  We study the effects of nonzero time delays in stochastic synchronization
problems with linear couplings in complex networks. 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.
