The thesis consists of several chapters that explore different aspects of the graph alignment problem:
Chapter 1 provides a general introduction to inference on random graphs, including an overview of the graph alignment problem and related concepts like correlation detection in random trees.
Chapter 2 investigates the information-theoretic limits for exact alignment in the Gaussian setting, where the graphs are complete and the signal lies on correlated Gaussian edge weights. The author proves a sharp fundamental threshold for the exact recovery task.
Chapter 3 studies a simple and natural spectral method for graph alignment in the Gaussian setting, providing theoretical guarantees for this algorithm.
Chapter 4 focuses on the sparse Erdős-Rényi graph alignment regime, where the mean degree of the nodes is constant. The author proves an information-theoretical result characterizing a regime where even partial alignment is impossible.
Chapter 5 proposes an algorithm for sparse graph alignment based on a measure of similarity between tree-like neighborhoods of the nodes, called the tree matching weight. The author also studies the related problem of correlation detection in random unlabeled trees.
Chapter 6 further explores the correlation detection in random trees problem, deriving an optimal test based on the likelihood ratio and characterizing regimes of performance. The author then proposes a message-passing algorithm for graph alignment inspired by the tree correlation detection results.
Chapter 7 presents recent improvements on the correlation detection in trees problem, providing a general understanding of the fundamental limits.
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by Luca Ganassa... at arxiv.org 04-22-2024
https://arxiv.org/pdf/2404.12418.pdfDeeper Inquiries