{i}\in \mathcal{M}}\frac{\sigma ^{\mathcal{j}}}{2} \left (c_{\mathbf{K}}\|\partial _x\zeta \|_\infty +\|\mathbf{K}^{\mathcal{ij}}\|_\infty \|\partial _{xx}\zeta \|_\infty \right )T|\Omega |, \end{align*}

that vanishes as $N\to \infty$ together with $\sigma _N^{\mathcal{i}}$ . We are now left in showing that the remainder, i.e.

\begin{align*} R_{k,k+1}^{\mathcal{i}}=&\frac{1}{2}\sum _{k\in \mathcal{N}\setminus \left \{N\right \}}\int _0^T\rho _k^{\mathcal{i}}(t)\dot{x}_{k+1}^{\mathcal{i}}(t)\int _{I_k^{\mathcal{i}}(t)}\partial _{xx}\zeta (t,\hat{x}_{k+1}^{\mathcal{i}})\left (x-x_{k+1}^{\mathcal{i}}(t))\right )^2\,dx\,dt\\ &-\frac{1}{2}\sum _{k\in \mathcal{N}\setminus \left \{N\right \}}\int _0^T\rho _k^{\mathcal{i}}(t)\dot{x}_{k}^{\mathcal{i}}(t)\int _{I_k^{\mathcal{i}}(t)}\partial _{xx}\zeta (t,\hat{x}_{k}^{\mathcal{i}})\left (x-x_{k}^{\mathcal{i}}(t))\right )^2\,dx\,dt, \end{align*}

goes to zero. We first notice that for all $h,k\in \mathcal{N}\setminus \left \{N\right \}$ , we have

\begin{equation*} \int _{I_k^{\mathcal {i}}(t)}\partial _{xx}\zeta (t,\hat {x}_{h}^{\mathcal {i}})\left (x-x_{k}^{\mathcal {i}}(t))\right )^2\,dx\leq \|\partial _{xx}\zeta \|_\infty \frac {|I_k^{\mathcal {i}}|^3(t)}{3}\leq C \end{equation*}

for some constant $C\gt 0$ , then

\begin{align*} |R_{k,k+1}^{\mathcal{i}}|\leq &\frac{\|\partial _{xx}\zeta \|_\infty }{2}\sum _{k\in \mathcal{N}\setminus \left \{N\right \}}\int _0^T\rho _k^{\mathcal{i}}(t)\left (|\dot{x}_{k+1}^{\mathcal{i}}(t)|+|\dot{x}_{k}^{\mathcal{i}}(t)|\right )\frac{|I_k^{\mathcal{i}}|^3(t)}{3}\,dt\\ \leq & C\|\partial _{xx}\zeta \|_\infty \sum _{k\in \mathcal{N}\setminus \left \{N\right \}}\int _0^T\rho _k^{\mathcal{i}}(t)|I_k^{\mathcal{i}}|^3(t)\,dt\\ =& \sigma _N^{\mathcal{i}} C\|\partial _{xx}\zeta \|_\infty |\Omega |T, \end{align*}

where the second inequality holds in view of Lemma 3.4 and the BV bound on $\rho _k^{\mathcal{i}}$ .

4.1. Modelling

As stated in the introduction, the objective of this model is to elucidate various aspects inherent in interactions on social networks and social media. Specifically, it focuses on elements related to network evolution, homophily and heterophobia. The aim is to examine whether polarisation of opinions and fragmentation of the population can be observed solely by modelling the processes governing network rewiring and the epistemic process. It is essential to note that we do not explicitly model polarisation; instead, we adhere to describing the dynamics of opinion formation that underlie the formation of echo chambers and epistemic bubbles.

In recent years, epistemologists have placed particular emphasis on studying and understanding the formation and evolution processes of the aforementioned social structures. Following Nguyen’s definition [Reference Thi Nguyen38], echo chambers and epistemic bubbles are social epistemic structures based on two distinct types of filtering. In the case of epistemic bubbles, filtering occurs at the network level, while for echo chambers, it takes place at the epistemic process level – specifically, in the attitude towards an interlocutor. In this work, we consider an attitude guided by the distance between the opinions of two interacting agents. Nevertheless, recent studies suggest that it is important to also take into account individuals’ biases for a more comprehensive analysis of the social dynamics based on distrust leading to social filtering, as demonstrated in the work by Pederneschi [Reference Pederneschi43].

In this work, we introduce the concepts of attitude areas and a Euclidean network to describe these dynamics.

Attitude Areas – Open-Mindedness

In this section, we model and simulate a possible choice of opinion dynamic. In particular, we focus our attention on a model that takes into account the heterophobia and homophilia dynamics. The model that we propose is based on five attitude areas: attraction/homophilia, curiosity, indifference, mistrust, repulsion/heterophobia. When two agents interact, their attitude depends on the distance between their respective opinions. We do not consider the attitude depending locally on the opinion itself, this could be a possible improvement that describes the low propensity of changing extreme beliefs. The interaction can be attractive, i.e. the agents move towards a position of consensus looking for a compromise, or can be repulsive, i.e. each agent changes its own opinion moving farther from the one of the other agent. We consider the DPA structure of equations (2.4a), (2.4b), (2.7), (2.8). The operator describing this phenomenon has the following structure

(4.1) \begin{equation} \mathbf{K}^{\mathcal{ij}}(w,v,a^{\mathcal{ij}}) = \omega (a^{\mathcal{ij}}) \, \zeta (\mu ^{\mathcal{i}} - \mu ^{\mathcal{j}})\, (v - w) \end{equation}

where the function $\omega$ depends on the network connections, and $\zeta$ is the attitude function that measures the distance between the agents’ mean opinion. Being $s$ the distance between the mean opinions, the five attitude areas coincide with the following intervals

\begin{equation*} \begin {aligned} &s \in (0,r_{\textit {f}}) \quad &&\text {strong attraction, homophilia} &&(r_{\textit {friends}})\\ &s \in (r_{\textit {f}},r_{\textit {a}}) \quad &&\text {curiosity} &&(r_{\textit {attraction}})\\ &s \in (r_{\textit {a}},r_{\textit {r}}) \quad &&\text {indifference} \\ &s \in (r_{\textit {r}},r_{\textit {l}}) \quad &&\text {mistrust} &&(r_{\textit {repulsion}})\\ &s \in (r_{\textit {l}},|\Omega |) \quad &&\text {repulsion, heterophobia}&&(r_{\textit {limit}})\,. \end {aligned} \end{equation*}

The function $\zeta$ is given by

\begin{equation*} \zeta (s) = \begin {cases} \begin {aligned} &1 - \frac {1}{10} \frac {|s|}{r_{\textit {f}}} \quad &if \quad & |s| \lt r_{\textit {f}}\\ &0.1 + \frac {8}{10} \bigg [1 - \frac {|s| - r_{\textit {f}}}{r_{\textit {a}} - r_{\textit {f}}} \bigg ] \quad &if \quad & r_{\textit {f}} \le |s| \lt r_{\textit {a}}\\ & -0.1 + \frac {2}{10} \bigg [1 - \frac {|s| - r_{\textit {a}}}{r_{\textit {r}} - r_{\textit {a}}} \bigg ] \quad &if \quad & r_{\textit {a}} \le |s| \lt r_{\textit {r}}\\ &-0.9 + \frac {8}{10} \bigg [1 - \frac {|s| - r_{\textit {r}}}{r_{\textit {l}} - r_{\textit {r}}} \bigg ] \quad &if \quad & r_{\textit {r}} \le |s| \lt r_{\textit {l}}\\ &-0.9 - \frac {1}{10} (|s| - r_{\textit {l}}) \quad &if \quad & r_{\textit {l}} \le |s| \,, \end {aligned} \end {cases} \end{equation*}

depending on the values of the extreme of the intervals it looks like those in Figure 1. The term open-mindedness refers to the individual’s inclination to positively consider opinions that differ from their own. In this article, the quantitative description of this term is associated with the value $s \in [0,2]$ such that $\zeta (s)=0$ . In other words, open-mindedness corresponds to the maximum distance within which an agent regards another opinion favourably.

Figure 1. Attraction/repulsion function.

The positive values of the function coincide with the attraction, on the other hand, while the function has negative values it describes repulsion between the agents’ opinion, which could bring to radicalisation or polarisation. Due to the choice of the domain, we have that $s\in [0,2]$ . The different colours coincide with the following definitions of the attitude intervals:

.

Diffusion

Moreover, the opinion dynamic is not ruled only by the direct interaction with the connected agents in the network. We continuously get inputs from all the media, this phenomenon is described in this model by the following operator

(4.2) \begin{equation} \mathbf{A}^{\mathcal{ij}}(w,v,a^{\mathcal{ij}}) = |\mu ^{\mathcal{j}} - w|^2 \,. \end{equation}

In this case, the interaction is not filtered by the network connections, i.e. by $\omega$ , and it is not affected by the attitude. This operator plays the role of the diffusion mobility, the opinion tends to diffuse the more the inputs are far from it, and it is not affected by the diffusion when the inputs coincide with the opinion itself. The underlying idea is that we cannot process actively – i.e. through $\zeta$ – all the information that we get. Those inputs that we cannot elaborate they influence our opinion distribution smoothing it depending on the distance between the input and our mean opinion.

Evolution of the Network

$\,$ As anticipated in the introduction, the Euclidean network relies on the distance between agents in $\mathbb{R}^2$ . The choice of employing $\mathbb{R}^d$ with $d=2$ instead of another dimension lacks specific applicative rationale in this case. Should a concrete application be considered, the dimensions of the network space might depend on factors such as biases, as discussed in [Reference Pederneschi43]. In other words, the network space could potentially capture factors of social segregation correlated with the subject of opinion evolution under consideration.

In this paper, the distance would play a singular role, specifically in defining the interaction radius between agents through the function $\zeta$ . This function is associated with an interaction resembling that of social networks and epistemic bubbles, where filtering occurs at the network level, and interaction in the network space takes place locally rather than globally. We consider $r_{\textit{loc}}$ the radius defining the ball of the local interaction,

(4.3) \begin{equation} \omega (a^{\mathcal{ij}}) = \begin{cases} 1 \quad if \quad a^{\mathcal{ij}} \le \rho _{\textit{loc}}\\ 0 \quad if \quad a^{\mathcal{ij}} \gt \rho _{\textit{loc}} \end{cases} \,. \end{equation}

The agents $a^{\mathcal{i}}$ and $a^{\mathcal{j}}$ interact if their distance on the network is smaller or equal to $\rho _{\textit{loc}}$ . The global interaction coincides with the radius $\rho _{\textit{loc}}=+\infty$ . The initial condition used for our simulation is the one given in Figure 2.

Figure 2. Initial network condition.

In this figure, the initial agent’s coordinates belong to $[0,10]^2$ . The colours describe the mean opinion of each agent, which belongs to the interval $[{-}1,1]$ . The number of agents is $N=40$ . The agent’s coordinates are uniformly randomly distributed on each axis. The dimension of each square is proportional to the social strength $\sigma ^{\mathcal{i}}$ of each agent, and in this case, they all almost coincide.

Considering 3 different radii of interaction the network connections change as in Figure 3.

The evolution of the agents is ruled by the following operator

(4.4) \begin{equation} \mathbf{V}(\mu _{\rho ^{\mathcal{i}}}, \mu _{\rho ^{\mathcal{j}}}, a^{\mathcal{ij}}) = \sum _{l=1}^{2} \sum _j \zeta |\mu _{\rho ^{\mathcal{i}}} - \mu _{\rho ^{\mathcal{j}}}| \, \omega (a^{\mathcal{ij}}) \, (a^{\mathcal{i}} - a^{\mathcal{j}}) \mathbf{e}_l\,, \end{equation}

with $\mathbf{e}_l$ being the $j$ -th normal vector. In this scenario, we have three terms to consider. The function $\omega$ evaluates the distance between two agents in the network space; if this distance is not greater than $\rho _{\textit{loc}}$ , interaction occurs between the agents. The function $\zeta$ determines whether there is attraction or repulsion and assesses the respective intensity by evaluating the distance between the opinion means of the two agents. The last term defines both the intensity and the direction of movement in the network space. Both the last term and the attitude function play a role in determining the interaction intensity. However, their roles should not be confused: the attitude function pertains to the distance between opinions, while the third term depends on the network.

Figure 3. Local initial network interaction.

The agents interact if they are connected by a link. The magnitude and the sign of the connection range from $[{-}1,1]$ and are described by the legend on the right of the pictures. In this case, the attitude areas are given by the parameters $r_{\textit{f}}=0.25$ , $r_{\textit{a}}=0.34$ , $r_{\textit{r}}=0.36$ , $r_{\textit{l}}=0.65$ , i.e. the black function in Figure 1.

4.2. Simulation

Radicalisation

The term radicalisation denotes an agent’s inclination to anchor their opinion within a defined range of opinions. In our model, this phenomenon is observed when the opinion distribution increases within an interval and decreases sharply outside of it.

Remark 4.1. Note that this dynamics is not observable if one opts for a description that does not consider the opinion distribution but only a singular value.

We consider the attitude areas given by the following intervals: $r_{\textit{f}}=0.25$ , $r_{\textit{a}}=0.34$ , $r_{\textit{r}}=0.36$ , $r_{\textit{l}}=0.65$ . This setting coincides with the $\zeta$ function described by the black line in Figure 1.

Given the initial opinion distribution as in Figure 4, we can observe that the distribution evolves towards a more and more radicalised society as the radius increases. As evident from Figure 5, contrary to common intuition, increasing the interaction radius—thus engaging with a greater number of agents—leads to distributions becoming more concentrated around either positive or negative values. As the interaction radius expands, fewer agents exhibit distributions encompassing both positive and negative values.

Figure 4. Initial opinion distribution.

In this picture are represented the opinion distributions at initial time of the 40 agents considered for the simulation. Each distribution is described by a truncated Gaussian function, mean and variance of the Gaussian functions are independently uniform random distributed respectively in the intervals $[-0.7,0.7]$ and $[0.07, 0.15]$ .

Figure 5. Final opinion distribution.

We observe how the distributions are more and more concentrated either on the positive or negative side as the radius increases. Due to the diffusion, the distributions tend to flatten once that they are concentrated on one of the two sides.The sharp oscillations close to the extreme values are due to the low resolution of the numerical partition of $\Omega$ .

The same phenomenon is described by the histogram of the mean opinions distribution in Figure 6.

Figure 6. Initial and final mean opinions distribution.

In this figure, in blue the distribution of the mean opinions at , and in orange the mean opinions’ distribution at (which corresponds to the time showing a quasi-stable status of the simulation result). We observe that the final distribution tends to have two peaks, which means that the opinions of the population are more and more split into two opinion groups. However, they are also more close to the centre. This means that we observe a sort of fragmentation and radicalisation, but there is no polarisation.

Figure 7. Final network distribution.

The olive function has not been plotted because it describes an extreme behaviour, all the agents collapse very fast into a unique point.

Opinion Polarisation and Network Fragmentation

It is intriguing to examine the final network while varying the parameters of the attitude areas. We set the interaction radius as $\rho _{\textit{loc}}= 5$ , and we compare the first three functions depicted in Figure 1. In this scenario, we note that a more open-minded society corresponds to a less fragmented network, as illustrated in Figure 7.

While focusing solely on the operator $\mathbf{K}^{\mathcal{ij}}$ by setting $\mathbf{A}^{\mathcal{i}}=0$ , we observe the phenomenon of polarisation, i.e. the tendency of agents towards an opinion distribution centred on more extreme values, in our case $\pm 1$ .

It is interesting to observe how the connectivity of the population is the real driver of the polarisation, instead of the open-mindedness. We now compare the results of the simulation with fixed open-mindedness. We chose the most close-minded population, i.e. the one described by the blue function in Figure 1. Increasing the radius of the interaction, the opinion gets more and more polarised. In Figure 8, we notice that a more connected society tends to cluster into extreme opinions, while a less connected society keeps a more sparse distribution of opinions. This is a not expected behaviour, usually a large or global interaction is related to a higher consensus. In the work by Tucker et al. [Reference Tucker, Guess, Barberá, Vaccari, Siegel, Sanovich, Stukal and Nyhan53], it is emphasised how in the literature on opinion polarisation, the crucial role of social networks in accelerating and intensifying the polarisation of opinions on highly debated topics online is now well-established. This dynamic aligns with the outcomes of our simulations, given that the operator $\mathbf{K}^{\mathcal{ij}}$ describes the interaction process on social networks.

Figure 8. Polarisation while increasing the radius of interaction. Attitude function , $r_{\textit{a}}=0.20$ , $r_{\textit{r}}=0.30$ , $r_{\textit{l}}=0.40$ .

If we observe the evolution of the network, it seems that increasing the radius of interaction does not really affect the fragmentation of the population, but it plays a role in the opinion homogeneity of the network clusters. In Figure 9, at the same fixed time, a larger radius of interaction brings to a wider network space, and the groups of connected agents show a stronger homogeneity of the opinion.

Figure 9. Network fragmentation and opinion homogeneity.

Network while increasing the radius of interaction and keeping the same attitude function, i.e. $r_{\textit{f}}=0.15$ , $r_{\textit{a}}=0.20$ , $r_{\textit{r}}=0.30$ , $r_{\textit{l}}=0.40$ .

4.3. Conclusions, interpretations and possible follow-up

The goal of this model is to introduce the study of the processes describing the interaction on social networks and social media. We mainly focused on the role of the attitude areas and on that of the radius of interaction. The dynamics ruling the opinion formation of agents interacting on social platforms is different from the one described by models based on alignment, averaged consensus or Cucker-Smale with positive communication rate.

Recently, sociologists and philosophers described the epistemic processes of the hyperconnected society typical of the last two decades. The high amount of interactions and notions, together with their high frequency, modified the way how we create and reinforce our beliefs. Authors like Nguyen, see [Reference Thi Nguyen38], explain how the network and the opinion distance are the discriminant for different epistemic processes. In our model, we describe these two aspects through the definition of the attitude areas and through the dynamic of the network, which takes into account the distance on the network and the distance of the opinions.

Drawing on the literature, e.g. [Reference Begby5, Reference Benatti, de Arruda, Silva, Comin and da Fontoura Costa8, Reference Lim and Bentley34, Reference Tucker, Guess, Barberá, Vaccari, Siegel, Sanovich, Stukal and Nyhan53], we observe that describing interactions through attitude areas and Euclidean network suggests that the approach is heading in the right direction to capture the dynamics of polarisation, radicalisation and fragmentation that arise through interactions on social networks and social media. A particularly intriguing phenomenon to observe is the fragmentation of society into numerous ‘mono-opinion’ groups.

Future steps could involve assuming that the social strength $\sigma ^{\mathcal{i}}$ of each agent evolves over time based on connections. Additionally, in our case, we used a single function $\zeta$ for the entire population; however, it would be interesting to consider a different function for each agent, reflecting varying levels of open-mindedness. Numerous possible avenues exist, but caution should be exercised to avoid overexposing the problem to parameters and external choices. The authors’ intention is to continue in the direction of general models based on epistemological theories and interacting group theories. Certainly, an intriguing insight for future work comes from Pederneschi [Reference Pederneschi43] concerning the possibility of introducing biases to explain the dynamics of trust and distrust.