Connectivity and Synchronization in Bounded Confidence Kuramoto Oscillators

Frequency synchronization of bounded confidence Kuramoto oscillators is analyzed. The dynamics of each oscillator is defined by the average of the phase differences with its neighbors, where any two oscillators are considered neighbors if their geodesic distance is less than a certain confidence threshold. A phase-dependent graph is defined whose nodes and edges represent the oscillators and their connections, respectively. It is studied how the connectivity of the graph influences steady-state behaviors of the oscillators. It is proved that the oscillators synchronize asymptotically if the subgraph of each partition, possibly not complete, eventually remains constant over time. Simulation results show the application of the theoretical findings also in the presence of oscillators having different natural frequencies.