  Synchronization is an important and prevalent phenomenon in natural and
engineered systems. In many dynamical networks, the coupling is balanced or
adjusted in order to admit global synchronization, a condition called Laplacian
coupling. Many networks exhibit incomplete synchronization, where two or more
clusters of synchronization persist, and computational group theory has
recently proved to be valuable in discovering these cluster states based upon
the topology of the network. In the important case of Laplacian coupling,
additional synchronization patterns can exist that would not be predicted from
the group theory analysis alone. The understanding of how and when clusters
form, merge, and persist is essential for understanding collective dynamics,
synchronization, and failure mechanisms of complex networks such as electric
power grids, distributed control networks, and autonomous swarming vehicles. We
describe here a method to find and analyze all of the possible cluster
synchronization patterns in a Laplacian-coupled network, by applying methods of
computational group theory to dynamically-equivalent networks. We present a
general technique to evaluate the stability of each of the dynamically valid
cluster synchronization patterns. Our results are validated in an electro-optic
experiment on a 5 node network that confirms the synchronization patterns
predicted by the theory.
