  The need to build a link between the structure of a complex network and the
dynamical properties of the corresponding complex system (comprised of multiple
low dimensional systems) has recently become apparent. Several attempts to
tackle this problem have been made and all focus on either the controllability
or synchronisability of the network --- usually analyzed by way of the master
stability function, or the graph Laplacian. We take a different approach. Using
the basic tools from dynamical systems theory we show that the dynamical
stability of a network can easily be defined in terms of the eigenvalues of an
homologue of the network adjacency matrix. This allows us to compute the
stability of a network (a quantity derived from the eigenspectrum of the
adjacency matrix). Numerical experiments show that this quantity is very
closely related too, and can even be predicted from, the standard structural
network properties. Following from this we show that the stability of large
network systems can be understood via an analytic study of the eigenvalues of
their fixed points --- even for a very large number of fixed points.
