  In this paper we consider networks of dynamical systems that evolve in
synchrony and investigate how dynamical information from the synchronization
dynamics can be effectively used to learn the network topology, i.e., identify
the time evolution of the couplings between the network nodes. To this aim, we
present an adaptive strategy that, based on a potential that the network
systems seek to minimize in order to maintain synchronization, can be
successfully applied to identify the time evolution of the network from limited
information. This strategy takes advantage of the properties of synchronism of
chaos and of the presence of different communication delays over the network
links. As a motivating example we consider a network of sensors surveying an
area, in which information regarding the time evolution of the network
connections can be used, e.g., to detect changes taking place within the area.
We propose two different setups for our strategy. In the first one,
synchronization has to be achieved at each node (as well as the identification
of the couplings over the network links), based solely on a single scalar
signal representing a superposition of signals from the other nodes in the
network. In the second one, we incorporate an additional node, termed the
maestro, having the function of maintaining network synchronization. We will
see that when such an arrangement is realized, it will become possible to
effectively identify the time evolution of networks that are much larger than
would be possible in the absence of a maestro.
