The study of dynamical processes running on top of complex networks has become a central issue in many research fields, ranging from the microscopic realm of genes and neurones to large technological and social systems. However, many times the information we can accede to about the actual topology of interactions is somehow incomplete, because of experimental limitations or lags on the details of the system. Moreover, a macroscopic observable usually is the only reflection of the dynamics, and many topologies are compatible with it, raising the problem of multi-valuation. Following this perspective, we propose to study a novel mapping problem: given a network structure and a dynamical process on top of it (the synchronization dynamics of coupled oscillators), we wonder how to transform the network into a different connectivity structure so that the collective behavior remains invariant. Such transformation must adjust the interactions weights in the new configuration to achieve an equivalent steady-state functionality to the original structure.

Inspired by the derivation of statistical mechanics from information theory as a particular case of statistical inference, we tackle the functional mapping as an optimization problem for the unknown weights subject to structural constraints. In this framework, we derive optimal transformations according to different states of available information that are able to preserve the collective behavior even if the mapped networks have very different connectivity patterns and the coupling function between units is highly non-linear. Also, we show that the mapping of homogeneous networks into heterogeneous ones is usually less accurate and requires more -costly- microscopic information than the reverse process, unveiling an interesting symmetry unbalance phenomenon.

Furthermore, we apply our theory in a realistic scenario of network degradation, where the system attemps to preserve its functionality under random removal of links. By tuning the weights of the remaining connections according to our proposed transformations, the system preserves the functionality for a very large fraction of removed links (which depends on the initial configuration) and shows a much higher resilience than other existing methods that do not exploit the strength of the weight-tuning as an efficient adaptive mechanism for complex networks.