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Spatial networks are ubiquitous in social, geographical, physical, andbiological applications. To understand the large-scale structure of networks,it is important to develop methods that allow one to directly probe the effectsof space on structure and dynamics. Historically, algebraic topology hasprovided one framework for rigorously and quantitatively describing the globalstructure of a space, and recent advances in topological data analysis (TDA)have given scholars a new lens for analyzing network data. In this paper, westudy a variety of spatial networks -- including both synthetic and naturalones -- using novel topological methods that we recently developed foranalyzing spatial networks. We demonstrate that our methods are able to capturemeaningful quantities, with specifics that depend on context, in spatialnetworks and thereby provide useful insights into the structure of thosenetworks, including a novel approach for characterizing them based on theirtopological structures. We illustrate these ideas with examples of syntheticnetworks and dynamics on them, street networks in cities, snowflakes, and websspun by spiders under the influence of various psychotropic substances.

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