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Opinion dynamics over complex networks: Kinetic modelling and numerical methods

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  • In this paper we consider the modeling of opinion dynamics over time dependent large scale networks. A kinetic description of the agents' distribution over the evolving network is considered which combines an opinion update based on binary interactions between agents with a dynamic creation and removal process of new connections. The number of connections of each agent influences the spreading of opinions in the network but also the way connections are created is influenced by the agents' opinion. The evolution of the network of connections is studied by showing that its asymptotic behavior is consistent both with Poisson distributions and truncated power-laws. In order to study the large time behavior of the opinion dynamics a mean field description is derived which allows to compute exact stationary solutions in some simplified situations. Numerical methods which are capable to describe correctly the large time behavior of the system are also introduced and discussed. Finally, several numerical examples showing the influence of the agents' number of connections in the opinion dynamics are reported.

    Mathematics Subject Classification: Primary: 35Q91, 91D30, 91B74; Secondary: 65M08.


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  • Figure 1.  Stationary states of (21) with relaxation coefficients ${V_r}={V_a}=1$, mean density of connectivity $\gamma=30$, ${{c}_{\max }}=1500$ and several values of the attraction parameters $\alpha$, and having fixed $\beta = 0$. Left: convergence toward the Poisson distribution for big values of $\alpha$. Right: convergence toward a power-law distribution in the limit $\alpha\rightarrow 0$, we indicated with $p_{\infty}^{(-k)}, k=1,2,3$ the $\alpha-$dependent stationary solutions for $\alpha=10^{-1},10^{-2},10^{-3}$, respectively.

    Figure 2.  Stationary solutions of type $f_\infty(w,c) = g_\infty(w)p_\infty(c)$, where $ g_\infty(w)$ is given by (51) with $\kappa =1$, $m_w = 0$, $\sigma^2 = 0.05$ and $p_\infty(c)$ defined by (25), with ${V_r}={V_a}=1$, $\gamma=30$ and $\alpha = 10$ on the left and $\alpha = 0.1$ on the right.

    Figure 3.  Stationary solutions captured via Monte Carlo simulations, with $N_s=2\times10^4$ samples. Parameters of the model are chosen as follows $\sigma^2 = 0.05$, ${V_r}={V_a}=1$, $\beta = 0$, $\alpha = 10$ on the right hand side and $\alpha =0.1$ on the left hand side.

    Figure 4.  Test $\#1$. One-dimensional setting: on the left, convergence of (58) to the stationary solution (51), of the Fokker-Planck equation, for decreasing values of the parameter $\varepsilon$, $g^N_0$ represents the initial distribution. On the right, convergence of the Monte-Carlo (55) to the reference solution (24) for increasing values of the the number of samples $N_s$.

    Figure 5.  Test $\#1.$ One-dimensional setting: on the left, the solution of the Chang-Cooper type scheme with the flux (70) is indicated with $g^N_T$ and compared with the stationary solution (51), also the initial data $g^N_0$ (72) is reported. On the right we report the decay of the $L^1$ relative error (73) for different choices of the quadrature rule, mid-point rule (67) and Milne's rule (respectively of $2^{nd}$ and $4^{th}$ order).

    Figure 6.  Test $\#2.$ From left to right and from the top to the bottom: evolution of the density $f(w,c,t)$ at different time steps. The plot $(a)$ represents the initial data $f_0(w,c)$ (74) and plot $(d)$ the stationary solution. On the plane $(z,c)$ we depict with a blu line the marginal distribution $p(c,t)$ of the solution at time $t$, with red line we represent the reference marginal distribution of the stationary solution.

    Figure 7.  Test $\#2.$ Decay of the $L^1$ relative error with respect to the stationary solution (51). On the left, fixed characteristic rates $V=\{10^3,10^4,10^5\}$, on the right, variable characteristic rates defined as in (76) with $U =\{10^3,10^4,10^5\}$. In both cases for increasing values of the characteristic rate $V$ and $U$ the stationary state is reached faster.

    Figure 8.  Test $\#2.$ Evolution at time $t = 1$ of the initial data $f_0(w, c)$ (74) as isoline plot. On the left in the case of constant characteristic rate on the right variable characteristic rates defined as in (76). The right plot shows that for lower opinion's density the evolution along the connection is faster and slower where the opinions are more concentrated.

    Figure 9.  Test $\#3.$ From left to right and from the first row to the second row, evolution of the initial data (78) in time frame $[0,T]$ with $T = 2$. The evolution shows how a small portion of density with high connectivity can bias the majority of the population towards their position. (Note: The density is scaled according to the marginal distribution $\rho(c,t)$ in order to better show its evolution, the actual marginal density $\rho(c,t)$ is depicted in the background, scaled by a factor 10).

    Figure 10.  Test $\#3.$ On the left-hand side final and initial state of the marginal distribution $g(w,t)$ of the opinion, the green line represents the evolution of the average opinion $\bar{m}(t)$, the red and blue dashed lines represent respectively the opinions $\bar{w}_L=0.75$ and $\bar{w}_F=-0.5$, which are the two leading opinions of the initial data (78).

    Figure 11.  Test $\#4.$ Evolution of the Fokker-Planck model (41) where the interaction are described by (80) with $\Delta=0.25$, in the time frame $[0,T]$ with $T = 100$. The evolution shows the emergence of three main opinion clusters, which are not affected by the connectivity variable. (Note: In order to better show its evolution, we represent the solution as $\log(f(w,c,t)+\epsilon)$, with $\epsilon = 0.001$.)

    Figure 12.  Test $\#4.$ Evolution of the solution of the Fokker-Planck model (41), where the interaction are described by (79) with $\Delta(c)=d_0c/{c_{\max }}$ and $d_0=1.01$, in the time frame $[0,T]$ with $T = 100$. The choice of $\Delta(c)$ reflects in the heterogeneous emergence of clusters with respect to the connectivity level: for higher level of connectivity consensus is reached, instead for lower levels of connectivity multiple opinion clusters are present. (Note: In order to better show its evolution, we represent the solution as $\log(f(w,c,t)+\epsilon)$, with $\epsilon = 0.001$.

    Table 1.  Parameters in the various test cases

    Test $\sigma^2$ $\sigma^2_F$ $\sigma^2_L$ ${{c}_{\max }}$ ${V_r}$ ${V_a}$ $\gamma_0$ $\alpha$ $\beta$
    #1 $5\times10^{-2}$ $6\times10^{-2}$ $-$ $250$ $1$ 1 $30$ $1\times10^{-1} $ $0$
    #2 $5\times10^{-2}$ $6\times10^{-2}$ $-$ $250$ $-$ $-$ $30$ $1\times10^{-1} $ $0$
    #3 $5\times10^{-3}$ $4\times10^{-2}$ $2.5\times10^{-2}$ 250 $1$ $1$ 30 $1\times10^{-4}$ $0$
    #4 $1\times10^{-3}$ $-$ $-$ 250 $1$ $1$ 30 $1\times10^{-1}$ $0$
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