Spatial scaling of network length and shared-node density in bilayer distribution grids
Abstract
Certain distribution networks are built in layers, each layer collecting a commodity from the layer above it and delivering it to the layer below. We propose a minimal model that gives the length of each layer per unit area from the local density of demand, here the population density. Starting from the Beardwood, Halton and Hammersley law for minimal networks, the length of a layer depends on the number of nodes it reaches and on the area occupied by the users. When two consecutive layers share this area, the gap between their two length exponents is fixed by the exponent of the density of their shared nodes, with no adjusted parameter. We test the model on the complete low and medium-voltage grid of metropolitan France, at the resolution of a population raster. The exponents are fitted by Poisson maximum likelihood on every pixel, with confidence intervals from a spatial block bootstrap. The length exponents imply an effective settled area that grows sublinearly with the density, and the gap between them agrees with the exponent of the substation density, counted independently. The agreement holds over the whole territory, so that the model can serve to estimate the length of such networks where only the density of demand is known.
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BibTeX
@misc{emery2026spatial,
title = {{Spatial scaling of network length and shared-node density in bilayer distribution grids}},
author = {Emile Emery and José Halloy},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31053},
eprint = {2610.31053},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31053},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Spatial scaling of network length and shared-node density in bilayer distribution grids
AU - Emile Emery
AU - José Halloy
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31053
UR - https://arxiv.org/abs/2610.31053
AB - Certain distribution networks are built in layers, each layer collecting a commodity from the layer above it and delivering it to the layer below. We propose a minimal model that gives the length of each layer per unit area from the local density of demand, here the population density. Starting from the Beardwood, Halton and Hammersley law for minimal networks, the length of a layer depends on the number of nodes it reaches and on the area occupied by the users. When two consecutive layers share this area, the gap between their two length exponents is fixed by the exponent of the density of their shared nodes, with no adjusted parameter. We test the model on the complete low and medium-voltage grid of metropolitan France, at the resolution of a population raster. The exponents are fitted by Poisson maximum likelihood on every pixel, with confidence intervals from a spatial block bootstrap. The length exponents imply an effective settled area that grows sublinearly with the density, and the gap between them agrees with the exponent of the substation density, counted independently. The agreement holds over the whole territory, so that the model can serve to estimate the length of such networks where only the density of demand is known.
ER -