Kinetic models of opinion formation

We introduce and discuss certain kinetic models of (continuous) opinion formation involving both exchange of opinion between individual agents and diffusion of information. We show conditions which ensure that the kinetic model reaches non trivial stationary states in case of lack of diffusion in correspondence of some opinion point. Analytical results are then obtained by considering a suitable asymptotic limit of the model yielding a Fokker-Planck equation for the distribution of opinion among individuals

A Rosenau-type approach to the approximation of the linear Fokker--Planck equation

{The numerical approximation of the solution of the Fokker--Planck equation is a challenging problem that has been extensively investigated starting from the pioneering paper of Chang and Cooper in 1970. We revisit this problem at the light of the approximation of the solution to the heat equation proposed by Rosenau in 1992. Further, by means of the same idea, we address the problem of a consistent approximation to higher-order linear diffusion equations

Renyi entropy and improved equilibration rates to self-similarity for nonlinear diffusion equations

We investigate the large-time asymptotics of nonlinear diffusion equations $u_t = \Delta u^p$ in dimension $n \ge 1$, in the exponent interval $p > n/(n+2)$, when the initial datum $u_0$ is of bounded second moment. Precise rates of convergence to the Barenblatt profile in terms of the relative R\'enyi entropy are demonstrated for finite-mass solutions defined in the whole space when they are re-normalized at each time $t> 0$ with respect to their own second moment. The analysis shows that the relative R\'enyi entropy exhibits a better decay, for intermediate times, with respect to the standard Ralston-Newton entropy. The result follows by a suitable use of the so-called concavity of R\'enyi entropy power

Hydrodynamics from kinetic models of conservative economies

In this paper, we introduce and discuss the passage to hy- drodynamic equations for kinetic models of conservative economies, in which the density of wealth depends on additional parameters, like the propensity to invest. As in kinetic theory of rarefied gases, the closure depends on the knowledge of the homogeneous steady wealth distribution (the Maxwellian) of the underlying kinetic model. The collision operator used here is the Fokker-Planck operator introduced by J.P. Bouchaud and M. Mezard in [4], which has been recently obtained in a suitable asymp- totic of a Boltzmann-like model involving both exchanges between agents and speculative trading by S. Cordier, L. Pareschi and one of the authors [11]. Numerical simulations on the fluid equations are then proposed and analyzed for various laws of variation of the propensity.Wealth and income distributions, Boltzmann equation, hy- drodynamics, Euler equations

Measure valued solutions of sub-linear diffusion equations with a drift term

In this paper we study nonnegative, measure valued solutions of the initial value problem for one-dimensional drift-diffusion equations when the nonlinear diffusion is governed by an increasing $C^1$ function $\beta$ with $\lim_{r\to +\infty} \beta(r)<+\infty$. By using tools of optimal transport, we will show that this kind of problems is well posed in the class of nonnegative Borel measures with finite mass $m$ and finite quadratic momentum and it is the gradient flow of a suitable entropy functional with respect to the so called $L^2$-Wasserstein distance. Due to the degeneracy of diffusion for large densities, concentration of masses can occur, whose support is transported by the drift. We shall show that the large-time behavior of solutions depends on a critical mass ${m}_{\rm c}$, which can be explicitely characterized in terms of $\beta$ and of the drift term. If the initial mass is less then ${m}_{\rm c}$, the entropy has a unique minimizer which is absolutely continuous with respect to the Lebesgue measure. Conversely, when the total mass $m$ of the solutions is greater than the critical one, the steady state has a singular part in which the exceeding mass ${m} - {m}_{\rm c}$ is accumulated.Comment: 30 page

Opinion modeling on social media and marketing aspects

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

A concavity property for the reciprocal of Fisher information and its consequences on Costa's EPI

We prove that the reciprocal of Fisher information of a log-concave probability density $X$ in ${\bf{R}}^n$ is concave in $t$ with respect to the addition of a Gaussian noise $Z_t = N(0, tI_n)$. As a byproduct of this result we show that the third derivative of the entropy power of a log-concave probability density $X$ in ${\bf{R}}^n$ is nonnegative in $t$ with respect to the addition of a Gaussian noise $Z_t$. For log-concave densities this improves the well-known Costa's concavity property of the entropy power

The dissipative linear boltzmann equation

AbstractWe introduce and discuss a linear Boltzmann equation describing dissipative interactions of a gas of test particles with a fixed background. For a pseudo-Maxwellian collision kernel, it is shown that, if the initial distribution has finite temperature, the solution converges exponentially for large time to a Maxwellian profile drifting at the same velocity as field particles and with a universal nonzero temperature which is lower than the given background temperature

Phase transitions in a two parameter model of opinion dynamics with random kinetic exchanges

Recently, a model of opinion formation with kinetic exchanges has been proposed in which a spontaneous symmetry breaking transition was reported [M. Lallouache et al, Phys. Rev. E, {\bf 82} 056112 (2010)]. We generalise the model to incorporate two parameters, $\lambda$, to represent conviction and $\mu$, to represent the influencing ability of individuals. A phase boundary given by $\lambda=1-\mu/2$ is obtained separating the symmetric and symmetry broken phases: the effect of the influencing term enhances the possibility of reaching a consensus in the society. The time scale diverges near the phase boundary in a power law manner. The order parameter and the condensate also show power law growth close to the phase boundary albeit with different exponents. Theexponents in general change along the phase boundary indicating a non-universality. The relaxation times, however, become constant with increasing system size near the phase boundary indicating the absence of any diverging length scale. Consistently, the fluctuations remain finite but show strong dependence on the trajectory along which it is estimated.Comment: Version accepted for PRE; text modified, new figures and references adde