Diffusive and Anti-Diffusive Behavior for Kinetic Models of Opinion Dynamics

Abstract

In the present paper, we study a class of nonlinear integro-differential equations of a kinetic type describing the dynamics of opinion for two types of societies: conformist ( σ = 1 ) and anti-conformist ( σ = - 1 ). The essential role is played by the symmetric nature of interactions. The class may be related to the mesoscopic scale of description. This means that we are going to statistically describe an individual state of an agent of the system. We show that the corresponding equations result at the macroscopic scale in two different pictures: anti-diffusive ( σ = 1 ) and diffusive ( σ = - 1 ). We provide a rigorous result on the convergence. The result captures the macroscopic behavior resulting from the mesoscopic one. In numerical examples, we observe both unipolar and bipolar behavior known in political sciences.

Modeling Asymmetric Interactions in Economy

“…The mathematical theory of such structures is much more complex than the corresponding theory for equations with symmetric kernels-cf. [6,18,[22][23][24]. In fact, at the formal level, we showed the possible singular behavior of the solutions.…”

“…In contrast to the previous attempts [6,18,[22][23][24], in the present paper, we study an asymmetric way of interactions. The symmetric interactions resulting in nonlinear diffusion equations were considered in reference [24].…”

“…The simplest choice could be γ = 1; however, the case γ > 1 leads to more difficult considerations involving possibilities of blow-ups in a finite time, see [22][23][24]. The previous studies of Equation (4) were concentrated on symmetric interactions, that is the symmetric function κ.…”

Cherry Picking: Consumer Choices in Swarm Dynamics, Considering Price and Quality of Goods

“…The KTAP has been applied to model several socio-economic phenomena, among others, propagation of opinion formation and credit risk over networks [6,7], idiosyncratic learning [8], opinion dynamics [9,10] and social inequality [11], while additional bibliography is provided in [12]. Another example about the use of kinetic theory to explain the market mechanisms is in [13] where the microscopic description leads to a system of linear Boltzmann-type equations.…”

Diauxic Growth at the Mesoscopic Scale

“…Thus it was excluded as it is stated in Case 1. The detailed information on the modeling leading to Case 1 can be found in [ 19 ] (see also references therein), where it was referred to the conformist society.…”