Statistical Topology of Embedded Graphs

Abstract

Complex systems with interesting topology and geometry abound in the physical and mathematical sciences. This thesis explores several methods for their analysis, namely swatches, persistent homology, and the knotting of embedded graphs. We begin by describing curvature flow on embedded graphs, and methods for simulating it in two and three dimensions. This complex system will serve as a case study for the remainder of the thesis.

The method of swatches characterizes the local topology of a regular cell complex in terms of probability distributions of local configurations. It is used to define a distance on cell complexes, which has many applications. Convergence in this distance is related to the concept of a Benjamini-Schramm graph limit, and we use it to formalize a universality conjecture about the long-term behavior of curvature-flow on graphs.

Persistent homology is a powerful tool for studying complex geometric structures. It establishes a correspondence between geometric features of a subset of a metric space and topological features of its $\epsilon$-neighborhoods. This correspondence is represented by a scatter plot of feature points, which may be used to define many interesting statistics including an analogue of fractal dimension. We also describe a tree structure for the $(n-1)$-dimensional persistent homology of a subset of $n$-dimensional space, and apply it to compare two types of embedded graphs.

We conclude by proposing a definition for unknotted embedded graphs, and examining the relationship between curvature flow and knottedness in graphs.