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Fundamental Limits and Efficient Algorithms for the Graph Alignment Problem

Core Concepts
This thesis focuses on statistical inference in graphs, particularly the graph alignment problem, which aims to recover a hidden underlying matching between the nodes of two correlated random graphs.
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.
"The exact recovery task exhibits a sharp information-theoretic threshold." "Only a fraction of the nodes can be correctly matched by any algorithm in the sparse Erdős-Rényi graph alignment regime."
"While most of the recent work on the subject was dedicated to recovering the hidden signal in dense graphs, we next explore graph alignment in the sparse regime, where the mean degree of the nodes are constant, not scaling with the graph size." "Our second contribution is an information-theoretical result which characterizes a regime where even this partial alignment is impossible, and gives upper bounds on the reachable overlap between any estimator and the true planted matching."

Deeper Inquiries

How do the results and algorithms presented in this thesis generalize to other types of random graph models beyond Erdős-Rényi and Gaussian?

The results and algorithms presented in the thesis can be generalized to other types of random graph models beyond Erdős-Rényi and Gaussian by adapting the underlying principles and methodologies to suit the specific characteristics of the new models. For example, the techniques used for graph alignment in Erdős-Rényi graphs can be extended to other models with different edge probabilities or correlations. The spectral methods and message-passing algorithms developed for alignment tasks can be applied to various random graph models by adjusting the parameters and assumptions accordingly. Furthermore, the information-theoretic limits and computational hardness results obtained for Erdős-Rényi and Gaussian models can serve as a foundation for analyzing similar problems in different graph structures. By understanding the fundamental limits and complexities of inference tasks in these models, researchers can apply similar frameworks to study inference problems in a wide range of random graph models, such as preferential attachment networks, small-world networks, or community-structured graphs.

What are the implications of the impossibility results for partial graph alignment on the design of practical algorithms?

The impossibility results for partial graph alignment, which characterize regimes where even partial alignment is impossible and provide upper bounds on the achievable overlap between estimators and the true planted matching, have significant implications for the design of practical algorithms in graph inference tasks. These results highlight the inherent challenges and limitations in recovering hidden structures in sparse graphs, where the mean degree of nodes remains constant. One implication is the need for algorithm designers to carefully consider the trade-offs between computational efficiency and the achievable accuracy of partial alignment. In regimes where partial alignment is impossible, algorithms may need to focus on identifying local structures or approximate solutions rather than aiming for exact recovery. This could lead to the development of heuristic methods or approximation algorithms that prioritize speed and scalability over optimality in sparse graph alignment tasks. Additionally, the impossibility results can guide the development of robust algorithms that incorporate uncertainty and noise in the inference process. By acknowledging the limitations of partial alignment in certain regimes, algorithms can be designed to provide reliable estimates and confidence intervals rather than striving for perfect matching in challenging scenarios.

How can the insights from the correlation detection in trees problem be further leveraged to tackle more complex graph inference tasks?

The insights from correlation detection in trees can be leveraged to tackle more complex graph inference tasks by providing a foundation for understanding the relationships and dependencies between nodes in random graphs. By studying correlation detection in trees, researchers can gain valuable insights into the underlying structures and patterns that exist in graph data, which can be extended to more intricate graph inference problems. One way to leverage these insights is by developing novel similarity measures and clustering techniques based on tree matching weights and correlation detection algorithms. These methods can be applied to identify communities, detect anomalies, or classify nodes in complex graphs with unknown structures. By leveraging the principles of correlation detection in trees, researchers can design efficient algorithms for graph partitioning, community detection, and anomaly detection in various types of networks. Furthermore, the study of correlation detection in trees can inspire the development of probabilistic models and hypothesis testing frameworks for analyzing graph data. By applying statistical methods and machine learning techniques informed by tree correlation detection, researchers can enhance the accuracy and robustness of graph inference tasks, leading to more reliable and interpretable results in real-world applications.