  Motivated by novel results in the theory of network synchronization, we
analyze the effects of nonzero time delays in stochastic synchronization
problems with linear couplings in an arbitrary network. We determine {\it
analytically} the fundamental limit of synchronization efficiency in a noisy
environment with uniform time delays. We show that the optimal efficiency of
the network is achieved for
$\lambda\tau={{\pi^{3/2}}\over{2\sqrt{\pi}+4}}\approx0.738$, where $\lambda$ is
the coupling strength (relaxation coefficient) and $\tau$ is the characteristic
time delay in the communication between pairs of nodes. Our analysis reveals
the underlying mechanism responsible for the trade-off phenomena observed in
recent numerical simulations of network synchronization problems.
