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 (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.
Highlights
Many complex systems have a natural embedding in a lowdimensional space or are otherwise influenced by space, and it is often insightful to study such spatial complex systems using the formalism of networks [1,2]
By using methods for computing persistent homology that take spatial information into account, we presented several applications of topological data analysis to spatial networks
We showed that topological methods are capable of characterizing network structures and detecting structural differences in images of various spatial systems
Summary
Many complex systems have a natural embedding in a lowdimensional space or are otherwise influenced by space, and it is often insightful to study such spatial complex systems using the formalism of networks [1,2]. One confounding factor in the use of PH to study spatial networks is that PH is able to capture information across scales, traditional distance-based PH constructions can have difficulty with applications in which differences in scale may not be meaningful. In a recent paper [21], we examined the shape of voting patterns in the state of California and observed that traditional methods for computing PH are more likely to capture disparities in population density than to detect the presence of interesting voting patterns To address this issue, we developed two PH constructions—one based on network adjacency and one based on the physical geometry of a map—that were successful at capturing these voting patterns. A public repository of the code that we use for our computations is available at Ref. [49]
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