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 and hierarchical networks reveal highly structured, intermittent spectral transfer episodes characterised by forward and inverse spectral cascades, 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.
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BibTeX
@misc{kowalczyk2026transfer,
title = {{Transfer Dynamics and Spectral Cascades in Graph-Coupled Kuramoto Networks}},
author = {Marcin Kowalczyk and Pietro Liò and Zbigniew R. Struzik},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28432},
eprint = {2609.28432},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28432},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Transfer Dynamics and Spectral Cascades in Graph-Coupled Kuramoto Networks
AU - Marcin Kowalczyk
AU - Pietro Liò
AU - Zbigniew R. Struzik
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28432
UR - https://arxiv.org/abs/2609.28432
AB - 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 and hierarchical networks reveal highly structured, intermittent spectral transfer episodes characterised by forward and inverse spectral cascades, 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.
ER -