Generalized connection graph method for synchronization in asymmetrical networks

We present a general framework for studying global complete synchronization in networks of dynamical systems with asymmetrical connections. We extend the connection graph stability method, originally developed for symmetrically coupled networks, to the general asymmetrical case. The principal new component of the method is the transformation of the directed connection graph into an undirected graph. In our method for symmetrically coupled networks we have to choose a path between each pair of nodes. The extension of the method to asymmetrical coupling consists in symmetrizing the graph and associating a weight to each path. This weight involves the “node unbalance” of the two nodes. This quantity is defined to be the difference between the sum of connection coefficients of the outgoing edges and the sum of the connection coefficients of the incoming edges to the node. The synchronization condition for this symmetrized-and-weighted network then also guarantees synchronization in the original asymmetrical network.