SPECTRAL STABILITY OF DYNAMIC WEIGHTED HYPERGRAPHS UNDER NONLINEAR EDGE PERTURBATIONS

Objective: Traditional graph models are inherently limited to pairwise dyadic relationships, rendering them insufficient for capturing the complex, higher-order group interactions pervasive in modern network systems. Method: In this paper, we establish a rigorous mathematical framework to analytically bound and evaluate the spectral stability of the hypergraph Laplacian under nonlinear edge dynamics. Diverging from conventional linear matrix approximations, we leverage multilinear algebra to model the dynamic network as a time-evolving tensor field. We present a generalized, higher-order multilinear extension of Weyl’s inequality, proving that the shift in the Z-eigenvalue spectrum is rigorously bounded by the fractional (m-1)-th root of the perturbation tensor's spectral norm. Results: This theoretical breakthrough mathematically formalizes the inherent "structural dampening" phenomenon in higher-order networks. Extensive empirical simulations on large-scale benchmarks (Cora, DBLP) demonstrate that our proposed multilinear safety radius reduces approximation errors by up to 34% compared to baseline linear methodologies. Ultimately, this framework provides robust, deterministic safety guarantees, significantly enhancing the operational reliability and resilience of hypergraph neural networks (HGNNs) in highly volatile, nonlinear environments. Novelty: While dynamic weighted hypergraphs resolve this topological limitation, their spectral properties which govern macroscopic system stability, algorithmic convergence, and synchronization exhibit profound sensitivity to non-convex, state-dependent perturbations. We present a generalized, higher-order multilinear extension of Weyl’s inequality, proving that the shift in the Z-eigenvalue spectrum is rigorously bounded by the fractional (m-1)-th root of the perturbation tensor's spectral norm.