The macroscopic description of dynamics of interacting particle systems is a classical problem in mathematical physics. Modern applications ranging from neuronal networks to power grids feature models with spatially structured interactions. The derivation of the continuum limit for such systems has to deal with the fact that in contrast to the classical setting the particles are no longer identical. It also has to take into account the limiting connectivity of the network.
In this talk, we describe a class of evolution equations, which are derived as a continuum limit of interacting dynamical systems on convergent graph sequences. For systems on random graphs, we present a dynamical law of large numbers and a large deviation principle. Furthermore, we discuss applications to the analysis of synchronization, pattern formation, and metastability in the Kuramoto model of coupled biological oscillators.