Extracting topological features from dynamical measures in networks of Kuramoto oscillators

The Kuramoto model for an ensemble of coupled oscillators provides a paradigmatic example of nonequilibrium transitions between an incoherent and a synchronized state. Here we analyze populations of almost identical oscillators in arbitrary interaction networks. Our aim is to extract topological features of the connectivity pattern from purely dynamical measures based on the fact that in a heterogeneous network the global dynamics is not only affected by the distribution of the natural frequencies but also by the location of the different values. In order to perform a quantitative study we focused on a very simple frequency distribution considering that all the frequencies are equal but one, that of the pacemaker node. We then analyze the dynamical behavior of the system at the transition point and slightly above it as well as very far from the critical point, when it is in a highly incoherent state. The gathered topological information ranges from local features, such as the single-node connectivity, to the hierarchical structure of functional clusters and even to the entire adjacency matrix.