Templates for Risk Informed Assurance with Curvature Embeddings (TRACE)
We investigate recovery of geometric structure from networks embedded in manifolds with spatially varying curvature, extending the constant-curvature framework of Lubold et al. (2023). Our work supports cascade risk assessment in critical infrastructure through the Templates for Risk-informed Assurance with Curvature Embeddings (TRACE) framework. Simulations on a bi-modal Gaussian surface show that constant-curvature methods yield weighted averages shaped by clique patterns, while hierarchical clustering identifies distinct regimes. Localized estimation, however, reveals boundary contamination in transitional regions. To address heterogeneity, we develop distance metrics for graphs with edge and node features, proving their metric validity, and validate them via deterministic graph generation from canonical tilings. We further propose a diffusion-based anomaly detection approach that treats networks as glued manifolds, using curvature discontinuities to detect structural anomalies. Employing the carré-du-champ operator and scalar curvature, we achieve robust anomaly discrimination, demonstrated on the Singapore Water Treatment (SWaT) dataset with joint network-traffic and sensor features. Integration with TRACE reveals how curvature shapes cascade dynamics: positive curvature impedes, while negative curvature accelerates propagation. This geometric perspective provides interpretable risk metrics and visualization tools for critical infrastructure managers. While full validation remains ongoing, our contributions establish a rigorous foundation for geometric analysis of network resilience and cascade vulnerability.