A continuum limit for dense spatial networks

TL;DR

This paper develops a continuum PDE framework for dense spatial networks, translating complex graph-based models into a macroscopic description with parameters derived from first principles, applicable to various network geometries.

Contribution

It introduces a systematic homogenization method to derive a PDE with an edge-conductivity tensor and other parameters from microscopic graph models, bridging discrete networks and continuous spaces.

Findings

Finite network models converge to the derived PDE as vertex density increases.

The framework captures the influence of network geometry on macroscopic diffusion.

Numerical examples validate the continuum limit across different network types.

Abstract

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.