  One of the hallmarks of real networks is their ability to perform
increasingly complex tasks as their topology evolves. To explain this, it has
been observed that as a network grows certain subsets of the network begin to
specialize the function(s) they perform. A recent model of network growth based
on this notion of specialization has been able to reproduce some of the most
well-known topological features found in real-world networks including
right-skewed degree distributions, the small world property, modular as well as
hierarchical topology, etc. Here we describe how specialization under this
model also effects the spectral properties of a network. This allows us to give
conditions under which a network is able to maintain its dynamics as its
topology evolves. Specifically, we show that if a network is intrinsically
stable, which is a stronger version of the standard notion of global stability,
then the network maintains this type of dynamics as the network evolves. This
is one of the first steps toward unifying the rigorous study of the two types
of dynamics exhibited by networks. These are the \emph{dynamics of} a network,
which is the study of the topological evolution of the network's structure,
modeled here by the process of network specialization, and the \emph{dynamics
on} a network, which is the changing state of the network elements, where the
type of dynamics we consider is global stability. The main examples we apply
our results to are recurrent neural networks, which are the basis of certain
types of machine learning algorithms.
