The essential synchronization backbone problem

Network optimization strategies for the process of synchronization have generally focused on the re-wiring or re-weighting of links in order to (1) expand the range of coupling strengths that achieve synchronization, (2) expand the basin of attraction for the synchronization manifold, or (3) lower the average time to synchronization. A new optimization goal is proposed in seeking the minimum subset of the edge set of the original network that enables the same essential ability to synchronize in that the synchronization manifolds have conjugate stability. We call this type of minimal spanning subgraph an essential synchronization backbone of the original system, and we present two algorithms: one is a strategy for an exhaustive search for a true solution, while the other is a method of approximation for this combinatorial problem. The solution spaces that result from different choices of dynamical systems and coupling schemes vary with the level of a hierarchical structure present and also the number of interwoven central cycles. Applications can include the important problem in civil engineering of power grid hardening, where new link creation may be costly, and the defense of certain key links to the functional process may be prioritized.