  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 (ESB) 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 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.
