Analytic treatment of the network synchronization problem with time delays

TL;DR

This paper analytically investigates how uniform time delays affect the efficiency of stochastic network synchronization with linear couplings, revealing a fundamental limit and underlying mechanisms.

Contribution

It provides an analytical determination of the maximum synchronization efficiency in networks with time delays, advancing understanding of delay effects in stochastic synchronization.

Findings

Optimal efficiency at λτ ≈ 0.738

Derived the fundamental limit of synchronization efficiency

Unveiled mechanisms behind delay-induced trade-offs

Abstract

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.