In the study of dynamical systems on networks/graphs, a key theme is how the
network topology influences stability for steady states or synchronized states.
Ideally, one would like to derive conditions for stability or instability that
instead of microscopic details of the individual nodes/vertices rather make the
influence of the network coupling topology visible. The master stability
function is an important such tool to achieve this goal. Here we generalize the
master stability approach to hypergraphs. A hypergraph coupling structure is
important as it allows us to take into account arbitrary higher-order
interactions between nodes. As for instance in the theory of coupled map
lattices, we study Laplace type interaction structures in detail. Since the
spectral theory of Laplacians on hypergraphs is richer than on graphs, we see
the possibility of new dynamical phenomena. More generally, our arguments
provide a blueprint for how to generalize dynamical structures and results from
graphs to hypergraphs