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Generating Random Hyperfractal Cities

Geoffrey Deperle, Philippe Jacquet

TL;DR

This work presents a hierarchical, hyperfractal approach to generating synthetic cities composed of interacting neighborhoods. By tiling the city into convex cells and assigning a per-cell hyperfractal network, it unifies geometry and traffic distribution, enabling scalable, parameterizable city models. It provides explicit models (including a Manhattan-style baseline) and practical methods for estimating hyperfractal dimensions from data, alongside a procedural pipeline incorporating anisotropic Gaussian-driven urban sprawl. The framework supports applications in routing algorithm testing, synthetic data generation for machine learning, and exploration of how fractal properties influence information propagation in urban networks.

Abstract

This paper focuses on the challenge of interactively modeling street networks. In this work, we extend the simple fractal model, which is particularly useful for describing small cities or individual districts, by constructing random cities based on a tiling structure over which hyperfractals are distributed. This approach enables the connection of multiple hyperfractal districts, providing a more comprehensive urban representation. Furthermore, we demonstrate how this decomposition can be used to segment a city into distinct districts through fractal analysis. Finally, we present tools for the numerical generation of random cities following this model.

Generating Random Hyperfractal Cities

TL;DR

This work presents a hierarchical, hyperfractal approach to generating synthetic cities composed of interacting neighborhoods. By tiling the city into convex cells and assigning a per-cell hyperfractal network, it unifies geometry and traffic distribution, enabling scalable, parameterizable city models. It provides explicit models (including a Manhattan-style baseline) and practical methods for estimating hyperfractal dimensions from data, alongside a procedural pipeline incorporating anisotropic Gaussian-driven urban sprawl. The framework supports applications in routing algorithm testing, synthetic data generation for machine learning, and exploration of how fractal properties influence information propagation in urban networks.

Abstract

This paper focuses on the challenge of interactively modeling street networks. In this work, we extend the simple fractal model, which is particularly useful for describing small cities or individual districts, by constructing random cities based on a tiling structure over which hyperfractals are distributed. This approach enables the connection of multiple hyperfractal districts, providing a more comprehensive urban representation. Furthermore, we demonstrate how this decomposition can be used to segment a city into distinct districts through fractal analysis. Finally, we present tools for the numerical generation of random cities following this model.

Paper Structure

This paper contains 13 sections, 3 theorems, 16 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Hyperfractal Model
  3. Hyperfractal Model: The General Case
  4. Hyperfractal Manhattan's Model
  5. Estimation of the Hyperfractal Dimension
  6. Tessellation and Hyperfractal Model
  7. The Model
  8. Generating Random Fractal Cities
  9. Generating Cities with This Model
  10. Urban Sprawling
  11. Gaussian Distribution for Neighborhood Centers.
  12. Eigenstructure of the Covariance Matrix.
  13. Discussion

Key Result

Theorem 2

Given a finite family of contraction similarities $f_i(x) = \lambda_i x + t_i$, $i \in I$, with $|\lambda_i| < 1$, and a corresponding probability vector $(p_i)_{i \in I}$, there is a unique Borel probability measure $\mu$ such that The dimension of $\mu$ is then given by

Figures (5)

  • Figure 1: Recursive procedure for constructing the Manhattan model.
  • Figure 2: Example of $1000$ points generated randomly on the Manhattan grid with the measure $\mu$ for the parameters $p = 0.1, 0.3, 0.5, 0.8$.
  • Figure 3: Example of a fractal city with a Voronoi tiling where each network is a Manhattan grid.
  • Figure 4: Example of cities generated with covariance matrices $0.1000.1$ and $0.5000.5$.
  • Figure 5: City generated with covariance matrix $0.02000.7$.

Theorems & Definitions (10)

  • definition thmcounterdefinition
  • Example 1
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  • Theorem 2: Dimension of a self-similar measure geronimo
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • Theorem 3
  • Theorem 4
  • definition thmcounterdefinition