Recent Advances in Opinion Modeling: Control and Social Influence

Abstract

We survey some recent developments on the mathematical modeling of opinion dynamics. After an introduction on opinion modeling through interacting multi-agent systems described by partial differential equations of kinetic type, we focus our attention on two major advancements: optimal control of opinion formation and influence of additional social aspects, like conviction and number of connections in social networks, which modify the agents’ role in the opinion exchange process.

Appendix: Numerical Simulation Methods

In this short appendix we sketch briefly some particular numerical technique used to produce the various simulation results presented in the manuscript. We omit the description of the Monte Carlo simulation approach for the Boltzmann equation describing the opinion exchange dynamics addressing the interested reader to [89]. For the development of Monte Carlo methods that works in the Fokker-Planck regime we refer to [5].

We first summarize the Monte Carlo approach used to deal with the evolution of the social network and then the steady state preserving finite-difference approach used for the mean-field models. More details can be found in [10].

1.1 Monte Carlo Algorithm for the Evolution of the Network

The evolution of the network is given by

$$\begin{aligned}{\left\{ \begin{array}{ll} \dfrac{d}{dt} f(w,c,t) + \mathscr {N}[f(w,c,t)]=0, \\ f(w,c,0)=f_0(w,c). \end{array}\right. }\end{aligned}$$

Let $f^n = f(w,c,t^n)$ the empirical density function for the density of agents at time $t^n$ with opinion w and connections c. For any given opinion w we approximate the solution of the above problem at time $t^{n+1}$ by

$$\begin{aligned} \begin{aligned} f^{n+1}(w,c) =&\left( 1-\varDelta t\dfrac{{{V_r}}(c+\beta )}{\gamma ^n+\beta }-\varDelta t\dfrac{ {{V_a}}(c+\alpha )}{\gamma ^n+\alpha }\right) f^n(w,c)\\&+\varDelta t \dfrac{{{V_r}}(c+\beta )}{\gamma ^n+\beta }f^n(w,c-1)+\varDelta t\dfrac{ {{V_a}}(c+\alpha )}{\gamma ^n+\alpha }f^n(w,c+1), \end{aligned}\end{aligned}$$

with boundary conditions

$$\begin{aligned}\begin{aligned} f^n(w,0)&= \left( 1-\varDelta t\dfrac{ {{V_a}}(c+\alpha )}{\gamma ^n+\alpha }\right) f^n(w,0)+\varDelta t\dfrac{ {{V_a}}(c+\alpha )}{\gamma ^n+\alpha }f^n(w,1), \\ f^n(w,c_{\text {max}})&= \left( 1-\varDelta t\dfrac{{{V_r}}(c+\beta )}{\gamma ^n+\beta }\right) f^n(w,c_{\text {max}})+\varDelta t \dfrac{{{V_r}}(c_{\text {max}}+\beta )}{\gamma ^n+\beta }f^n(w,c_{\text {max}}-1), \end{aligned}\end{aligned}$$

and temporal discretization such that

$$\begin{aligned} \varDelta t\le \min \left\{ \dfrac{\gamma ^n+\beta }{{{V_r}}(c_{\text {max}}+\beta )},\dfrac{\gamma ^n+\alpha }{{{V_a}}(c_{\text {max}}+\alpha )} \right\} . \end{aligned}$$

(A3)

The algorithm to simulate the above equation reads as follows

Algorithm 1

 

• 1. Sample $(w^0_i,c^0_i)$, with $i=1,\ldots ,N_s$, from the distribution $f^{0}(w,c)$.

• 2. for $n=0$ to $n_{tot}-1$

  • a. Compute $\gamma ^n =\frac{1}{N_s} \sum _{j=1}^{N_s}c^n_j$;

  • b. Fix $\varDelta t$ such that condition (A3) is satisfied.

  • c. for $k=1$ to $N_s$

    • i. Compute the following probabilities rates

      $$\begin{aligned} p_k^{(a)} =\frac{\varDelta t V_a(c_k^n+\alpha )}{\gamma ^n+\alpha },\qquad p_k^{(r)} = \frac{\varDelta t V_r(c_k^n+\beta )}{\gamma ^n+\beta }, \end{aligned}$$

    • ii. Set $c^*_k =c_k^n$.

    • iii. if    $0 \le c_k^*\le c_{\text {max}}-1$,

         with probability $p_k^{(a)}$ add a connection: $c_k^* = c_k^* +1$;

    • iv. if    $1 \le c_k^*\le c_{\text {max}}$,

         with probability $p_k^{(r)}$ remove a connection: $c_k^* = c_k^* -1$;

    • end for

  • d. set $c^{n+1}_i = c^*_i$, for all $i= 1,\ldots , N_s$.

  • end for

1.2 Chang-Cooper Type Numerical Schemes

In the domain $(w,c)\in I\times {\mathscr {C}}$ we consider the Fokker-Planck system

$$\begin{aligned} \dfrac{\partial }{\partial t}f(w,c,t)+\mathscr {N}\left[ f(w,c,t)\right] =\dfrac{\partial }{\partial w}\mathscr {F}[f], \end{aligned}$$

(A4)

with zero flux boundary condition on w, initial data $f(w,c,0)=f_0(w,c)$ and

$$\begin{aligned} \mathscr {F}[f] = \left( \mathscr {P}[f] + \sigma ^2 D'(w,c)D(w,c)\right) f(w,c,t)+\dfrac{\sigma ^2}{2}D(w,c)^2 \dfrac{\partial }{\partial w}f(w,c,t), \end{aligned}$$

where $\mathscr {P}[f]$ is given by (101). Let us introduce a uniform grid $w_{i}=-1+i\varDelta w$, $i=0,\ldots ,N$ with $\varDelta w = 2/N$, we denote by $w_{i \pm 1/2}=w_i \pm \varDelta w/2$ and define

$$ f_{i}(c,t)=\frac{1}{\varDelta w}\int _{w_{i+1/2}}^{w_{i-1/2}} f(w,c,t)\,dw. $$

Integrating equation (A4) yields

$$\begin{aligned}\begin{aligned} \dfrac{\partial }{\partial t}f_{i}(c,t)+\mathscr {N}\left[ f_{i}(c,t)\right] =\frac{\mathscr {F}_{i+1/2}[f]-\mathscr {F}_{i-1/2} [f]}{\varDelta w}, \end{aligned}\end{aligned}$$

where $\mathscr {F}_{i}[f]$ is the flux function characterizing the numerical discretization. We assume the Chang-Cooper flux function

$$\begin{aligned}\nonumber \begin{aligned} \mathscr {F}_{i+1/2}[f]=&\left( (1-\delta _{i+1/2})(\mathscr {P}[f_{i+1/2}] + \sigma ^2 D'_{i+1/2}D_{i+1/2})+\frac{\sigma ^2}{2\varDelta w}D^2_{i+1/2} \right) f_{i+1}\\&+\left( \delta _{i+1/2} (\mathscr {P}[f_{i+1/2}] + \sigma ^2 D'_{i+1/2}D_{i+1/2})-\frac{\sigma ^2}{2\varDelta w} D^2_{i+1/2} \right) f_{i}, \end{aligned} \end{aligned}$$

where $D_{i+1/2}=D(w_{i+1/2},c)$ and $D'_{i+1/2}=D'(w_{i+1/2},c)$. The weights $\delta _{i+1/2}$ have to be chosen in such a way that a steady state solution is preserved. Moreover this choice permits also to preserve nonnegativity of the numerical density. The choice

$$\begin{aligned} \delta _{i+1/2}=\frac{1}{\lambda _{i+1/2}}+\frac{1}{1-\exp (\lambda _{i+1/2})}, \end{aligned}$$

(A5)

where

$$\begin{aligned}\nonumber \lambda _{i+1/2}=\frac{2\varDelta w}{\sigma ^2}\frac{1}{D^2_{i+1/2}}\left( \mathscr {P}[f_{i+1/2}] + \sigma ^2 D'_{i+1/2}D_{i+1/2}\right) , \end{aligned}$$

leads to a second order Chang-Cooper nonlinear approximation of the original problem. Note that here, at variance with the standard Chang-Cooper scheme [39], the weights depend on the solution itself as in [76]. Thus we have a nonlinear scheme which preserves the steady state with second order accuracy. In particular, by construction, the weight in (A5) are nonnegative functions with values in [0, 1].

Higher order accuracy of the steady state can be recovered using a more general numerical flux [10].