Events

Date: 24 September 2025   Time: 13:00 - 14:00

Location: MB-503

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 boundary conditions at the nodes. We propose a continuum-limit framework which replaces edgewise differential equations with a coarse-grained partial differential equation 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. These 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. These high-density networks encode emergent material and functional properties. They reflect the ability of many real-world, space-filling networks to function simultaneously at multiple scales, using the continuum as a feature.