Synchronization: An obstacle to identification of network topology

Identification of network topology is an important issue for physics, biology, engineering, and other science disciplines. Under the application of an adaptive-feedback control algorithm, we explore a process through which the topology of an unknown dynamical network can be identified. Analytical results show that the key to guaranteeing successful topology identification is the existence of the condition that all coupling terms of the unknown network are linearly independent of the synchronization manifold between the drive (unknown) network and a response network. When complete synchronization is achieved in the unknown network, the linearly independent condition is no longer satisfied, making its topology unidentifiable. We also find that partial synchronization in the unknown network implies a part of topology being unidentifiable. The results can be extended to projection synchronization and some generalized synchronization. Furthermore, we show that when the network dynamical equation is stable, its topology can be identified subject to some limitations. Finally, a method of avoiding synchronization for a network is presented and verified by numerical simulations.