# American Institute of Mathematical Sciences

January  2021, 14(1): 45-76. doi: 10.3934/krm.2020048

## Opinion formation systems via deterministic particles approximation

 1 DISIM, Università degli Studi dell'Aquila, via Vetoio 1 (Coppito), 67100 L'Aquila (AQ), Italy 2 EPFL SB, Station 8, CH-1015 Lausanne, Switzerland

* Corresponding author: Simone Fagioli

Received  April 2020 Revised  August 2020 Published  September 2020

We propose an ODE-based derivation for a generalized class of opinion formation models either for single and multiple species (followers, leaders, trolls). The approach is purely deterministic and the evolution of the single opinion is determined by the competition between two mechanisms: the opinion diffusion and the compromise process. Such deterministic approach allows to recover in the limit an aggregation/(nonlinear)diffusion system of PDEs for the macroscopic opinion densities.

Citation: Simone Fagioli, Emanuela Radici. Opinion formation systems via deterministic particles approximation. Kinetic & Related Models, 2021, 14 (1) : 45-76. doi: 10.3934/krm.2020048
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Convergence for different initial data to the stationary state (47). The left column shows the evolution in time for the reconstructed density, where the initial data are (48), (49) and (50) respectively, while the right column shows the opinions evolution in time. The (magenta) stars-line represent the mean opinion $m_1(t)$ in each case. Note that, in the second simulation, $m_1$ is not conserved in time but still converges to zero
Convergence for different initial data to the stationary state (51), where the initial data are (48), (49) and (50) respectively. Note that also in this case the mean opinion $m_1$ (star magenta line in the right column) converges to zero asymptotically
Comparison of different stationary states (46) for different values of $\alpha$. On the left the diffusion coefficient $\lambda^2 = 0.5$, on the right $\lambda^2 = 0.03$
Convergence for the initial data (48), (49) and (50) to the stationary state (53), with $\alpha = 1$. Note that the nonlinearity in the diffusion produces stationary solutions with supports that are smaller than the ones we have seen in the linear diffusion case
Stationary states in (53) for different values of $\alpha$
Opinion dynamics in presence of equally-strong leaders. Top-left initial data for followers(black), left leaders (red) and right leaders (blue). Top-right and bottom-left opinions evolutions for all the groups in short and long term-respectively. Bottom-right opinions evolutions for the followers
Opinion dynamics in presence of one strong group of leaders and one weak group of leaders. Top-left initial data for followers(black), left leaders (red) and right leaders (blue). Top-right and bottom-left opinions evolutions for all the groups in short and long term-respectively. Bottom-right opinions evolutions for the followers
Evolution of system (12). The left column concerns the case of two equally strong groups of leaders, the right column instead describes the situation where the right leader is weaker. Trolls are plotted in green and are associated to the right leaders. Top: initial data. Centre: opinions dynamic for all species. Bottom: comparison between the followers paths with or without trolls
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