Inferring Latent Geometries in Weighted Spatio-Functional Networks

Abstract

Physical constraints and functional demands constantly co-evolve across real-world networks, yet empirical topologies rarely conform to idealized dichotomies of pure planar grids or unconstrained relational spaces. Here, we formulate a scalable generative framework to solve the inverse problem of socio-spatial inference in weighted networks. By superimposing explicit metric decay with an unobserved latent functional geometry, we jointly reconstruct hidden node coordinates and infer a coupling parameter $λ\in [0, 1]$ quantifying the share of connectivity driven by functional affinity versus geographic distance decay. Our deterministic pipeline combines physics-informed spectral initialization on spatial residuals with hybrid optimization (Adam and L-BFGS) and profile-likelihood confidence intervals derived from empirical Fisher curvature. Across diverse empirical systems, this framework resolves a continuous spectrum of socio-spatial topologies: in the \textit{Drosophila melanogaster} whole-brain connectome, unsupervised inference uncovers an architectural hierarchy from metric sensory peripheries to unconstrained associative neuropils; in the European power grid, metric coupling remains invariant under severe line pruning, whereas topological rewiring triggers an explosive surge in $λ$ that exposes deceptive percolation robustness; and in global aviation, temporal tracking throughout the COVID-19 pandemic captures an abrupt systemic spatialization toward localized metric connectivity. Together, these results establish our framework as a principled, scalable diagnostic of structural organization, physical degradation, and dynamic resilience in complex networked systems.