  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 (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 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.
