Structure preserving schemes for Fokker-Planck equations with nonconstant diffusion matrices

 

Nadia Loy, Mattia Zanella

Mathematics and Computers in Simulation, 188: 342-362, 2021. (Preprint arXiv)

In this work we consider an extension of a recently proposed structure preserving numerical scheme for nonlinear Fokker-Planck-type equations to the case of nonconstant full diffusion matrices. While in existing works the schemes are formulated in a one-dimensional setting, here we consider exclusively the two-dimensional case. We prove that the proposed schemes preserve fundamental structural properties like nonnegativity of the solution without restriction on the size of the mesh and entropy dissipation. Moreover, all the methods presented here are at least second order accurate in the transient regimes and arbitrarily high order for large times in the hypothesis in which the flux vanishes at the stationary state. Suitable numerical tests will confirm the theoretical results.

Multiple-interaction kinetic modelling of a virtual-item gambling economy

Giuseppe Toscani, Andrea Tosin, Mattia Zanella

Physical Review E, 100(1): 012308, 2019.

In recent years, there has been a proliferation of online gambling sites, which made gambling more accessible with a consequent rise in related problems, such as addiction. Hence, the analysis of the gambling behaviour at both the individual and the aggregate levels has become the object of several investigations. In this paper, resorting to classical methods of the kinetic theory, we describe the behaviour of a multi-agent system of gamblers participating in lottery-type games on a virtual-item gambling market. The comparison with previous, often empirical, results highlights the ability of the kinetic approach to explain how the simple microscopic rules of a gambling-type game produce complex collective trends, which might be difficult to interpret precisely by looking only at the available data.

 

Uncertainty damping in kinetic traffic models by driver-assist controls

Andrea Tosin, Mattia Zanella

Mathematical Control and Related Fields, 11(3): 681-713, 2021. (Preprint arXiv)

In this paper, we propose a kinetic model of traffic flow with uncertain binary interactions, which explains the scattering of the fundamental diagram in terms of the macroscopic variability of aggregate quantities, such as the mean speed and the flux of the vehicles, produced by the microscopic uncertainty. Moreover, we design control strategies at the level of the microscopic interactions among the vehicles, by which we prove that it is possible to dampen the propagation of such an uncertainty across the scales. Our analytical and numerical results suggest that the aggregate traffic flow may be made more ordered, hence predictable, by implementing such control protocols in driver-assist vehicles. Remarkably, they also provide a precise relationship between a measure of the macroscopic damping of the uncertainty and the penetration rate of the driver-assist technology in the traffic stream.

 

Hydrodynamic models of preference formation in multi-agent societies

Lorenzo Pareschi, Giuseppe Toscani, Andrea Tosin, Mattia Zanella

Journal of Nonlinear Science, 29(6): 2761-2796, 2019.  Preprint arXiv,

In this paper, we discuss the passage to hydrodynamic equations for kinetic models of opinion formation. The considered kinetic models feature an opinion density depending on an additional microscopic variable, identified with the personal preference. This variable describes an opinion-driven polarisation process, leading finally to a choice among some possible options, as it happens e.g. in referendums or elections. Like in the kinetic theory of rarefied gases, the derivation of hydrodynamic equations is essentially based on the computation of the local equilibrium distribution of the opinions from the underlying kinetic model. Several numerical examples validate the resulting model, shedding light on the crucial role played by the distinction between opinion and preference formation on the choice processes in multi-agent societies.

Kinetic-controlled hydrodynamics for traffic models with driver-assist vehicles

Andrea Tosin, Mattia Zanella

Multiscale Modeling and Simulation, 17(2): 716-749, 2019. (Preprint arXiv)

We develop a hierarchical description of traffic flow control by means of driver-assist vehicles aimed at the mitigation of speed-dependent road risk factors. Microscopic feedback control strategies are designed at the level of vehicle-to-vehicle interactions and then upscaled to the global flow via a kinetic approach based on a Boltzmann-type equation. Then first and second order hydrodynamic traffic models, which naturally embed the microscopic control strategies, are consistently derived from the kinetic-controlled framework via suitable closure methods. Several numerical examples illustrate the effectiveness of such a hierarchical approach at the various scales.

 

Opinion modeling on social media and marketing aspects

Giuseppe Toscani, Andrea Tosin, Mattia Zanella

Physical Review E, 98(2): 022315, 2018. (Preprint arXiv)

We introduce and discuss kinetic models of opinion formation on social networks in which the distribution function depends on both the opinion and the connectivity of the agents. The opinion formation model is subsequently coupled with a kinetic model describing the spreading of popularity of a product on the web through a social network. Numerical experiments on the underlying kinetic models show a good qualitative agreement with some measured trends of hashtags on social media websites and illustrate how companies can take advantage of the network structure to obtain at best the advertisement of their products.

Related popularization article for the Italian blog Madd:Math!: La popolarità delle opinioni

 

 

Kinetic models of collective decision-making in the presence of equality bias

BC_initial_t0

Lorenzo Pareschi, Pierluigi Vellucci, Mattia Zanella 

Physica A: Statistical Mechanics and its Applications, 10(1):1-32, 2017. (Preprint arXiv)

We introduce and discuss kinetic models describing the influence of the competence in the evolution of decisions in a multi-agent system. The original exchange mechanism, which is based on the human tendency to compromise and change opinion through self-thinking, is here modified to include the role of the agents’ competence. In particular, we take into account the agents’ tendency to behave in the same way as if they were as good, or as bad, as their partner: the so-called equality bias. This occurred in a situation where a wide gap separated the competence of group members. We discuss the main properties of the kinetic models and numerically investigate some examples of collective decision under the influence of the equality bias. The results confirm that the equality bias leads the group to suboptimal decisions.

Opinion dynamics over complex networks: kinetic modelling and numerical methods

CC2D_0

Giacomo Albi, Lorenzo Pareschi, Mattia Zanella

Kinetic and Related Models, 10(1): 1-32, 2017. (Preprint arXiv)

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.