Spectral Prediction of Synchronization Clusters in Commuting Multilayer Networks

Cluster synchronization, where subsets of nodes evolve coherently while others remain desynchronized, underlies a wide range of coordinated phenomena in complex systems. In networks with multiple types of interactions—so-called hypernetworks—each coupling layer can promote distinct synchronization patterns, making it challenging to predict how coherent groups form and remain stable. Existing symmetry-based methods often require explicit group-theoretic identification, which becomes infeasible for large or structurally heterogeneous systems. Here, we introduce a spectral framework that predicts the hierarchical formation and stability of synchronization clusters in hypernetworks composed of commuting interaction layers. By exploiting the shared eigenspace of layer-specific Laplacian matrices, the framework analytically determines the sequence of cluster formation and decouples their stability via a generalized Master Stability Function. Numerical experiments with Chen and Lorenz oscillators confirm the theoretical predictions, revealing a one-to-one correspondence between spectral organization and the hierarchical emergence of synchronization clusters. This approach provides a unified, computationally efficient method for analyzing and controlling clustered coordination in systems governed by multiple, structurally compatible interaction mechanisms.