  Network dynamics is nowadays of extreme relevance to model and analyze
complex systems. From a dynamical systems perspective, understanding the local
behavior near equilibria is of utmost importance. In particular, equilibria
with at least one zero eigenvalue play a crucial role in bifurcation analysis.
In this paper, we want to shed some light on nilpotent equilibria of network
dynamical systems. As a main result, we show that the blow-up technique, which
has proven to be extremely useful in understanding degenerate singularities in
low-dimensional ordinary differential equations, is also suitable in the
framework of network dynamical systems. Most importantly, we show that the
blow-up technique preserves the network structure. The further usefulness of
the blow-up technique, especially with regard to the desingularization of a
nilpotent point, is showcased through several examples including linear
diffusive systems, systems with nilpotent internal dynamics, and an adaptive
network of Kuramoto oscillators.
