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Synchronization is an important and prevalent phenomenon in natural andengineered systems. In many dynamical networks, the coupling is balanced oradjusted in order to admit global synchronization, a condition called Laplaciancoupling. Many networks exhibit incomplete synchronization, where two or moreclusters of synchronization persist, and computational group theory hasrecently proved to be valuable in discovering these cluster states based uponthe topology of the network. In the important case of Laplacian coupling,additional synchronization patterns can exist that would not be predicted fromthe group theory analysis alone. The understanding of how and when clustersform, merge, and persist is essential for understanding collective dynamics,synchronization, and failure mechanisms of complex networks such as electricpower grids, distributed control networks, and autonomous swarming vehicles. Wedescribe here a method to find and analyze all of the possible clustersynchronization patterns in a Laplacian-coupled network, by applying methods ofcomputational group theory to dynamically-equivalent networks. We present ageneral technique to evaluate the stability of each of the dynamically validcluster synchronization patterns. Our results are validated in an electro-opticexperiment on a 5 node network that confirms the synchronization patternspredicted by the theory.

Complete Characterization of Stability of Cluster Synchronization in Complex Dynamical Networks