Date of Award

2025

Document Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematical Sciences

First Advisor

Alice Patania

Abstract

The study of higher-order structures in complex networks has become an increasingly important topic recently for applications in neuroscience, where traditional methods based on graph theory - often limited to pairwise interactions - fail to capture the full richness of neural connectivity. Topological data analysis (TDA) offers an alternative framework, providing tools to extract quantitative local descriptors which help us to understand global structural features such as loops and voids in data. Traditional spectral methods use the graph Laplacian, but recently, the Hodge Laplacian has been used for uncovering topological and spectral properties across multiple scales.

In this thesis, we examine the spectral decomposition of the Hodge Laplacian using alpha-filtrations on simplicial complexes derived from a point cloud. We classify the resulting eigenvectors into harmonic, gradient and curl spaces and compare these structural features with the dynamical behaviour of the system.

The work is built on the framework by Vincent P. Grande and Michael T. Schaub, where, unlike traditional eigenvalue-based clustering methods, an approach to explicitly track eigenvectors across different steps of filtration is used. This gives a better insight into the structural properties of the simplicial complexes.

To represent the dynamics of the system we are interested in, we implement the Kuramoto model, a well-studied model of a network of coupled oscillators. Our results establish a connection between the Hodge spectral properties observed and the synchronization dynamics, revealing that spectral topological features have a notable correlation with the synchronization behaviour of the system. This study provides a framework for comparing spectral topology with nonlinear dynamics of a network.

Language

en

Number of Pages

62 p.

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