arXiv subjects

Sidney Holden

Publications and source records attributed to Sidney Holden.

2 recordsLinked to original sources

arXiv2026

Length-scale selection in adaptive transport networks

Adaptive transport networks in biological and physical systems exhibit hierarchical organization, characteristic channel spacing, and robust scaling relations. Existing adaptive network models, formulated on a lattice, successfully reproduce many observed topologies and conduit scaling laws; however, the mechanism that selects network density and spatial spacing remains unclear. We address this in a continuum formulation where conductivity evolves as a tensor field coupled to pressure-driven flow. Linearizing about a homogeneous conducting state, we identify a finite-wavelength instability with a $-1/4$ preferred wavelength scaling in the control parameter. Simulations of the full equations confirm the analytical predictions and demonstrate the formation of anisotropic conducting structures above threshold. These results establish a scale-selection principle for adaptive transport network formation which arises from a pattern-forming instability rather than solely from relaxation within a nonconvex energy landscape. The instability mechanism places adaptive transport systems within a broader class of nonequilibrium pattern-forming media in which constitutive transport feedback generates spatial organization. Beyond reproducing hierarchical scaling laws, the theory additionally predicts the intrinsic density of transport networks and the spatial scale of resource delivery.

nlin.AO↗

arXiv2025

A continuum limit for dense spatial networks

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.

math-ph↗