  Given a reaction-diffusion system interacting via a complex network, we
propose two different techniques to modify the network topology while
preserving its dynamical behaviour. In the region of parameters where the
homogeneous solution gets spontaneously destabilized, perturbations grow along
the unstable directions made available across the networks of connections,
yielding irregular spatio-temporal patterns. We exploit the spectral properties
of the Laplacian operator associated to the graph in order to modify its
topology, while preserving the unstable manifold of the underlying equilibrium.
The new network is isodynamic to the former, meaning that it reproduces the
dynamical response (pattern) to a perturbation, as displayed by the original
system. The first method acts directly on the eigenmodes, thus resulting in a
general redistribution of link weights which, in some cases, can completely
change the structure of the original network. The second method uses
localization properties of the eigenvectors to identify and randomize a
subnetwork that is mostly embedded only into the stable manifold. We test both
techniques on different network topologies using the Ginzburg-Landau system as
a reference model. Whereas the correlation between patterns on isodynamic
networks generated via the first recipe is larger, the second method allows for
a finer control at the level of single nodes. This work opens up a new
perspective on the multiple possibilities for identifying the family of
discrete supports that instigate equivalent dynamical responses on a
multispecies reaction-diffusion system.
