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Synchronization is crucial for the correct functionality of many natural andman-made complex systems. In this work we characterize the formation ofsynchronization patterns in networks of Kuramoto oscillators. Specifically, wereveal conditions on the network weights and structure and on the oscillators'natural frequencies that allow the phases of a group of oscillators to evolvecohesively, yet independently from the phases of oscillators in differentclusters. Our conditions are applicable to general directed and weightednetworks of heterogeneous oscillators. Surprisingly, although the oscillatorsexhibit nonlinear dynamics, our approach relies entirely on tools from linearalgebra and graph theory. Further, we develop a control mechanism to determinethe smallest (as measured by the Frobenius norm) network perturbation to ensurethe formation of a desired synchronization pattern. Our procedure allows us toconstrain the set of edges that can be modified, thus enforcing the sparsitystructure of the network perturbation. The results are validated through a setof numerical examples.

Synchronization Patterns in Networks of Kuramoto Oscillators: A Geometric Approach for Analysis and Control