Part XV: Topological Data Analysis
Algebraic topology of urban form — simplicial homology, Betti numbers distinguishing five city types, persistent homology for multi-scale structure, and Mapper for topological skeletons. A rigorous mathematical lens for classifying and comparing urban morphologies.
Part Overview
Builds from the simplicial complex and boundary operator through homology groups \(H_k = \ker\partial_k / \text{im}\,\partial_{k+1}\) to Betti numbers \(\beta_0, \beta_1\) that classify urban forms. Extends to Vietoris-Rips persistent homology with stability guarantees, and the Mapper algorithm for extracting topological skeletons of high-dimensional urban data.
Key Topics
- • Simplicial complex and boundary operator
- • Homology groups \(H_k = \ker\partial_k / \text{im}\,\partial_{k+1}\)
- • Betti numbers \(\beta_0, \beta_1\) for urban forms
- • Vietoris-Rips persistent homology
- • Stability theorem
- • Mapper algorithm
- • Fortran union-find
3 chapters | From simplices to persistence diagrams | Topology reveals urban shape
Chapters
Chapter 1: Simplicial Homology
Constructs simplicial complexes from urban point clouds, defines the boundary operator and chain groups, and computes homology groups \(H_k = \ker\partial_k / \text{im}\,\partial_{k+1}\). Betti numbers \(\beta_0\) (connected components) and \(\beta_1\) (loops) distinguish five canonical city types.
Chapter 2: Persistent Homology
Vietoris-Rips filtration tracks how topological features are born and die across scales. Persistence diagrams and barcodes capture multi-scale urban structure, with the stability theorem guaranteeing robustness to noise in spatial data.
Chapter 3: Mapper Algorithm
The Mapper algorithm constructs topological skeletons of high-dimensional urban data via filter functions, covering, and clustering. Implemented with Fortran union-find for efficient connected component computation on large urban datasets.