2023/01/17 by Sidney Holden, Holden, Sidney, Geoffrey M. Vasil +1
Mathematics · #FOS: Physical sciences #Graph theory and applications #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2301.07086
openalex publication_date 2023/01/17 · openalex created_date 2023/01/20 · openalex updated_date 2026/07/28
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.