Network Structured Kinetic Models of Social Interactions

Abstract

The aim of this paper is to study the derivation of appropriate meso- and macroscopic models for interactions as appearing in social processes. There are two main characteristics the models take into account, namely a network structure of interactions, which we treat by an appropriate mesoscopic description, and a different role of interacting agents. The latter differs from interactions treated in classical statistical mechanics in the sense that the agents do not have symmetric roles, but there is rather an active and a passive agent. We will demonstrate how a certain form of kinetic equations can be obtained to describe such interactions at a mesoscopic level and moreover obtain macroscopic models from monokinetics solutions of those. The derivation naturally leads to systems of nonlocal reaction-diffusion equations (or in a suitable limit local versions thereof), which can explain spatial phase separation phenomena found to emerge from the microscopic interactions. We will highlight the approach in three examples, namely the evolution and coarsening of dialects in human language, the construction of social norms, and the spread of an epidemic.

Keywords: Kinetic models; Network structured interactions; Nonlinear nonlocal equations; Reaction-diffusion equations; Social networks.

Fig. 1

Evolution of u (left) and v (right) at time steps t = 1,2,3,4,5 with the network structured model in full lines and the nonlocal reaction-diffusion model in dash-dotted lines