Spectral Effects Of Heavy-Tailed Vertex Noise In Geometric Graphs

Abstract: We characterize which local matrix structures saturate Weyl's eigenvalue perturbation bound for graph Laplacians under geometrically constrained vertex displacements. Geometric graphs with heavy-tailed vertex noise arise across sensor networks, biological imaging, and spatial omics, yet tractable predictions for noise-induced spectral error remain limited. We study geometric graphs abstracted from biophysical systems, incorporating clearance, planarity, and identifiability constraints that govern physically realizable embeddings. Within this constrained setting, we identify witness motifs, small subgraphs in maximally noise-sensitive geometric configurations, that dominate weighted-degree and graph Laplacian spectral perturbations under tempered power-law vertex displacements. This motif decomposition reduces global spectral sensitivity to a finite catalog of local extremal structures and identifies configurations that attain Weyl-tight bounds. We then lift these constrained-graph results to general straight-line embedded graphs in arbitrary dimension via local repair operations producing a constrained surrogate graph that preserves sensitivity-relevant structure. To quantify noise-induced spectral variation in both strong-oracle and weak-oracle regimes, we introduce stochastic co-spectrality (SC) and the stochastic spectral separation index (S3I), which characterize when observed spectral distances are noise-driven and when noise parameters are separable. Together, these results provide a principled pathway from local geometric noise to global spectral error in graph Laplacian matrices, enabling estimation of spectral fragility from graph structure without exhaustive eigenvalue computation or restrictive distributional assumptions beyond moment bounds.