  We study the local complete synchronization of discrete-time dynamical
networks with time-varying couplings. Our conditions for the temporal variation
of the couplings are rather general and include both variations in the network
structure and in the reaction dynamics; the reactions could, for example, be
driven by a random dynamical system. A basic tool is the concept of Hajnal
diameter which we extend to infinite Jacobian matrix sequences. The Hajnal
diameter can be used to verify synchronization and we show that it is
equivalent to other quantities which have been extended to time-varying cases,
such as the projection radius, projection Lyapunov exponents, and transverse
Lyapunov exponents. Furthermore, these results are used to investigate the
synchronization problem in coupled map networks with time-varying topologies
and possibly directed and weighted edges. In this case, the Hajnal diameter of
the infinite coupling matrices can be used to measure the synchronizability of
the network process. As we show, the network is capable of synchronizing some
chaotic map if and only if there exists an integer T>0 such that for any time
interval of length T, there exists a vertex which can access other vertices by
directed paths in that time interval.
