Abstract

This dissertation investigates the intersection of topological data analysis and network science, applying and extending persistent homology, Betti numbers, and simpliciality measures to study temporal, evolving, and coevolving networked systems. We introduce a multi-layer zigzag persistent homology framework for analyzing temporal hypergraphs across multiple timescales simultaneously, and validate it on the DARPA Operationally Transparent Cyber dataset, where it produces topological signatures that distinguish malicious from benign network activity by source IP. We then apply topological tools to classical generative network models and their hypergraph extensions, characterizing how Betti numbers, filling efficiencies, and simpliciality measures evolve as these models grow or their parameters vary. Applying the same methodology to the LittleSIS political-financial network database reveals a robust topological transition in the inter-organizational network around 2008 that persists after the removal of a data artifact, consistent with structural reorganization following the global financial crisis. We further introduce a generalized preferential attachment hypergraph model that allows for varied hyperedge sizes and node additions, and prove analytically that its stationary hyperdegree distribution follows a power law whose exponent depends only on the ratio of the expected number of new nodes to the expected hyperedge size, independent of the shapes of the underlying distributions. Fitting this model to real-world hypergraph datasets demonstrates that preferential attachment encourages the formation of simplicial structures, with its importance modulated by the degree to which simpliciality is already encoded in the hyperedge size and degree distributions. Finally, we introduce a coevolving voter model on graphs and hypergraphs using a double-edge swap rewiring mechanism that preserves the degree sequence exactly, isolating the structural effects of the rewiring rule from degree heterogeneity. The initial graph topology is found to determine which rewiring mechanism produces the richest higher-order terminal topology, with structured lattice and hub-dominated initial conditions amplifying complementary rewiring regimes. In the hypergraph setting, majority and proportional voting produce qualitatively distinct consensus-to-fragmentation transitions, and triangle closure is found to reduce simpliciality more than opinion-driven rewiring, contrary to naive expectation.

Publication Date

6-16-2026

Document Type

Dissertation

Student Type

Graduate

Degree Name

Mathematical Modeling (Ph.D)

Department, Program, or Center

Mathematics and Statistics, School of

College

College of Science

Advisor

Brendan Rooney

Advisor/Committee Member

Ivona Bezáková

Advisor/Committee Member

Tony Wong

Campus

RIT – Main Campus

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