In this paper we study a dynamical optimal transport problem on a network that allows
for transport of mass between different edges if a penalty κ is paid. We show existence
of minimisers using duality and discuss the relationships of the distance-functional to
other metrics such as the Fisher-Rao and the classical Wasserstein metric and analyse the
resulting distance functional in the limiting case κ → ∞.
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.