  In the recently developed theory of isospectral transformations of networks
isospectral compressions are performed with respect to some chosen
characteristic (attribute) of nodes (or edges) of networks. Each isospectral
compression (when a certain characteristic is fixed) defines a dynamical system
on the space of all networks. It is shown that any orbit of such dynamical
system which starts at any finite network (as the initial point of this orbit)
converges to an attractor. Such attractor is a smaller network where a chosen
characteristic has the same value for all nodes (or edges). We demonstrate that
isospectral contractions of one and the same network defined by different
characteristics of nodes (or edges) may converge to the same as well as to
different attractors. It is also shown that spectrally equivalent with respect
to some characteristic networks could be non-spectrally equivalent for another
characteristic of nodes (edges). These results suggest a new constructive
approach to analysis of networks structures and to comparison of topologies of
different networks.
