Abstract
The analysis of network data, describing relationships, interactions and dependencies between entities, often begins with embedding: the process of mapping these entities into a low-dimensional vector space in a way which preserves salient information present in the data. Spectral embedding is a family of embedding algorithms in which representations are obtained using the eigenvectors of a specially designed matrix constructed from the network. It has emerged as a simple yet effective approach, which is both highly scalable and interpretable.In this thesis, we provide a statistical lens into spectral embedding, shining light on how various network structures manifest themselves as geometric patterns in the vector space, and how certain algorithmic choices influence the information which is extracted from the network. We present new methodology which exploits these insights and provide statistical theory, as well as simulated and real data studies to support them.
Chapter 2 introduces some statistical models for network data and reviews existing estimation theory for spectral embedding; Chapter 3 studies spectral embedding using the random walk Laplacian matrix, developing estimation theory which illuminates a key inferential difference between it and other popular matrix constructions; Chapter 4 elucidates the geometric structure which emerges in the spectral embeddings of multipartite networks and develops bespoke statistical methodology which exploits it for dimension reduction; and Chapter 5 presents an algorithmic framework for spectral embedding of dynamic networks which produces representations that evolve in continuous time and reflect the changing structural roles of the nodes in the network.
| Date of Award | 23 Jan 2024 |
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| Original language | English |
| Awarding Institution |
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| Supervisor | Patrick Rubin-Delanchy (Supervisor) |