Analytic treatment of the network synchronization problem with time delays

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.