Graph-based criteria for synchronization of diffusively coupled oscillators

We extend the connection graph stability method for complete synchronization, originally developed for symmetrically coupled networks, to the asymmetrical case. First, we study synchronization in 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 obtain the same criterion as for the network with a symmetrized connection matrix, provided that the condition of node balance is satisfied. Then, we derive a general synchronization criterion for arbitrary asymmetrical networks. The criterion is obtained by symmetrizing the connection graph and associating a weight to the path between any two nodes.