When Symmetrization Guarantees Synchronization in Directed Networks

We review and illustrate our recent results on globally stable synchronization in directed oscillator networks. We consider asymmetrically connected networks with node balance, the property that the sum of the coupling coefficients of all edges directed to a node equals the sum of the coupling coefficients of all the edges directed outward from the node. We show that for such directed but node balanced networks, it is sufficient to symmetrize all connections by replacing a unidirectional coupling with a bidirectional coupling of half the coupling strength. The synchronization condition for the symmetrized network then guarantees synchronization in the original directed network. By considering an example of coupled driven pendula, we show how to prove global stability of synchronization in a concrete unidirectional network. We also discuss the relation between local and global synchronization.