2.1. The kinetic model for the evolution of social media contacts

We aim now in deriving an evolutionary model for the formation of a network on a social media platform resorting to the contact law (7) derived previously. To that aim, let us observe that the variation of the density $h_\varepsilon (c,t)$ , also rescaled through the parameter $\varepsilon$ , obeys a linear Boltzmann-like equation [Reference Pareschi and Toscani45] whose weak form corresponds to

(8) \begin{equation} \frac{\mathrm{d}}{\mathrm{d}t}\int _{\mathbb{R}_+} \varphi (c) h_\varepsilon (c,t) \mathrm{d}c = \big \langle \int _{\mathbb{R}_+} \chi (c)\!\left ( \varphi (c_*) - \varphi (c) \right ) h_\varepsilon (c,t) \mathrm{d}c \big \rangle, \end{equation}

for all smooth test functions $\varphi (c)$ . These functions are the so-called observable quantities of the underlying random process. For example, taking $\varphi (c)=1$ leads equation (8) to an equation for the time evolution of the number of individuals in the network which can be easily inferred from equation (8) is constant in time. Instead, the case $\varphi (c)=c$ leads to an evolution equation for the average number of connections in the network which can be inferred, contrary to the previous case, not to be conserved in time. The positive function $\chi (c)$ measures the frequency of the interactions with $c$ followers and the expectation $\langle \cdot \rangle$ takes into account the presence of the random variable $\eta _\varepsilon$ . More in detail $\langle \cdot \rangle$ gives the expected value with respect to the random space in which $\eta _{\varepsilon }$ lives. In order to preserve the positivity of the connections in (1) and in the rescaled equation (6) based on the bounds of the value function (5), we require the random variable $\eta _\varepsilon$ to be uniformly distributed and to take values in

(9) \begin{equation} { \eta _\varepsilon \in \left [-\left |\frac{e^{-\varepsilon/\delta }+1}{(1-\mu )e^{-\varepsilon/\delta } + 1 + \mu }\right |,\left |\frac{e^{-\varepsilon/\delta }+1}{(1-\mu )e^{-\varepsilon/\delta } + 1 + \mu }\right |\right ].} \end{equation}

In the sequel, we consider collision kernels in the form [Reference Furioli, Pulvirenti, Terraneo and Toscani35]

(10) \begin{equation} \chi (c) = c^\beta \alpha, \end{equation}

for some multiplicative positive constants $\alpha \gt 0$ and for exponents $\beta \geq 0$ . In order to establish reasonable values for $\chi (c)$ , one can observe that, if $s\gt 0$ , the individual rate of growth ${\partial \Psi _\delta ^\varepsilon (s)}/{\partial s}$ vanishes as $\varepsilon \to 0$ (cfr. [Reference Dimarco and Toscani25]). So, to maintain a collective growth different than 0 for all values of the scaling parameter $\varepsilon$ , one suitable choice is to take

(11) \begin{equation} \alpha = \frac{1}{\kappa \varepsilon }, \end{equation}

corresponding to a frequency of interaction proportional to $1/\varepsilon$ with instead $\kappa$ an order 1 constant. Concerning the second parameter $\beta$ , we will consider two different situations. The first consists in taking $\beta \gt 0$ which implies that the frequency of interaction becomes greater if the number of connections is higher. This situation is encountered in social platforms where the type of exchanges are typically one to one and consequently one shares contents proportionally to the number of its connections. The second case we will consider is $\beta =0$ which corresponds to the situation in which interactions are independent of the number of connections meaning that the activity of each individual is independent of the other and the content sharing is independent of the size of the relative network. In the case $\beta \gt 0$ , a rational choice would consist in setting $\beta =\delta$ when $\delta$ is positive. Choices of $\beta \lt \delta$ will imply very high variations of the collective growth of the number of social media connections when $c$ is small compared to the case with $c$ large. However, it is not reasonable to expect that individuals with few connections can reach an influential position easily. Conversely, $\beta \gt \delta$ implies very small variations of collective growth for small values of $c$ , excluding consequently the opposite situation, that is, the possibility to increase the number of followers for individuals having few connections. Hence, choosing $\beta = \delta$ is a good compromise between the two scenarios. With the above-discussed choices, then equation (8) can be rewritten as:

(12) \begin{equation} \frac{\mathrm{d}}{\mathrm{d}t}\int _{\mathbb{R}_+} \varphi (c) h_\varepsilon (c,t) \mathrm{d}c = \frac{1}{\varepsilon \kappa } \big \langle \int _{\mathbb{R}_+}c^\delta \!\left ( \varphi (c_*) - \varphi (c) \right ) h_\varepsilon (c,t) \mathrm{d}c \big \rangle. \end{equation}

To get some insight into the time evolution of the model above, we use a standard procedure borrowed from the theory of gases and so we expand in Taylor series $\varphi (c')$ around $\varphi (c)$ supposing $\varphi (c)$ smooth enough. We have that

\begin{equation*} \langle c' -c\rangle = -\Psi _\delta ^\varepsilon (c/\bar c) c, \qquad \langle (c' - c)^2 \rangle = \left (\Psi _\delta ^\varepsilon (c/\bar c)\right )^2 c^2 + \varepsilon \nu ^2 c^2, \end{equation*}

One can also observe that for $0\lt \delta \leq 1$ , it holds that

(13) \begin{equation} \lim _{\varepsilon \to 0} \frac{1}{\varepsilon } \Psi _\delta ^\varepsilon \!\left ( \frac{c}{\bar c} \right ) = \frac{\mu }{2\delta } \!\left (1- \!\left ( \frac{\bar c}{c}\right )^\delta \right ). \end{equation}

This gives

\begin{equation*} \langle \varphi (c') - \varphi (c) \rangle = \varepsilon \!\left (-\varphi '(c) \frac {1}{\varepsilon }\Psi _\delta ^\varepsilon (c/\bar c) c + \frac {\nu ^2}{2} \varphi ''(c) c^2\right ) + R_\varepsilon (c), \end{equation*}

where $ R_\varepsilon (c)$ is a remainder of the Taylor expansion such that $R_\varepsilon (c) = o(\varepsilon )$ thanks to equation (13). Therefore, using for the interaction frequency (11), we get for the evolution of the observable $\varphi (c)$ :

\begin{equation*} \frac {\mathrm {d}}{\mathrm {d}t} \int _{\mathbb {R}_+} \varphi (c) h_\varepsilon (c,t) \mathrm {d}c = \int _{\mathbb {R}_+}\frac {c^\delta }{\kappa }\!\left ( -\varphi '(c) \frac {1}{\varepsilon }\Psi _\delta ^\varepsilon (c/\bar c) c + \frac {\nu ^2}{2} \varphi ''(c) c^2 \right )h_\varepsilon (c,t) \mathrm {d}c + \frac {1}{\kappa \varepsilon }\mathcal{R}_\varepsilon (c,t) \end{equation*}

where

\begin{equation*} \mathcal{R}_\varepsilon (c,t) = \int _{\mathbb {R}_+} R_\varepsilon (c)h_\varepsilon (c,t) \mathrm {d}c \end{equation*}

and by using equation (13), we get the following approximation:

(14) \begin{equation} \frac{\mathrm{d}}{\mathrm{d}t} \int _{\mathbb{R}_+} \varphi (c) h (c,t) \mathrm{d}c = \int _{\mathbb{R}_+}\!\left ( -\varphi '(c)\frac{\tilde \mu }{2\delta }\!\left (1- \!\left ( \frac{\bar c}{c} \right )^\delta \right )c^{1+\delta } + \frac{\tilde \nu ^2}{2} \varphi ''(c) c^{2+\delta } \right )h(c,t) \mathrm{d}c, \end{equation}

in which we have set $\tilde \mu = \mu/\kappa$ and $\tilde \nu ^2 = \nu ^2/\kappa$ . Under the additional hypothesis that the boundary terms produced by the integration by parts vanish, that is, a zero flux condition, equation (14) is the weak form of the following Fokker–Planck equation:

(15) \begin{equation} \frac{\partial h(c,t)}{\partial t} = \frac{\tilde \mu }{2\delta } \frac{\partial }{\partial c} \!\left ( \!\left (1- \!\left ( \frac{\bar c}{c} \right )^\delta \right )c^{1+\delta }h(c,t)\right ) + \frac{\tilde \nu ^2}{2} \frac{\partial ^2}{\partial c^{2}}\!\left (c^{2+\delta }h(c,t)\right ), \end{equation}

that describes the evolution of the density of contacts $c\in \mathbb{R}_+$ in the limit of the quasi-invariant variations of followers.

We are now interested in a steady state solution of equation (15). In fact, for the time of dynamics which we aim to study, that is, the one related to the formation of opinions through social interactions on online platforms, one can reasonably suppose that the connectivity network is stationary being the time at which opinions about a given subject are shaped much faster than the changes in the network. Thus, one can observe that the equilibrium solution of equation (15) is a function solving the first-order differential equation:

(16) \begin{equation} \frac{\tilde \nu ^2}{2} \frac{\mathrm{d}}{\mathrm{d}c}\!\left (c^{2+\delta }h(c)\right ) + \frac{\tilde \mu }{2\delta } \!\left (1- \!\left ( \frac{\bar c}{c} \right )^\delta \right )c^{1+\delta }h(c)= 0 \end{equation}

To find a solution to (16), we perform the change of variable $\rho (c) = c^{2+\delta } h(c)$ and by setting $\gamma = \tilde \mu/\tilde \nu ^2 = \mu/ \nu ^2$ , one can easily observe that the function $\rho (c)$ solves the following equation:

(17) \begin{equation} \frac{\mathrm{d}\rho (c)}{\mathrm{d}c}= -\frac{\gamma }{\delta }\!\left (\frac{1}{c} - \frac{\bar c^\delta }{c^{1+\delta }}\right ) \rho (c). \end{equation}

The unique solution of (17) is then given by:

(18) \begin{equation} h_\infty (c) = h_\infty (\bar c)\!\left (\frac{\bar c}{c}\right )^{2+\delta +\gamma/\delta }\text{exp} \left \{ -\frac{\gamma }{\delta ^2}\!\left (\!\left (\frac{\bar c}{c} \right )^\delta -1 \right )\right \}, \end{equation}

which is known as Amoroso distribution, corresponding to a particular class of the generalised Gamma distribution.

We focus now on a particular case, that is, the case in which $\delta \to 0$ and consequently $\beta =0$ , that is, the collision kernel is independent on the number of contacts. In this situation, the value function (7) degenerates to

(19) \begin{equation} \Psi ^\varepsilon _0(s) =- \mu \frac{s^{-\varepsilon } - 1}{(1-\mu )s^{-\varepsilon } + 1 + \mu }=\!\left (\frac{\mu }{1-\mu }\right ) \frac{s^{\varepsilon }-1}{\frac{1+\mu }{1-\mu }s^{\varepsilon } +1}, \end{equation}

where now the following limit holds true

\begin{equation*} \lim _{\varepsilon \to 0} \frac {1}{\varepsilon }\Psi ^\varepsilon _0\!\left ( \frac {c}{\bar c}\right ) =\frac {\mu/(1-\mu )}{\frac {1+\mu }{1-\mu }+1} \text {ln}\!\left ( \frac {c}{\bar c}\right )=\frac {\mu }{2} \text {ln}\!\left ( \frac {c}{\bar c}\right ), \end{equation*}

so that the evolution of the observables, as $\varepsilon \to 0$ and in a case of a collision kernel which does not depend on the number of connections, is well described by the following Fokker–Planck type of equation in weak form:

(20) \begin{equation} \frac{\mathrm{d}}{\mathrm{d}t} \int _{\mathbb{R}_+} \varphi (c) h (c,t) \mathrm{d}c = \int _{\mathbb{R}_+}\!\left ( -\varphi '(c)\frac{\tilde \mu }{2} \text{ln}\!\left ( \frac{c}{\bar c}\right )c + \frac{\tilde \nu ^2}{2} \varphi ''(c) c^2 \right )h(c,t) \mathrm{d}c, \end{equation}

where $\tilde \mu = \mu/\kappa$ and $\tilde \nu ^2 = \nu ^2/\kappa$ . Under again the zero flux hypothesis at the boundary, one gets a strong form of a Fokker–Planck type of equation:

(21) \begin{equation} \frac{\partial h(c,t)}{\partial t} = \frac{\tilde \mu }{2} \frac{\partial }{\partial c} \!\left ( c \text{ln} \!\left (\frac{c}{\bar c}\right )h(c,t)\right ) + \frac{\tilde \nu ^2}{2} \frac{\partial ^2}{\partial c^2}\!\left (c^2h(c,t)\right ), \end{equation}

which equilibrium state now reads

(22) \begin{equation} h_\infty (c) = \frac{1}{\sqrt{2 \pi \sigma } c}\text{exp} \left \{ -\frac{(\text{ln} c - \lambda )^2}{2\sigma }\right \}, \end{equation}

where $\gamma = \tilde \mu/\tilde \nu ^2$ and where we denoted $\sigma = 1/\gamma$ and $\lambda = \text{ln}\bar c - \sigma$ . Equation (22) is a log-normal probability distribution with mean and variance, respectively, given by:

\begin{equation*} m(h_\infty ) = \bar c e^{-\sigma/2}, \quad \text {Var}(h_\infty ) = \bar c^2(1-e^{-\sigma }). \end{equation*}

2.2. Contact distribution on Twitter and fitting

We discuss now the capability of our contact model to describe real networks. To that aim, we first collected data from Twitter in order to reconstruct a typical ensemble of connections. Successively, we estimated the parameters appearing in the general equilibrium state derived previously, (18) and (22), in such a way for our model to be as close as possible to real observable networks. The choice of using Twitter among other possible online platforms is motivated by the fact that one key characteristic of Twitter is to be more focused on staying informed and updated with respect, for instance, to Facebook which mainly aims at making friends. Thus, the first seemed more adapted to the study of opinion formation and modification with respect to the latter.

At the time we started this research, Twitter allowed academics to retrieve information about its users, given the IDs or usernames. Thanks to this possibility, the data have been collected through the user IDs of some of the most followed politicians from around the world, and successively by accessing their network, we retrieved the IDs of a million followers from each of their profiles. We then merged all these data, ignoring possible repetitions, and extracted one million profiles from such a set. We then gathered the number of followers of each participant in order to have a statistical representation of the distribution of connections over the platform. During this operation, we eliminated profiles with zero followers, assuming them as inactive. In Figure 2, we represent our dataset using a sample of $N_{\texttt{s}}=400$ accounts from the $N=10^6$ accounts extracted from Twitter. Sizes of the bubbles are proportional to the logarithm of agents’ contacts, where the information of the edges connection is reconstructed based on the statistical distribution of the connections. The fitting of the contact distribution arising from the data with the steady state solution (18)–(22) has been obtained by solving a nonlinear least-squares problem, through the Matlab function lsqcurvefit. The analysis of the best fit has been done using different choices for the contact distribution, namely log-normal, Amoroso, and Inverse Gamma distributions have been tested (each one corresponding to different values of the parameter $\delta$ appearing in the value function). In Figure 3 and in Table 1 we show the results of such a study obtained with different fitting functions. We can conclude that the best fit is obtained in the case in which the steady state distribution of the number of social media connections is distributed as a log-normal density, meaning that the parameter $\delta$ in Section 2 is equal to $0$ .

Figure 2. Representation of the social network using a sample of $N_{\texttt{s}}=400$ accounts from the $N=10^6$ dataset extracted from Twitter. Sizes of the bubbles are proportional to the logarithm of agents’ contacts, where edges are reconstructed based on the statistical distribution of the connections.

Figure 3. Comparison between the tails of the data distribution and the different possible equilibrium distributions of the Fokker–Planck models of Section 2.1.

We should remark that the log-normal distribution only has two parameters, while both the Amoroso and the Inverse Gamma have three parameters, and that the process of fitting is easier when fewer parameters have to be identified. Notice also that both the mean and the variance of the steady state depend on $\nu$ , the ratio between the variance of the random percentage of variation of followers, and $\mu$ , the maximal percentage allowed of possible variation of followers per interaction. Referring to (22), the values of the parameters resulting from the fitting process are $\lambda = 7.165 \times 10^{-1}$ and $\sigma = 8.882$ . In the sequel, we will then use a log-normal distribution to characterise the structure of the network.

3.1. The binary interaction

We start by associating the opinion of each agent with a variable $v \in I = [\!-\!1,1]$ . At the microscopic scale, we then suppose that binary interactions between individuals obey the following law:

(23) \begin{equation} \begin{split} v' & = v + \alpha P(v,v_*,c,c_*)(v_*-v) +\xi D(v,c),\\[5pt] v'_{\!\!\ast} & = v_* + \alpha P(v_*,v,c_*,c)(v-v_*)+\xi _* D(v_*,c_*), \end{split} \end{equation}

where $v$ and $v_*$ are the agents’ opinions before the interaction, while $v'$ and $v'_{\!\!\ast}$ are their opinions after interacting. In (23), the function $P$ can be seen as the compromise propensity of the agent. In other words, as a consequence of the exchange of relative information, the two interacting agents change their opinions, in a symmetric or more in general non-symmetric way, approaching one the opinion of the other and vice versa. The function $D$ instead is responsible for diffusion effects, and it models the unpredictable role played by the environment. It is indeed multiplied by the random variable $\xi$ , with $\langle \xi \rangle =0$ , and $\langle \xi ^2 \rangle =\sigma ^2$ . Let us observe that in the general case depicted in (23), the post-interaction opinions depend on the number of connections of both participants to the interaction. We will detail this dependence later. We remark for the moment that the restriction of the binary interaction to the case in which the values assumed by the couple $(v',v'_{\!\!\ast})$ are independent of the number of contacts $(c,c_*)$ can be considered quite classical and it is discussed for instance in [Reference Albi, Pareschi, Toscani and Zanella5].

Let now $f(v,c,t)$ be the density of agents which at time $t \gt 0$ are represented by their opinion $v$ and have connection $c$ . The time evolution of the distribution of opinions/connections $f(v,c,t)$ , consequence of the binary interactions of type (23) among individuals acting on a social platform, is obtained by resorting to kinetic collision-like models [Reference Pareschi and Toscani45, Reference Toscani49]. This reads in weak form as:

(24) \begin{equation} \begin{aligned} & \displaystyle \frac{d}{dt}\int _{I \times{\mathbb R}_+}f(v,c,t)\varphi (v,c)\,dv\,dc =\displaystyle \frac 12 \Big \langle \int _{I^2\times{\mathbb R}_+^2} \bigl (\varphi (v',c')+ \varphi (v'_{\!\!\ast},c'_{\!\!\ast})\cr &\qquad \qquad \qquad \qquad \displaystyle -\varphi (v,c)-\varphi (v_*,c_*) \bigr ) f(v_*,c_*,t)f(v,c,t) \,dv\,dv_*\,dc\,dc_* \Big \rangle. \end{aligned} \end{equation}

In (24), the post-interaction opinions $v'$ and $v'_{\!\!\ast}$ are given by (23), while the post-interaction connections are given by (1). The operator $\langle \cdot \rangle$ represents the mathematical expectation with respect to the random variables $\xi$ and $\eta$ . Let us observe that here we do not consider an interaction kernel depending on the number of contacts as done for instance in (8). This choice is driven by the fact that we aim to represent a specific situation when comparing the model to the experiments, namely the case in which the network is well described by a log-normal distribution as shown in Section 2.2. However, we stress that the extension to the case of kernels depending on $c$ is possible even if not discussed in the present work.

The opinion variable $v$ belongs to the bounded domain $[\!-\!1,1]$ , so it is important to only consider interactions that do not produce values outside of such domains. A sufficient condition to preserve the bounds is given by the following proposition.

Proposition 3.1. The binary interaction (23) preserves the bounds, that is, $v',v'_{\!\!\ast}\in [\!-\!1,1]$ if $v,v_*\in [\!-\!1,1]$ and if

(25) \begin{equation} 0 \lt P(v,v_*,c,c_*) \leq 1, \quad 0\lt \alpha \leq 1/2, \quad |\xi | \leq (1-\gamma ^*)d \end{equation}

where

(26) \begin{equation} \gamma ^* = \alpha \min _{\substack{v,v_* \in [\!-\!1,1], \\ c,c_*\gt 0}} P(v,v_*,c,c_*), \quad d = \min _{\substack{v\in [\!-\!1,1], \\ c\gt 0}} \left \{ \frac{1-|v|}{D(v,c)}, D(v,c) \neq 0\right \}. \end{equation}

Proof. Let us define $\gamma = \alpha P(v,v_*,c,c_*)$ . We first consider the case in which there is no diffusion, that is, $\xi = 0$ : we have that

\begin{equation*} |v'| = | v + \gamma (v_* - v)| \leq (1-\gamma )|v| + \gamma |v_*| \leq 1, \end{equation*}

since $|v|, |v_*| \leq 1$ and, under the hypothesis (25), we have that $0\lt \gamma \leq 1$ .

Let us now assume that $\xi \neq 0$ : we can write, using that $|v_*|\leq 1$ ,

\begin{equation*} |v'| = |v + \gamma (v_* - v) + \xi D(v,c)| \leq (1-\gamma )|v| + \gamma |v_*| + |\xi | D(v,c) \leq (1-\gamma )|v| + \gamma + |\xi | D(v,c). \end{equation*}

So, in order to have $|v'|\leq 1$ it is sufficient to require

\begin{equation*} |\xi | \leq \frac {(1-\gamma )(1-|v|)}{D(v,c)}, \end{equation*}

with $D(v,c)\neq 0$ , for all the possible values of $v$ and $c$ . Thus, defining $\gamma _*$ and $d$ as in (25), we get the result.

3.2. Fokker–Planck asymptotics

In order to model the fact that the formation of the opinions are due to a large number of interactions, each one producing a small change in the point of view of individuals up to the moment in which the final opinion is formed, we regularise the dynamics of the Boltzamnn-like equation (24) by relying on a quasi-invariant scaling. This computation permits us to get some insights on the behaviour of the model (24) by retrieving a Fokker–Planck equation for the asymptotic combined evolution of contacts and opinions. The quasi-invariant scaling is as follows:

(27) \begin{equation} \alpha \to \varepsilon \alpha,\qquad \sigma ^2 \to \varepsilon \sigma ^2, \end{equation}

for $\varepsilon \ll 1$ , similarly to the relation (6) for the sole contacts dynamics.

We assume that the scaled random variables $\eta _\varepsilon$ , $\xi _{\varepsilon }$ , and $\xi _{\varepsilon *}$ are independent with zero mean and bounded moments at least of order $n=3$ . We also assume that $\xi _{\varepsilon },\xi _{\varepsilon *}$ are identically distributed, and that the following holds

(28) \begin{equation} \langle \xi _{\varepsilon }\rangle =\langle \xi _{\varepsilon *}\rangle = 0,\quad \langle \xi _{\varepsilon }^2\rangle =\langle \xi _{\varepsilon *}^2\rangle = \varepsilon \sigma ^2,\quad \langle \xi _{\varepsilon }^3\rangle =\langle \xi _{\varepsilon *}^3\rangle = \varepsilon ^{3/2}\varrho, \end{equation}

and for $\eta _\varepsilon$ we have recalled the following:

(29) \begin{equation} \langle \eta _{\varepsilon }\rangle = 0,\quad \langle \eta _{\varepsilon }^2\rangle = \varepsilon \nu ^2,\quad \langle \eta _{\varepsilon }^3\rangle = \varepsilon ^{3/2}\varkappa. \end{equation}

with $\rho$ and $\varkappa$ two assigned constants. To ease the notation, we now rewrite the value function (4) multiplied by the number of contacts as follows:

(30) \begin{equation} c\Psi _\delta ^\varepsilon (c/\bar c) = \mu{L}_\varepsilon (c), \end{equation}

and the interaction function in (23) as:

\begin{equation*} E(v,v_*,c,c_*) = P(v,v_*,c,c_*)(v_*-v). \end{equation*}

Using the previous properties of the random quantities $\xi, \xi _*, \eta$ , the equations (1) for the contacts and (23) for the opinions we have the following:

(31) \begin{equation} \begin{aligned} & \langle c' -c\rangle = \langle -\mu{L}_\varepsilon (c) + \eta _\varepsilon c \rangle = -\mu{L}_\varepsilon (c), \\[5pt] &\langle v' -v \rangle = \langle \alpha P(v,v_*,c,c_*)(v_*-v) +\xi D(v,c) \rangle = \varepsilon \alpha E(v,v_*,c,c_*), \end{aligned} \end{equation}

and

(32) \begin{equation} \begin{aligned} & \langle (c' - c)^2 \rangle =\mu ^2{L}_\varepsilon (c)^2 + \varepsilon \nu ^2 c^2, \\[5pt] & \langle (v' -v)^2 \rangle = \varepsilon ^2\alpha ^2 E(v,v_*,c,c_*)^2 + \varepsilon \sigma ^2 D^2(v,c), \\[5pt] & \langle (c' -c)(v'-v) \rangle = -\varepsilon \alpha \mu{L}_\varepsilon (c) E(v,v_*,c,c_*) \end{aligned} \end{equation}

while the third-order terms are

(33) \begin{equation} \begin{aligned} & \langle (c' - c)^3 \rangle = -\mu ^3{L}_\varepsilon (c)^3 + \varepsilon ^{3/2}\varkappa c^3 - 3\varepsilon \mu \nu ^2 L_\varepsilon (c) c^2, \\[5pt] & \langle (v' -v)^3 \rangle = \varepsilon ^3\alpha ^3E(v,v_*,c,c_*)^3 + \varepsilon ^{3/2}\varrho D^3(v,c) +3\varepsilon ^2 \alpha \sigma ^2 E(v,v_*,c,c_*) D(v,c)^2, \\[5pt] & \langle (c' -c)^2(v'-v) \rangle = \varepsilon \alpha E(v,v_*,c,c_*)(\mu ^2 L_\varepsilon (c)^2 + \varepsilon \nu ^2 c^2), \\[5pt] & \langle (c' -c)(v'-v)^2 \rangle =-\mu L_\varepsilon (c)( \varepsilon ^2\alpha ^2E(v,v_*,c,c_*)^2 +\varepsilon \sigma ^2 D^2(v,c) ). \end{aligned} \end{equation}

By expanding the smooth function $\varphi (x^*,v^*)$ in Taylor series up to order two, we have

(34) \begin{equation} \begin{aligned} &\langle \varphi (v',c')-\varphi (v,c) \rangle = \cr &\quad \displaystyle \varepsilon \!\left ( \alpha E(v,v_*,c,c_*)\frac{\partial \varphi }{\partial v} - \mu \frac{L_\varepsilon (c)}{\varepsilon }\frac{\partial \varphi }{\partial c} + \frac 12 \sigma ^2 D(v,c)^2 \frac{\partial ^2 \varphi }{\partial v^2} + \frac 12 \nu ^2 c^2 \frac{\partial ^2 \varphi }{\partial c^2} \right ) \cr &+ \frac{\varepsilon ^2}2 \!\left ( \alpha ^2E(v,v_*,c,c_*)^2 \frac{\partial ^2 \varphi }{\partial v^2} + \mu ^2 \frac{L_\varepsilon (c)^2}{\varepsilon ^2} \frac{\partial ^2 \varphi }{\partial c^2} - \mu \alpha \frac{L_\varepsilon (c)}{\varepsilon } E(v,v_*,c,c_*)\frac{\partial ^2 \varphi }{\partial v \partial c} \right )\cr &\qquad \qquad +R_\varepsilon (v,v_*,c,c_*), \end{aligned} \end{equation}

where the remainder of the Taylor expansion $R_\varepsilon (v,v_*,c,c_*)$ is expressed as follows:

(35) \begin{equation} \begin{aligned} &R_\varepsilon (v,v_*,c,c_*) = \displaystyle \frac{\varepsilon ^2}{6} \frac{\partial ^3 \varphi }{\partial c^3}(\hat v, \hat c) \!\left ( -\mu ^3 \frac{{L}_\varepsilon (c)^3}{\varepsilon ^2} + \varepsilon ^{-1/2}\varkappa c^3 - 3 \mu \nu ^2 \frac{ L_\varepsilon (c)}{\varepsilon } c^2 \right ) \\[5pt] & \, + \frac{\varepsilon ^2}{6} \frac{\partial ^3 \varphi }{\partial v^3} (\hat v, \hat c) \!\left ( \varepsilon \alpha ^3E(v,v_*,c,c_*)^3 + \varepsilon ^{-1/2}\varrho D^3(v,c) +3 \alpha \sigma ^2 E(v,v_*,c,c_*) D(v,c)^2\right )\\[5pt] & \quad + \frac{\varepsilon ^2}{2}\frac{\partial ^3 \varphi }{\partial v\partial c^2}(\hat v, \hat c)\!\left ( \alpha E(v,v_*,c,c_*)\!\left (\mu ^2 \frac{L_\varepsilon (c)^2}{\varepsilon } + \nu ^2 c^2\right )\right ) \\[5pt] & \quad \quad - \frac{\varepsilon ^2}2 \frac{\partial ^3 \varphi }{\partial v^2\partial c}(\hat v, \hat c) \!\left (\mu \frac{L_\varepsilon (c)}{\varepsilon }\!\left ( \varepsilon \alpha ^2E(v,v_*,c,c_*)^2 +\sigma ^2 D^2(v,c) \right )\right ), \end{aligned} \end{equation}

for $\hat v = \theta _v v'+(1-\theta _v)v$ with $\theta _v \in [0,1]$ and $\hat c=\theta _c c'+(1-\theta _c)c$ with $\theta _c \in [0,1]$ .

Hence, scaling the time variable $\tau = \varepsilon t$ and using the expansion (34) in (24), the solution $f_\varepsilon$ satisfies the weak relation:

(36) \begin{equation} \begin{aligned} &\displaystyle \frac{d}{d\tau }\int _{I \times{\mathbb R}_+}f_\varepsilon (v,c,\tau )\varphi (v,c)\,dv\,dc =\displaystyle \int _{I\times{\mathbb R}_+} \!\left ( \mathcal{E}[f_\varepsilon ](v,c,\tau )\frac{\partial \varphi }{\partial v} - \frac{\mu }{\varepsilon } \Psi _\delta ^\varepsilon (c/\bar c) c \frac{\partial \varphi }{\partial c} \right. \\[5pt] &\qquad \quad \displaystyle \left .+ \frac 12 \sigma ^2 D^2(v,c) \frac{\partial ^2 \varphi }{\partial v^2} + \frac 12 \nu ^2 c^2 \frac{\partial ^2 \varphi }{\partial c^2} \right )f_\varepsilon (v,c,\tau )\,dv\,dc+ \mathcal{R}_\varepsilon (\varphi ), \end{aligned} \end{equation}

where we introduced the following notation for the non-local operator:

\begin{equation*} \mathcal{E}[f_\varepsilon ](v,c,\tau ) =\alpha \int _{I\times {\mathbb R}_+} E(v,v_*,c,c_*) f_\varepsilon (v_*,c_*,\tau )\,dv_*\,dc_*, \end{equation*}

and where the scaled reminder is

(37) \begin{equation} \begin{aligned} &\mathcal{R}_\varepsilon (\varphi ) = \frac{\varepsilon }2\int _{I^2\times{\mathbb R}_+^2} \!\left ( \alpha ^2E(v,v_*,c,c_*)^2 \frac{\partial ^2 \varphi }{\partial v^2} + \mu ^2 \frac{L_\varepsilon (c)^2}{\varepsilon ^2} \frac{\partial ^2 \varphi }{\partial c^2}\right .\cr &\left .\quad \quad - \mu \alpha \frac{L_\varepsilon (c)}{\varepsilon } E(v,v_*,c,c_*)\frac{\partial ^2 \varphi }{\partial v \partial c} \right )f_\varepsilon (v,c,\tau )f_\varepsilon (v_*,c_*,\tau ) \,dv\,dv_*\,dc\,dc_*\cr &\qquad \qquad + \frac{1}{\varepsilon }\int _{I^2\times{\mathbb R}_+^2} R_\varepsilon (v,v_*,c,c_*)f_\varepsilon (v,c,\tau )f_\varepsilon (v_*,c_*,\tau ) \,dv\,dv_*\,dc\,dc_*. \end{aligned} \end{equation}

For $\varepsilon \to 0$ we recall that from (30) and (35), we have the following:

\begin{equation*} \begin {aligned} &L_\varepsilon (c)\to 0,\qquad {L_\varepsilon (c)}/{\varepsilon } \to \Phi _\delta (c),\qquad {R_\varepsilon (v,v_*,c,c_*)}/{\varepsilon } \to 0, \end {aligned} \end{equation*}

where

(38) \begin{equation} \Phi _\delta (c)\;:\!=\; \begin{cases} \displaystyle \frac{\mu }{2\delta } \!\left (1- \!\left ( \frac{ c}{\bar c}\right )^{-\delta } \right )c,\quad 0\lt \delta \leq 1\\[5pt] \\[5pt] \displaystyle \frac{\mu }{2} \text{ln}\!\left ( \frac{c}{\bar c}\right )c,\,\qquad \delta \to 0. \end{cases} \end{equation}

Hence, in the limit $\varepsilon \to 0$ , the reminder (37) vanishes and the equation (36) collapses to the weak form of the following equation:

(39) \begin{equation} \displaystyle \frac{\partial f}{\partial \tau }=-\frac{\partial \!\left (\mathcal{E}[f](v,c,\tau ) f \right )}{\partial v} + \frac{\partial ( \Phi _\delta (c)f)}{\partial c}+ \frac 12 \sigma ^2 \frac{\partial ^2 (D^2(v,c)f)}{\partial v^2} + \frac 12 \nu ^2 \frac{\partial ^2( c^2f)}{\partial c^2}. \end{equation}

Equation (39) is the model we will use in the sequel to describe the time evolution of opinion formation over a social network. In particular, in the final part of Section 4 focusing on a specific platform, namely Twitter, we will fit the parameter in our model with experimental data with the scope of describing a realistic phenomenon.

3.3. On the steady state solution for the opinion distribution

In the general case, the steady state of equation (39) is not known. However, it is possible to compute an explicit formula for the asymptotic solution under some particular assumptions. Let us assume that $D(v,c) = 1 - v^2$ and $P(v,v_*,c,c_*)=1$ so that

\begin{equation*}\begin {array}{rcl} \mathcal{E}[f](v,c,\tau ) &=& \displaystyle \alpha \!\left (\int _{I\times {\mathbb R}_+} v_* f(v_*,c_*,\tau )\,dv_*\,dc_* - v \right ) =\alpha \!\left ( m_v(\tau ) - v \right ). \end {array} \end{equation*}

In this situation, one can look to solutions of type $f(v,c,t) = g(v,t)h(c,t)$ leading to asymptotic of the form $f_\infty (v,c)=g_\infty (v) h_\infty (c)$ , where $h_\infty$ is the asymptotic state of the social contact distribution derived in Section 2. Under the above hypothesis, the Fokker–Planck equation (39) can be rewritten as:

(40) \begin{equation} \begin{array}{rcl} \displaystyle \frac{\partial g}{\partial \tau }h + \displaystyle \frac{\partial h}{\partial \tau }g &=&-\displaystyle \alpha \frac{\partial ( (m_v(\tau ) - v) g)}{\partial v}h + \frac{\mu }{2} \frac{\partial \!\left (c \text{ln} \!\left (\frac{c}{\bar c}\right )h\right )}{\partial c}g \\[5pt] & & \displaystyle + \frac 12 \sigma ^2 \frac{\partial ^2 ((1-v^2)^2g)}{\partial v^2} h + \frac 12 \nu ^2 \frac{\partial ^2( c^2h)}{\partial c^2}g, \end{array} \end{equation}

leading to

(41) \begin{multline} \!\left (\frac{\partial g}{\partial \tau } + \alpha \frac{\partial ( (m_v(\tau ) - v) g)}{\partial v}- \frac 12 \sigma ^2 \frac{\partial ^2 ((1-v^2)^2g)}{\partial v^2} \right )h \\[5pt] +\displaystyle \!\left (\frac{\partial h}{\partial \tau } - \frac{\mu }{2} \frac{\partial \!\left (c \text{ln} \!\left (\frac{c}{\bar c}\right )h\right )}{\partial c} -\frac 12 \nu ^2 \frac{\partial ^2( c^2h)}{\partial c^2} \right )g= 0. \end{multline}

Non-trivial solution for equation (41) are retrieved for

(42) \begin{equation} \frac{\partial g}{\partial \tau } = - \alpha \frac{\partial ( (m_v(\tau ) - v) g)}{\partial v} + \frac 12 \sigma ^2 \frac{\partial ^2 ((1-v^2)^2g)}{\partial v^2} \end{equation}

and

(43) \begin{equation} \frac{\partial h}{\partial \tau } = \frac{\mu }{2} \frac{\partial \!\left (c \text{ln} \!\left (\frac{c}{\bar c}\right )h\right )}{\partial c} + \frac 12 \nu ^2 \frac{\partial ^2( c^2h)}{\partial c^2}. \end{equation}

Thus stationary solutions for (39) are of the form $f_\infty (v,c) = g_\infty (v)h_\infty (c)$ as claimed before, where for (43) we obtain the log-normal distribution (22), while in order to compute the stationary solution to (42), one has to solve

(44) \begin{equation} \frac{\mathrm{d}((1-v^2)^2g_\infty )}{\mathrm{d}v} = \frac{2\alpha }{\sigma ^2}( (\bar m_v - v) g_\infty ),\qquad \bar m_v = \int _{I\times{\mathbb R}_+} v_* g_\infty (v_*)\,dv_*. \end{equation}

The solution to (44) is given by:

\begin{equation*} g_\infty (v) = K_\infty (1+v)^{-2+\alpha \bar m_v/2\sigma ^2}(1-v)^{-2-\alpha \bar m_v/2\sigma ^2}\text {exp}\left \{ -\frac {\alpha (1-\bar m_v v)}{\sigma ^2(1-v^2)}\right \}, \end{equation*}

where $K_\infty$ is a normalisation constant, such that the total mass of $g_\infty$ is equal to $1$ . Figure 4 shows the comparison between the analytical profile of $g_\infty (v)$ obtained in the case $\alpha = 0.1$ , $\sigma ^2 = 0.1$ and $\alpha = 0.25$ , $\sigma ^2 = 0.05$ , $\bar c = 1$ , $\mu = 0.1$ , $\nu ^2 = 0.0125$ and the numerical simulations of equation (24) through a Monte Carlo method which details are outlined in the Appendix A. In the simulation, we choose the scaling parameter $\varepsilon = 0.01$ in order to retrieve the Fokker–Planck asymptotic from the Boltzmann-type equation (24).

Figure 4. Profiles of the steady state solution $g_\infty (v)$ and its numerical approximation in the case of $\sigma ^2/\alpha = 1$ (left) and $\sigma ^2/\alpha = 0.2$ (right), both with scaling parameter $\varepsilon = 0.01$ .

4.1. Qualitative behaviour

We start by discussing some qualitative behaviours of the model presented through the use of numerical simulations. We first consider a bounded confidence model, in which the propensity to consensus is influenced by the number of connections. Furthermore, in the second test, we consider a confidence bound depending also on the number of contacts, where we compare homogeneous with heterogeneous cases. In a final test, we study a case in which a part of the population acting on a social network is composed of almost inflexible individuals.

4.1.1. Test 1: Bounded confidence model

In this first test, we build up a bounded confidence model [Reference Deffuant, Neau, Amblard and Weisbuch21, Reference Hegselmann and Krause39] by performing the following choices:

(46) \begin{equation} H(v,v_*,c,c_*)=\chi _{\{|v-v_*|\lt \Delta \}}(v_*), \qquad K(c,c_*)=\frac{c_*^2}{c^2+c_*^2}, \end{equation}

where $\chi (\!\cdot\!)$ is the indicator function and $\Delta$ is a positive constant. The diffusion part is weighted by $D(v,c,c_*) = 1-v^2$ . In the chosen setting, the interactions take place only if two individuals have sufficiently close opinions. Moreover, we give a higher relevance, in driving the opinions process, to the agents having more connections, that is, the influencers. Instead, individuals with few followers are less likely to be able to change the point of view of the other participants while prone to change their own opinion. In order to stress the importance of the presence of contacts and their relevance in modifying the evolution of the joint density $f(v,c,t)$ , we start from two different initial data: in the first simulation, the initial data is given by:

\begin{equation*} f_0(v,c) = \frac {1}{2}h_\infty (c), \end{equation*}

meaning that the opinion is uniformly distributed in the interval $I=[\!-\!1,1]$ , and the contacts are at the equilibrium (22) with parameters $\lambda = 5$ and $\sigma = 1.56\cdot 10^{-2}$ . In Figure 5, the initial data is shown in the first image, followed by the images of the resulting density $f(v,c,t)$ for different times, namely $t = 4, 8, 12, 16, 20$ . We sample $N_s=10^5$ agents to simulate model (24), with scaling parameter $\varepsilon = 0.01$ . In Figure 5, we clearly see the segmentation of the opinion which is due to the presence of a bounded confidence interaction with $\Delta = 0.55$ , but over time the two clusters merge again, the consensus is reached and we notice the formation of a single cluster in $v=0$ .

Figure 5. Test $1$ , $\sigma ^2/\alpha = 0.005$ . The pictures show the time evolution of the distribution function $f(v,c,t)$ for $t=0,4,8,12,16,20$ for a homogeneous distribution of the number of connections with respect to opinions. After the emergence of two clusters, the agents reach consensus at the final time.

In the second simulation, we suppose that contacts are still at the equilibrium (22) with the same parameters $\lambda$ and $\sigma$ as in the previous simulation, but the opinions are now distributed so that agents with a low number of connections have an opinion closer to $-1$ , while agents with a higher number of contacts have an opinion closer to $+1$ . In Figure 6, the initial data is shown in the first image, followed by the images of the resulting density $f(v,c,t)$ for $t = 4, 8, 12, 16, 20$ . We see the emergence of two clusters, but the symmetry of the previous case is lost: agents with a lower number of contacts are influenced by agents with a higher bound of contacts and behave like followers, whereas agents with a higher bound of contacts are less influenced by lower-contact agents. We stress that the dynamics observed in these two cases are different from the standard dynamics obtained with a bounded confidence opinion model where connections do not play a role. In fact, in this case, the steady state solution is represented by a bimodal distribution similar to the one obtained in Figure 6 for $t=8$ .

Figure 6. Test $1$ , $\sigma ^2/\alpha = 0.005$ . The pictures show the time evolution of the distribution function $f(v,c,t)$ for $t=0,4,8,12,16,20$ in the case of a non-homogeneous distribution of the number of connections with respect to opinions. After the emergence of two clusters, the agents reach consensus at the final time in the positive opinion region.

4.1.2. Test 2: Heterogeneous confidence bound

In this second test, we modify the previous experimental setting introducing confidence bound $\Delta \equiv \Delta (c,c_*)$ in (46) as a function of the contact numbers of the interacting agents. We assume that the confidence bound is above the threshold $1/2$ if the number of contacts of the interacting agent is larger, that is, $c_*\gt c$ , on the other hand, for a lower number of contacts $c_*\lt c$ the confidence bound becomes lower than $1/2$ . To model this behaviour, we consider the following function:

(47) \begin{equation} \Delta (c,c_*) = \frac{c_*}{c+c_*}. \end{equation}

We assume as initial datum $f^0(v,c)$ the uniform distribution on $ [\!-\!1,1] \times [0,1]$ . Simulations of the opinion dynamics (24) is performed using $N_s= 10^5$ agents and scaling parameter $\varepsilon = 0.01$ . The evolution of social contacts (1), which initially is not at the stationary state, is performed with parameters $\mu = 10^{-1}$ and $\nu ^2 = 0.0125$ .

We compare the case with heterogeneous confidence bound as in (47), with the case with homogeneous confidence bound equal to $\Delta (c,c_*)=1/2$ . In Figure 7, we report the resulting densities $f(v,c,t)$ for $t = 2$ on the left, $t=4$ in the centre, and $t=8$ on the right, where the rows are relative to the homogeneous and to the heterogeneous case, respectively, for top row and bottom row. In the top row, we observe the emergence of two clusters one centred around $-0.5$ and the second one centred around $0.5$ , independent of the number of contacts. The bottom row reports the case with heterogeneous bound $\Delta (c,c_*)$ , where we observe that agents with a high level of connections tend to maintain their opinions in time due to the lower values of $\Delta (c,c_*)$ , instead agents with lower connections reach consensus around the central opinion $v=0$ , since $\Delta (c,c_*)$ is larger.

Figure 7. Test $2$ , $\sigma ^2/\alpha = 0.005$ . The pictures show the distribution $f(v,c,t)$ for $t=4$ (left), $t=6$ (centre), and $t=8$ (right). Top row: constant bound of contacts ( $\Delta (c,c_*)=0.5)$ ), two main clusters emerge at any bound of contacts. Bottom row: heterogeneous confidence bound ( $\Delta (c,c_*)=$ in (47)), consensus is reached for agents with a low number of contacts, whereas for higher bound of contacts two main clusters emerge.

4.1.3. Test 3: Sznajd-type dynamics

We start now with a distribution of opinions given by the sum of two Gaussian distributions centred at different locations in $[\!-\!1,1]$ mimicking a clustering of opinion with respect to a given subject. The initial condition is

\begin{equation*} f_0(v,c) = K_0\begin {cases} \frac {1}{30} \text {exp} \{-(v + \frac {3}{4})^2/(2\sigma _1^2)\},\quad & \text {if } 100 \leq c \leq 130 \\[5pt] \frac {1}{30} \text {exp} \{-(v - \frac {3}{4})^2/(2\sigma _2^2)\},& \text {if } 170 \leq c \leq 200 \\[5pt] 0 &\text {otherwise,}\end {cases} \end{equation*}

with $K_0$ positive constant such that $f_0(v,c)$ has total mass equal to $1$ . The interaction kernels are such that

\begin{equation*}H(v,v_*,c,c_*) = (1-v^2), \qquad K(c,c_*) = \frac {c_*^2}{(c+c_*)^2},\end{equation*}

and with a diffusion term proportional to $D(v,c,c_*) = 1-v^2$ . The interaction function is chosen according to an approximation of Sznajd dynamics [Reference Sznajd-Weron and Sznajd48, Reference Toscani49] in such a way that individuals may interact with everyone on the social network, while alignment towards a given opinion is more frequent for people having weak opinions and less probable for individuals having strong believes (positive or negative). The role played by the number of connections in the function $K(c,_*)$ is similar to the previous cases even if now its impact is more important: agents with a higher number of social media contacts tend to only slightly modify their opinion over time, while the ones with a low number of contacts are more influenced and tend to align their opinion with one of the most popular persons. The simulation uses $N_s = 10^5$ agents in the time interval $[0,T] = [0,6]$ . Figure 8 shows the initial condition on the left and the resulting density $f(v,c,t)$ for $t=3$ and $t=6$ , respectively, on the centre and the right, using $\sigma _1^2=\sigma _2^2=0.005$ and a scaling parameter $\varepsilon = 0.01$ . For the evolution of the social contact dynamics (1), we account for the following parameters $\mu = 10^{-2}$ and $\nu = 3.54\cdot 10^{-2}$ . We observe that agents with a lower number of contacts are strongly influenced by agents with a higher number of contacts and behave like followers, whereas agents with higher bound of contacts are mildly influenced by lower-contact agents. The result of such dynamics is that individuals with a negative opinion at the beginning are driven towards a positive one over time, while people with a large number of contacts only slightly move towards the centre without really modifying their believes.

Figure 8. Test $2$ , $\sigma ^2/\alpha = 0.005$ . The pictures show the time evolution of the distribution function $f(v,c,t)$ for $t=0$ (left), $t=3$ (centre), and $t=6$ (right). Agents with lower bound of connections are strongly influenced by agents with a large number of connections.

4.2. Quantitative analysis

In this last part, we use our model to quantitatively represent the opinion dynamics for what concerns some extrapolated real data taken from Twitter. To that aim, we start by describing the way in which the data are pre-processed through a so-called SA with the scope of obtaining a set of information that can be effectively used in comparison with the model outcomes.

4.2.2. Test 4: Trump re-admission on Twitter

For this simulation, we used the Application Programming Interfaces (API) of Twitter to obtain the content of a certain number of tweets on a given topic. More specifically, we used the words ‘Donald Trump’ and some related hashtags to select tweets written (in English) some days after the re-admittance of Donald Trump on Twitter on the 20 November 2022, after an almost two-year-long ban from the platform. We then employed VADER to analyse the texts of the tweets and we obtained a rating between $-1$ and $1$ for each tweet, which we considered to be the agents’ opinions on the subject. To outline the polarised situation we remove tweets, which have scored exactly ‘0’ through VADER analysis.

To perform the model calibration, we introduce a class of interacting functions, where we make explicit the dependency with respect to a new set of parameters $\theta \in \Theta \subseteq \mathbb{R}^4_+$ as follows:

(48a) \begin{equation} P(v,v_*,c,c_*;\;\theta ) = H(v,v_*,c,c_*;\;\theta )K(c,c_*;\;\theta ), \end{equation}

with

(48b) \begin{equation} H(v,v_*,c,c_*;\;\theta ) = \chi (|v-v_*|\lt \Delta (c,c_*;\;\theta )), \end{equation}

where

(48c) \begin{equation} \Delta (c,c_*;\;\theta ) = \theta _1\!\left (\frac{ \log\!(1+c_*)}{\log\!(1+c_*)+\log\!(1+c)}\right )^{\theta _2}, \end{equation}

and

(48d) \begin{equation} K(c,c_*;\;\theta ) = \left (\frac{ \log\!(1+c_*)}{\log\!(1+c_*)+\log\!(1+c)}\right )^{\theta _3}, \end{equation}

while the diffusion is weighted by:

(48e) \begin{equation} D(v,c;\;\theta ) = \theta _4\sqrt{1-|v|^2}. \end{equation}

The initial data is well prepared, assuming that the distribution of connections is at the stationary state (22) with parameters estimated in 1, and the joint distribution of opinions and contacts is such that

\begin{equation*} f_0(v,c) = \begin {cases} \frac {h_\infty (c)}{2}, \quad &\mbox {if } 50 \lt c \leq 7500\\[5pt] \frac {h_\infty (c)}{0.2\sqrt {2 \pi }}e^{-\frac {1}{2}\!\left (\frac {v + 0.5}{0.2} \right )^2}, &\mbox {if } c \gt 7500. \end {cases} \end{equation*}

Finally, we identify the distribution obtained from the data with $\hat g(v,t)$ , while $g(v,t)$ is the one obtained simulating the virtual dynamics of particles. Then, we search the optimal value of the parameters $(\theta _1,\theta _2,\theta _3,\theta _4)\in \Theta$ , where $ \Theta =[0.5,1.5]\times [0.2,0.7]\times [1.5,2.5]\times [8,11],$ by minimising the $\ell _1$ distance at final time $T= \{20\}$ of the marginal distribution of the simulated opinions $g(v,T)$ and the one reconstructed from data $\hat g(v,T)$ .

The minimisation is performed using patternsearch() routine in matlab with initial guess $\theta ^{(0)}=(1.1,0.4, 2.0, 10)$ and reaching convergence after $k =60$ iterations, where the estimated parameter is $\theta ^{(k)}=(0.6313,0.2047, 2.3125; 9.7510)$ . Here, we minimise the discrepancy measure as the $\ell _1$ distance between the marginal distribution $g(\cdot,T|\theta ^{(k)})$ and the marginal distribution reconstructed from data $\hat g(\cdot,T)$ , the final value is $\mathcal{D}_1(g(\cdot,T|\theta ^{(k)}),\hat g(\cdot,T)) = 6.376\times 10^{-2}$ . The minimisation procedure is further detailed in the Appendix.

In Figure 9, we depict the comparison between our results and the data: on the left, we have the marginal of the opinions at time $t=20$ and the data, while on the right we have $f(v,c,t)$ (represented as $\log\!(f(v,c,t)+0.025)$ ) compared with the actual data extracted from Twitter (represented by the orange dots). We can claim that the model is capable of fitting the results obtained from a SA with good accuracy.

Figure 9. Test $4$ : marginal distribution of opinions at the final time (left) and the comparison between the reconstructed density of opinions and contacts and the real dataset.

4.2.3. Test 5: Climate change trends on Twitter

For this last simulation, we used the information coming from a pre-existing dataset ([Reference Littman and Wrubel43]) containing the IDs of tweets discussing the climate change. Such data were collected from Twitter’s API between 21 September 2017 and 17 May 2019, using as track parameters some keywords related to the subject, such as ‘climate change’, ‘global warming’ and hashtags like ‘ $\#$ climatechangeisreal’, ‘ $\#$ climatechangeisfalse’ or ‘ $\#$ globalwarminghoax’. We used VADER to perform the SA on the tweets collected on 13, 14, 15, 16 and 18 August 2018, and we remove from the dataset tweets with SA score exactly equal to “ $0$ ”, to reconstruct the data trend starting from a fictitious initial distribution of the opinions.

Similarly to the previous test 4.2.2, we consider the interaction functions reported in (48), and with well-prepared initial data, where we assume that the marginal distribution of the contacts is at the stationary state (22), and that the joint initial density of opinions and contacts is given by:

\begin{equation*} f_0(v,c) = \begin {cases}\frac {h_\infty (c)}{154 \sqrt {2\pi }}\!\left (55\sqrt {2\pi } + 200e^{-\frac {1}{2}\!\left (\frac {v + 0.35}{0.15} \right )^2} + 28e^{-\frac {1}{2}\!\left (\frac {v - 0.25}{0.5} \right )^2}\right ), \quad &\mbox {if } c \lt 40\\[5pt] \frac {h_\infty (c)}{6\sqrt {2\pi }}\!\left (250e^{-\frac {1}{2}\!\left (\frac {v + 0.8}{0.004} \right )^2} + 20e^{-\frac {1}{2}\!\left (\frac {v + 0.3}{0.2} \right )^2} + 5e^{-\frac {1}{2}\!\left (\frac {v - 0.3}{0.2} \right )^2} \right ), &\mbox {if } 40 \leq c \leq 400, \\[5pt] \frac {5 h_\infty (c)}{\sqrt {2 \pi }}e^{-\frac {1}{2}\!\left (\frac {v - 0.4}{0.2} \right )^2}, &\mbox {if } c \gt 400. \end {cases} \end{equation*}

We assume that the data are referred to the following numerical time $t_m = \{1, 2, 3, 4, 11\}$ and we denote by $\hat g(v,t)$ the empirical marginal distribution of the opinions obtained from Twitter, meaning that $\hat g(v,t_1)$ refers to 13 August, $\hat g(v,t_2)$ refers to 14 August and so on.

Hence, to obtain the optimal value of the parameters $(\theta _1,\theta _2,\theta _3,\theta _4)\in \Theta$ in the admissible space:

\begin{equation*} \Theta = [0.5, 1]\times [0.1, 1.5]\times [0.01, 2]\times [0,0.05]. \end{equation*}

we use the fmincon() matlab routine to minimise the discrepancy between data-reconstructed and simulated marginal distribution of opinions computing the sum over $t_m, m = 1,\ldots,5,$ of the $1-$ Wasserstein distances $\mathcal{W}^1_1(g(\cdot,t_m|\theta ),\hat g(\cdot,t_m))$ as follows:

\begin{equation*}\mathcal{D}_1(g(\cdot |\theta ),\hat g(\!\cdot\!)) = \frac {1}{M}\sum _{m=1}^M\mathcal{W}^1_1(g(\cdot,t_i|\theta ),\hat g(\cdot,t_i)), \qquad M=5.\end{equation*}

The chosen method reaches convergence after $k=11$ iterations with initial guess $\theta ^{(0)}=(0.75,1.25,0.65,0.03)$ and the estimated values of the parameters are $\theta ^{(k)} = (0.7432,1.0735,0.9295,0.0306)$ , resulting in a discrepancy of value $\mathcal{D}_1(g(\cdot |\theta ),\hat g(\!\cdot\!)) =4.893\times 10^{-1}$ . We outline that the choice of a good initial guess, in this case, is of paramount importance.

In Figure 10, we depict the evolution of the marginal distribution of opinions compared to the data, while Figure 11 shows the initial and terminal density of opinions and contacts. Again we can claim that we are able to follow the main trend of the opinion during the time even if compared to the previous situation of Test 4 in which the fitting was only about a given, supposed steady, state, here the differences are quite large in some time frameworks.

Figure 10. Test 5: comparison between the marginal distribution of the opinions reconstructed using the presented model and the data at each time step $t \in \{1,2,3,4,11\}$ corresponding to data relative to the 13, 14, 15, 16, and 18 August 2018.

Figure 11. Test $5$ : initial (left) and final (right) joint density of opinions and contacts (the images show $\log\!(f(v,c,t) + 0.025)$ ).