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

We study the effects of nonzero time delays in stochastic synchronization with linear couplings in a network. First, using the known exact threshold value from the theory of differential equations with delays, we establish the synchronizability threshold for an arbitrary network with uniform time delays. Scaling the underlying fluctuations, we derive the absolute limit of synchronization efficiency in a noisy environment with uniform time delays that is the minimum attainable width of the synchronization landscape. We consider two types of time delays: transmission delays between interacting nodes and local delays at each node (due to processing, cognitive, or execution delays). By investigating the underlying fluctuations for several delay schemes, we obtain the synchronizability threshold (phase boundary) and the scaling behavior of the width of the synchronization landscape, in some cases for arbitrary networks and in others for specific weighted networks. Numerical computations allow the behavior of these networks to be explored when direct analytical results are not available. We comment on the implications of these findings for simple locally or globally weighted network couplings and possible trade-offs present in such systems. In case of synchronization between two nodes, we consider two types of delays: transmission delays between interacting elements and processing, cognitive, or execution delays at each element. Our results imply the potential for optimization and trade-offs in synchronization problems with time delays.