Network Synchronization in a Noisy Environment with Time Delays: Fundamental Limits and Trade-Offs

TL;DR

This paper investigates the fundamental limits and trade-offs in network synchronization affected by noise and time delays, providing thresholds for synchronizability and bounds on synchronization efficiency.

Contribution

It introduces an exact synchronizability threshold for networks with delays and develops a scaling theory to determine the minimal synchronization fluctuations in noisy environments.

Findings

Derived the exact synchronizability threshold for delayed networks

Established the fundamental limit of synchronization efficiency in noisy settings

Provided insights into optimization trade-offs in network synchronization

Abstract

We study the effects of nonzero time delays in stochastic synchronization problems with linear couplings in an arbitrary network. Using the known exact threshold value from the theory of differential equations with delays, we provide the synchronizability threshold for an arbitrary network. Further, by constructing the scaling theory of the underlying fluctuations, we establish the absolute limit of synchronization efficiency in a noisy environment with uniform time delays, i.e., the minimum attainable value of the width of the synchronization landscape. Our results have also strong implications for optimization and trade-offs in network synchronization with delays.