  We consider the problem of global synchronization in a large random network
of Kuramoto oscillators where some of them are subject to an external
periodically driven force. We explore a recently proposed dimensional reduction
approach and introduce an effective two-dimensional description for the
problem. From the dimensionally reduced model, we obtain analytical predictions
for some critical parameters necessary for the onset of a globally synchronized
state in the system. Moreover, the low dimensional model also allows us to
introduce an optimization scheme for the problem. Our main conclusion, which
has been corroborated by exhaustive numerical simulations, is that for a given
large random network of Kuramoto oscillators, with random natural frequencies
$\omega_i$, such that a fraction of them is subject to an external periodic
force with frequency $\Omega$, the best global synchronization properties
correspond to the case where the fraction of the forced oscillators is chosen
to be those ones such that $|\omega_i-\Omega|$ is maximal. Our results might
shed some light on the structure and evolution of natural systems for which the
presence or the absence of global synchronization are desired properties. Some
properties of the optimal forced networks and its relation to recent results in
the literature are also discussed.
