Synchronizability of dynamical networks: different measures and coincidence

In this paper, we list four different interpretations of synchronizability, study their profile in different network structures and give evidence that they do not coincide, in general. By changing the network topological properties, their behavior is tracked and compared with each other. It is shown that their trend goes in different directions in heterogeneous networks such as scale-free ones, whereas in homogeneous ones such as random networks they go hand in hand, as networks' structural parameters change. We also consider networks whose synchronization properties are enhanced through proper weighting or rewiring. The weighting procedure considers information on the node-degrees, node and edges betweenness centralities to assign proper weights for the links. In this way, an undirected and unweighted graph is changed to a weighted and directed one and with enhanced synchronizability. The rewiring algorithm uses information on the eigenvectors corresponding to the second smallest and the largest eigenvalues of the Laplacian matrix to perform efficient rewirings to enhance the synchronizability It is shown that in these networks (weighted or rewired), different synchronizability interpretations have the same trend as the network structural parameter changes.