Synchronization in Noisy Networks with Multiple Time Delays

We expand our previous work of uniform time delays in stochastic, linearly-coupled synchronization problems\footnote{D. Hunt, G. Korniss, B.K. Szymanski, Phys. Rev. Lett. \textbf{105}, 068701 (2010)} by including descriptions of networks with multiple delays. Non-uniform time delays can arise when there are multiple sources of delay, e.g. the time to transmit and the time to process information. In this particular two-delay case\footnote{D. Hunt, G. Korniss, B.K. Szymanski, Phys. Lett. A \textbf{375}, 880 (2011)}, the primary limitation on the network to synchronize without any centralized direction does not come from restrictions in the transmission of a node's state to its neighbors; rather it depends on the ability for each node to process and respond to the information about itself in the context of its local environment. Furthermore, given a network's structure, there are optimal transmission delays for which the network remains synchronizable for longer processing delays. As a result, synchronization is not always improved--and in some cases it can be totally destroyed--by minimizing the transmission delays. For special cases we also study the scaling function that quantifies the synchronization of the system. This shows the limitation of synchronization in a noisy network