Abstract: Isospectral transformations (IT) of matrices and networks allow for compression of either object, a matrix or network, while keeping all the information about their eigenvalues and eigenvectors. We analyze what happens to the generalized eigenvectors under isospectral transformations. Isospectral compressions are performed with respect to some chosen characteristic of nodes or edges of the network. Each isospectral compression defines a dynamical system on the space of all networks. We show that any orbit of such a dynamical system with a finite network as the initial point converges to an attractor. Such an 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 may converge to the same as well as to different attractors. We also show that networks that are spectrally equivalent with respect to some characteristic could be non-spectrally equivalent for another characteristic.