A continuum limit for dense spatial networks

Many physical systems -- such as optical waveguide lattices and dense neuronal or vascular networks -- can be modeled by metric graphs, where slender "wires" (edges) support wave or diffusion equations subject to Kirchhoff conditions at the nodes. This work proposes a continuum-limit framework that replaces edge-based equations with a global coarse-grained partial differential equation (PDE) defined on the continuous space occupied by the network. The derivation naturally introduces an edge-conductivity tensor, an edge-capacity function, and a vertex number density to encode how each microscopic patch of the graph contributes to the macroscopic phenomena. The results have interesting similarities and differences with the Riemannian Laplace-Beltrami operator. We calculate all macroscopic parameters from first principles via a systematic discrete-to-continuous local homogenization, finding an anomalous effective embedding dimension resulting from a homogenized diffusivity. Numerical examples -- including an axisymmetric "spiderweb", several periodic lattices, random Delaunay triangulations, nearest-neighbor geometric graphs, and aperiodic monotiles -- demonstrate that each finite model converges to its corresponding PDE (posed on different manifolds like tori, disks, and spheres) in the limit of increasing vertex density.