Hidden synchrony in networks through Laplacian-spectrum-based En matrices.

Understanding cluster synchronization in complex networks is essential for characterizing hierarchical and partial coherence in coupled systems. This study investigates synchronization pathways using both the cumulative Sn matrix method and En matrices derived from the Laplacian spectrum. The En matrix analysis reveals clusters that are not apparent in the Sn representation, including clusters that remain hidden under certain dynamical conditions. Through networks of coupled oscillators, it is demonstrated that hidden clusters can remain dynamically hidden when their stability coincides with global synchronization, as in chaotic systems with class-II master stability functions. However, for systems with stable fixed points, the same clusters may remain synchronized independently and become directly observable as coupling varies. The results demonstrate that the En framework facilitates the systematic detection of both visible and hidden synchronization structures in complex networks.