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Observing rurality of a geographical area from road graph geometry -- a qualitative study

Rami Luisto

TL;DR

This study investigates whether rurality correlates with road-graph geometry by treating Finnish road networks as graphs and measuring hyperbolicity through geodesic triangles. It constructs triplets of road-endpoint geodesics in postal-code areas using the DIGIROAD K and Paavo datasets, computing Polsby-Popper compactness, a triangle-area ratio, and a relative-distortion measure. The results from three representative areas and a broader sampling indicate higher hyperbolicity in rural areas with a nonlinear link to population density, and show general agreement between distance- and travel-time geodesics. The work provides qualitative evidence for geometry-driven rural-urban differences in road networks and outlines methodological refinements and avenues for future research.

Abstract

In this paper we analyze the Finnish road network as a graph in order to measure whether the "rurality" or "urbanity" of an area correlates with local geometrical properties of the graph. Our primary motivation is the observation that the road systems in rural areas look similar to hyperbolic graphs, while in large cities they resemble more the Cayley graph of $\mathbb{Z}^2$. We do not aim for a comprehensive analysis, but rather wish to demonstrate that this observation can be measured and analyzed through looking at various "hyperbolicity measures" of randomly sampled geodesic triangles in the road graph.

Observing rurality of a geographical area from road graph geometry -- a qualitative study

TL;DR

This study investigates whether rurality correlates with road-graph geometry by treating Finnish road networks as graphs and measuring hyperbolicity through geodesic triangles. It constructs triplets of road-endpoint geodesics in postal-code areas using the DIGIROAD K and Paavo datasets, computing Polsby-Popper compactness, a triangle-area ratio, and a relative-distortion measure. The results from three representative areas and a broader sampling indicate higher hyperbolicity in rural areas with a nonlinear link to population density, and show general agreement between distance- and travel-time geodesics. The work provides qualitative evidence for geometry-driven rural-urban differences in road networks and outlines methodological refinements and avenues for future research.

Abstract

In this paper we analyze the Finnish road network as a graph in order to measure whether the "rurality" or "urbanity" of an area correlates with local geometrical properties of the graph. Our primary motivation is the observation that the road systems in rural areas look similar to hyperbolic graphs, while in large cities they resemble more the Cayley graph of . We do not aim for a comprehensive analysis, but rather wish to demonstrate that this observation can be measured and analyzed through looking at various "hyperbolicity measures" of randomly sampled geodesic triangles in the road graph.
Paper Structure (13 sections, 2 equations, 14 figures)

This paper contains 13 sections, 2 equations, 14 figures.

Table of Contents

  1. Introduction
  2. A few notes about hyperbolicity in metric spaces
  3. Some heuristical observations on geometry and curvature
  4. Measuring hyperbolicity in a metric space
  5. Numerical measurements
  6. Our roads and methods
  7. The data
  8. The methods
  9. Results
  10. Results in three representative areas
  11. Results from a larger sampling
  12. Final thoughts
  13. Further possible avenues of research

Figures (14)

  • Figure 1: Various graph geometries found in road systems and groups.
  • Figure 2: Examples of mushroom caps illustrating different geometries: spherical (fly agaric), flat (amethyst deceiver) and hyperbolic (chantarelle).
  • Figure 3: Basic images of triangles in hyperbolic, flat and spherical spaces.
  • Figure 4: Basic images of triangles in hyperbolic, flat and spherical spaces.
  • Figure 5: A comparison of the Hausdorff distance of a given edge of a triangle to the two other edges.
  • ...and 9 more figures

Theorems & Definitions (3)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3