Network Structured Kinetic Models of Social Interactions

Opinion polarization in social networks

In this paper, we propose a Boltzmann-type kinetic description of opinion formation on social networks, which takes into account a general connectivity distribution of the individuals. We consider opinion exchange processes inspired by the Sznajd model and related simplifications but we do not assume that individuals interact on a regular lattice. Instead, we describe the structure of the social network statistically, assuming that the number of contacts of a given individual determines the probability that their opinion reaches and influences the opinion of another individual. From the kinetic description of the system, we study the evolution of the mean opinion, whence we find precise analytical conditions under which a polarization switch of the opinions, i.e. a change of sign between the initial and the asymptotic mean opinions, occurs. In particular, we show that a non-zero correlation between the initial opinions and the connectivity of the individuals is necessary to observe polarization switch. Finally, we validate our analytical results through Monte Carlo simulations of the stochastic opinion exchange processes on the social network. This article is part of the theme issue 'Kinetic exchange models of societies and economies'.

On a kinetic opinion formation model for pre-election polling

Motivated by recent successes in model-based pre-election polling, we propose a kinetic model for opinion formation which includes voter demographics and socio-economic factors like age, sex, ethnicity, education level, income and other measurable factors like behaviour in previous elections or referenda as a key driver in the opinion formation dynamics. The model is based on Toscani's kinetic opinion formation model (Toscani G. 2006 Kinetic models of opinion formation. Commun. Math. Sci. 4, 481-496.) and the leader-follower model of Düring et al. (Düring B. et al. 2009 Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders. Proc. R. Soc. A 465, 3687-3708.), and leads to a system of coupled Boltzmann-type equations and associated, approximate Fokker-Planck-type systems. Numerical examples using data from general elections in the UK show the effect different demographics have on the opinion formation process and the outcome of elections. This article is part of the theme issue 'Kinetic exchange models of societies and economies'.

Mean-Field Selective Optimal Control via Transient Leadership

A mean-field selective optimal control problem of multipopulation dynamics via transient leadership is considered. The agents in the system are described by their spatial position and their probability of belonging to a certain population. The dynamics in the control problem is characterized by the presence of an activation function which tunes the control on each agent according to the membership to a population, which, in turn, evolves according to a Markov-type jump process. In this way, a hypothetical policy maker can select a restricted pool of agents to act upon based, for instance, on their time-dependent influence on the rest of the population. A finite-particle control problem is studied and its mean-field limit is identified via Γ -convergence, ensuring convergence of optimal controls. The dynamics of the mean-field optimal control is governed by a continuity-type equation without diffusion. Specific applications in the context of opinion dynamics are discussed with some numerical experiments.

A viral load-based model for epidemic spread on spatial networks

In this paper, we propose a Boltzmann-type kinetic model of the spreading of an infectious disease on a network. The latter describes the connections among countries, cities or districts depending on the spatial scale of interest. The disease transmission is represented in terms of the viral load of the individuals and is mediated by social contacts among them, taking into account their displacements across the nodes of the network. We formally derive the hydrodynamic equations for the density and the mean viral load of the individuals on the network and we analyse the large-time trends of these quantities with special emphasis on the cases of blow-up or eradication of the infection. By means of numerical tests, we also investigate the impact of confinement measures, such as quarantine or localised lockdown, on the diffusion of the disease on the network.