  Synchronization, in which individual dynamical units keep in pace with each
other in a decentralized fashion, depends both on the dynamical units and on
the properties of the interaction network. Yet, the role played by the network
has resisted comprehensive characterization within the prevailing paradigm that
interactions facilitating pair-wise synchronization also facilitate collective
synchronization. Here we challenge this paradigm and show that networks with
best complete synchronization, least coupling cost, and maximum dynamical
robustness, have arbitrary complexity but quantized total interaction strength
that constrains the allowed number of connections. It stems from this
characterization that negative interactions as well as link removals can be
used to systematically improve and optimize synchronization properties in both
directed and undirected networks. These results extend the recently discovered
compensatory perturbations in metabolic networks to the realm of oscillator
networks and demonstrate why "less can be more" in network synchronization.
