Date of Award

Spring 1-1-2025

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Gilbert, Anna

Abstract

This thesis bridges the ideas of hyperbolic geometry and geometric group theory with practical applications in graph theory and machine learning. A metric space $(X,d)$ is said to be Gromov $\delta$-hyperbolic if, for any points $x,y,z,w \in X$, the inequality $d(x,y)+d(z,w) \leq \max(d(x,z)+d(y,w), d(x,w)+d(y,z)) + 2 \delta$ holds. This definition generalizes key properties of hyperbolic spaces and serves as a fundamental measure of tree-likeness in a given space, making it a topic of interest across various fields. Recent advances propose evaluating the expectation of the hyperbolicity condition rather than relying solely on worst-case scenarios, offering a more global perspective on the geometric behavior of spaces. To complement this approach, we present a simple combinatorial tree fitting algorithm, demonstrating that for data sets with low average hyperbolicity, structured trees with low distortion can be constructed, thereby avoiding the high computational cost of previous methods. Beyond algorithmic contributions, we investigate the concept of average hyperbolicity. While multiple definitions of $\delta$-hyperbolicity exist, they are typically considered equivalent up to a constant factor in the traditional setting. We explore their average behavior, which has remained largely unexamined. Through counterexamples, we show that these average formulations are no longer equivalent, revealing fundamental discrepancies in their interpretations. We further illustrate this phenomenon using random graph models, suggesting that such inconsistencies may be prevalent in real-world data. Finally, we empirically demonstrate that analyzing average hyperbolicity provides a more nuanced understanding of geometric structures in real-world networks. Unlike traditional worst-case measures, which may overestimate deviations from small samples, average hyperbolicity better captures underlying structural properties in graphs. This insight is particularly relevant in modern machine learning and data set analysis, where understanding the geometric nature of input data can significantly impact model performance and interpretability.

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