High‑frequency localization scaling for multidimensional diffusively coupled networks

Prove the conjectured high‑frequency scaling law for sinusoidally driven networks of N diffusively coupled identical units with n‑dimensional nodal dynamics, where the amplitude of the sinusoidal response in the m‑th time derivative D_t^m x_i of node i at graph‑theoretic distance d from the driven node k satisfies 𝛛A_{i,m}^{(k)} ~ |Ψ_{ki}^{[d]}| · ω^{-n(d+1)+m} as the driving frequency ω → ∞, with |Ψ_{ki}^{[d]}| independent of ω.

Background

The paper derives localized steady‑state response patterns for the oscillator model of power grids and proves high‑frequency algebraic decay of response amplitudes with graph‑theoretic distance for n=2 in the power‑grid setting. It then proposes a generalization to networks of diffusively coupled identical units with n‑dimensional nodal dynamics.

The conjecture specifies an explicit scaling exponent −n(d+1)+m for the amplitude of the sinusoidal response in D_tm x_i at node i when node k is driven at high frequency. The authors note that this law holds for special cases (e.g., n=1 Kuramoto phase oscillators and n=2 oscillator model with m=1) but remains unproven in general.

References

If unit k is sinusoidally driven with frequency ω→∞, we conjecture that the amplitude \tilde{A}{i,m}{(k)} of the sinusoidal response in state variable D_tm x_i of unit i is given by \begin{equation} \tilde{A}{i,m}{(k)}\overset{\omega\rightarrow \infty}{\sim}\left|\Psi{[d]}_{ki}\right|\omega{-n(d+1)+m}, \label{conj:localize_pattern} \end{equation} where $|\Psi{[d]}_{ki}|$ is a distance- and node-specific prefactor but independent on the perturbation frequency.

— Fluctuation Response Patterns of Network Dynamics -- an Introduction (2403.05746 - Zhang et al., 9 Mar 2024) in Remark “Localized response patterns in networks of multi-dimensional dynamical systems,” Section 4.1.3