  We introduce a method that can be used to evolve the topology of a network in
a way that preserves both the network's spectral as well as local structure.
This method is quite versatile in the sense that it can be used to evolve a
network's topology over any collection of the network's elements. This
evolution preserves both the eigenvector centrality of these elements as well
as the eigenvalues of the original network. Although this method is introduced
as a tool to model network growth, we show it can also be used to compare the
topology of different networks where two networks are considered similar if
their evolved topologies are the same. Because this method preserves the
spectral structure of a network, which is related to the network's dynamics, it
can also be used to study the interplay of network growth and function. We show
that if a network's dynamics is intrinsically stable, which is a stronger
version of the standard notion of stability, then the network remains
intrinsically stable as the network's topology evolves. This is of interest
since the growth of a network can have a destabilizing effect on the network's
dynamics, in general. In this sense the methods developed here can be used as a
tool for designing mechanisms of network growth that ensure a network remains
stabile as it grows.
