Dynamic of frustrated Kuramoto oscillators with modular connections

Synchronization and collective movement are phenomena of a highly interdisciplinary nature, with examples ranging from neuronal activation to walking pedestrians. As of today, the Kuramoto model stands as the quintessential framework for investigating synchronization phenomena, displaying a second order phase transition from disordered motion to synchronization as the coupling between oscillators increases. The model was recently extended to higher dimensions allowing for the coupling parameter to be promoted to a matrix, leading to generalized frustration and new synchronized states. This model was previously investigated in the case of all-to-all and homogeneous interactions. Here, we extend the analysis to modular graphs, which mimic the community structure presented in many real systems. We investigated, both numerically and analytically, the matrix coupled Kuramoto model with oscillators divided into two groups with distinct coupling parameters to understand in which conditions they synchronize independently or globally. We discovered a very rich and complex dynamic, including an extended region in the parameter space in which the interactions between modules were destructive, leading to a global disordered motion even tough the uncoupled dynamic presented higher levels of synchronization. Additional simulations considering synthetic modular networks were performed to assess the robustness of our findings.