Transfer dynamics and spectral cascades in graph-coupled Kuramoto networks

Abstract This work introduces a spectral transfer formalism for nonlinear synchronisation dynamics to resolve how internal dynamical activity redistributes across network scales, moving beyond traditional global observables like the Kuramoto order parameter. By projecting Kuramoto phase dynamics onto the eigenbasis of the graph Laplacian, we construct a time-dependent transfer matrix and derive novel network metrics, including the spectral flux, which quantifies directional redistribution between structural dynamical scales rather than physical transport across nodes. Numerical simulations on modular networks reveal structured, intermittent cascade-like redistribution in graph-spectral space, directional reversals, and complex interaction geometries. Crucially, we demonstrate a clear separation between macroscopic coherence and microscopic spectral interactions, showing that highly organised, fluctuating internal redistribution dynamics persist even beneath nearly stationary and featureless global synchronisation states. This framework extends conventional synchronisation analysis and establishes a multiscale approach for interpreting network dynamics as an evolving process of spectral reorganisation.