Spatial Applications of Topological Data Analysis: Cities, Snowflakes, Random Structures, and Spiders Spinning Under the Influence

Abstract

Spatial networks are ubiquitous in social, geographic, physical, and biological applications. To understand their large-scale structure, 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 (TDA) 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 novel topological methods that we recently developed specifically for analyzing spatial networks. 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, including a novel approach for characterizing them based on their topological structures. We illustrate these ideas with examples of synthetic networks and dynamics on them, street networks in cities, snowflakes, and webs spun by spiders under the influence of various psychotropic substances.

Topological Machine Learning Methods for Power System Responses to Contingencies

“…Essentially, the application of AI tools for resilience quantification of power grid networks is essentially in its infancy. Topological Data Analysis (TDA) The recent decade has witnessed a steep rise in implementation of TDA machinery in diverse application settings including image detection (Asaad and Jassim 2017), system robustness analysis (Li, Ryerson, and Balakrishnan 2019) and spatial classification (Feng and Porter 2020). Most recently, Islambekov et al (2018); Li et al (2020) considered topological summaries as alternative metrics for transmission grid resilience assessment.…”

Hypergraphs for predicting essential genes using multiprotein complex data

“…Other mathematical structures are potentially also suited to represent highorder interactions. Simplicial complexes, for example, have been used to investigate time-series [14,15], and many other systems [16,17,18]. For our purposes, we argue that hypergraphs are a more suited mathematical framework because simplicial complexes require set inclusion 1 , which for our application implied that for every multiprotein complex, all subsets of constituent proteins would also form a multiprotein complex, which in general is not the case.…”

The shape of cancer relapse: Topological data analysis predicts recurrence in paediatric acute lymphoblastic leukaemia

“…PH is readily computable ( 22 ), robust to noise ( 23 ) and its outputs are interpretable. In recent years, improved computational feasibility of PH has increased its applications to (high-dimensional) data in many contexts, including studies of the shape of brain arteries ( 25 ), neurons ( 26 ), the neural code ( 27 ), airways ( 28 ), stenosis ( 29 ), zebrafish patterns ( 30 ), ion aggregation ( 31 ), contagion dynamics ( 32 ), spatial networks ( 33,35–37 ), and geometric anomalies ( 38 ). In oncology, PH has been used to construct new biomarkers ( 35, 39, 40 ), to classify tumours ( 41, 42 ) and genetic alterations ( 43 ) and to quantify patterns of immune cell infiltration into solid tumours ( 44 ).…”