Spatial applications of topological data analysis: Cities, snowflakes, random structures, and spiders spinning under the influence

Abstract

Spatial networks are ubiquitous in social, geographical, physical, and biological applications. To understand the large-scale structure of networks, it is important to develop methods that allow one to directly probe the effects of space on structure and dynamics. Historically, algebraic topology has provided one framework for rigorously and quantitatively describing the global structure of a space, and recent advances in topological data analysis have given scholars a new lens for analyzing network data. In this paper, we study a variety of spatial networks-including both synthetic and natural ones-using topological methods that we developed recently for analyzing spatial systems. We demonstrate that our methods are able to capture meaningful quantities, with specifics that depend on context, in spatial networks and thereby provide useful insights into the structure of those networks. We illustrate these ideas with examples of synthetic networks and dynamics on them, street networks in cities, snowflakes, and webs that were spun by spiders under the influence of various psychotropic substances.

The (homological) persistence of gerrymandering

“…Finally, in the example closest to our own approach, Feng and Porter introduce in [34] a filtration for producing a topological signature from districting and vote data which is similar to filtration we employ below. (See also the broader paper [35] for spatial applications.) The applications considered in [34] are quite different from those considered here: Feng and Porter primarily used topological signatures to study the vote patterns at the precinct level for fixed regions, whereas our goal is to understand the variation among districting plans, so that a single map does not suffice.…”

A persistent homology model of street network connectivity

“…In this article we propose a novel model of street network connectivity based on persistent homology. Feng and Porter (2020) previously used persistent homology to model street networks. However, their model does not model connectivity but instead models the shape of the areas enclosed by streets.…”

The right tools for the job: The case for spatial science tool‐building

“…Martínez Mori and Samaranayake (2019) model several cities’ street networks to empirically demonstrate heuristic approximation algorithms that make certain network analyses computationally tractable. Feng and Porter (2020) model cities around the world to explore their topology through persistent homology. Samson, Velez, Nobleza, Sanchez, and Milan (2018) model Filipino cities to develop a genetic algorithm for optimizing paratransit services in developing countries.…”