{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-continuum-limit-for-the-laplace-operator-on","title":"A continuum limit for dense spatial networks","arxiv_id":"2301.07086","date":"2023-01-17","proceeding":null,"authors":["Sidney Holden","Geoffrey Vasil"],"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.","url_abs":"https://arxiv.org/abs/2301.07086v4","url_pdf":"https://arxiv.org/pdf/2301.07086v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-continuum-limit-for-the-laplace-operator-on","repo_url":"https://github.com/sidneyholden1/laplace_operator_metric_graph","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}