  We propose a graphical notation by which certain spectral properties of
complex systems can be rewritten concisely and interpreted topologically.
Applying this notation to analyze the stability of a class of networks of
coupled dynamical units, we reveal stability criteria on all scales. In
particular, we show that in systems such as the Kuramoto model the Coates graph
of the Jacobian matrix must contain a spanning tree of positive elements for
the system to be locally stable.
