Fully solvable lower dimensional dynamics of Cartesian product of Kuramoto models

Implementing a positive correlation between the natural frequencies of nodes and their connectivity on a single star graph leads to a pronounced explosive transition to synchronization, additionally presenting hysteresis behavior. From the viewpoint of network connectivity, a star has been considered as a building motif to generate a big graph by graph operations. On the other hand, we propose to construct complex synchronization dynamics by applying the Cartesian product of two Kuramoto models on two star networks. On the product model, the lower dimensional equations describing the ensemble dynamics in terms of collective order parameters are fully solved by the Watanabe-Strogatz method. Different graph parameter choices lead to three different interacting scenarios of the hysteresis areas of two individual factor graphs, which further change the basins of attraction of multiple fixed points. Furthermore, we obtain coupling regimes where cluster synchronization states are often present on the product graph and the number of clusters is fully controlled. More specifically, oscillators on one star graph are synchronized while those on the other star are not synchronized, which induces clustered state on the product model. The numerical results agree perfectly with the theoretic predictions.