A continuum limit for dense spatial networks

Many physical systems—such as dense neuronal or vascular networks and optical waveguide lattices—can be modeled as spatial networks, where slender "wires" (edges) support wave or diffusion equations subject to conservation conditions at nodes. We propose a continuum-limit framework that replaces edgewise differential equations with a coarse-grained partial differential equation defined on the ambient 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. A discrete-to-continuous local homogenization produces an anomalous effective embedding dimension resulting from a homogenized diffusivity. High-density networks encode emergent material and functional properties, reflecting the ability of many real-world, space-filling networks to operate simultaneously at multiple scales, using the continuum as a feature. Applications include modeling large-scale coherent waves in vascular networks and effective transport properties in metamaterials.