  This paper deals with the global stability of time-delayed dynamical
networks. We show that for a time-delayed dynamical network with
non-distributed delays the network and the corresponding non-delayed network
are both either globally stable or unstable. We demonstrate that this may not
be the case if the network's delays are distributed. The main tool in our
analysis is a new procedure of dynamical network restrictions. This procedure
is useful in that it allows for improved estimates of a dynamical network's
global stability. Moreover, it is a computationally simpler and much more
effective means of analyzing the stability of dynamical networks than the
procedure of isospectral network expansions introduced in [Isospectral graph
transformations, spectral equivalence, and global stability of dynamical
networks. Nonlinearity, 25 (2012) 211-254]. The effectiveness of our approach
is illustrated by applications to various classes of Cohen-Grossberg neural
networks.
