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 laws at their nodes. These high-density networks encode effective material and functional properties. To understand how these properties emerge, we propose a continuum-limit framework that replaces edgewise differential equations with a 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, which together describe how each microscopic patch of the graph contributes to macroscopic behavior. All macroscopic parameters are computed from first principles via a systematic discrete-to-continuous homogenization, revealing an anomalous effective embedding dimension arising from the homogenized diffusivity. This framework captures how real-world, space-filling networks operate simultaneously across system-wide and local scales, using the continuum as a feature. Numerical examples—including periodic lattices and random graphs—demonstrate convergence of each finite network to its corresponding PDE (posed on general manifolds) in the limit of increasing vertex density. We expect these results to aid modeling of growth and function in biological transport networks.