Linear stability of edge-based topological synchronization with finite spectral dimension

Investigate the linear stability of the synchronized state for the edge-based topological Kuramoto model governed by dφ/dt = \hat{ω} − σ L_[1] φ on networks with finite spectral dimension, and determine under what conditions the fully synchronized edge-signal state is stable.

Background

The Dirac Topological Synchronization (DTS) model decouples node and edge dynamics when derived from a cosine free energy using the Dirac operator, leading to linearized equations dθ/dt = ω − σ L_[0] θ for nodes and dφ/dt = \hat{ω} − σ L_[1] φ for edges. Prior work has analyzed the linear stability of the node dynamics in terms of spectral dimension, but a corresponding analysis for the edge dynamics under the 1-Hodge Laplacian has not been performed.

The authors explicitly note the absence of a paper on how finite spectral dimension influences the stability of the edge-signal synchronized state in the linearized edge equation, highlighting a gap necessary for understanding conditions for edge-based synchronization in complex networks.