3 Dynamical Networks

The theory of dynamical networks is concerned with systems of dynamical units coupled according to an underlying graph structure. It therefore investigates the interplay between dynamics and structure, between the temporal processes going on at the individual units and the static spatial structure linking them. In order to analyse that spatial structure, formalized as a graph, we discuss an essentially complete system of graph invariants, the spectrum of the graph Laplacian, and how it relates to various qualitative properties of the graph. We also describe various stochastic construction schemes for graphs with certain qualitative features. We then turn to dynamical aspects and discuss systems of oscillators with diffusive coupling according to the graph Laplacian and analyse their synchronizability. The analytical tool here are local expansions in terms of eigenmodes of the graph Laplacian. This is viewed as a first step towards a general understanding of pattern formation in systems of coupled oscillators.