On deep learning for graph-structured data

Graphs are a powerful and flexible data structure for representing relational data and are widely used to describe complex real-world systems. Graph Neural Networks (GNNs) have emerged as the deep learning paradigm for such relational data, yet their inherent limitations and advantages over alternative models are under investigation. This thesis provides a multifaceted interrogation of GNNs to clarify their capabilities and boundaries. First, to enhance the interpretability of neural networks, we develop a thermodynamic framework that analyzes weight dynamics during training and quantifies parameter importance through gradient activity. This framework enables structured pruning and we apply it to prune input features in GNNs, identifying prunable features as those that induce low mean squared gradient on the weights. We further pioneer the use of GNNs for the complex task of 3D multi-phase microstructure segmentation, a domain typically dominated by Convolutional Neural Networks. Our results demonstrate that the relational inductive bias of GNNs enables effective handling of irregular geometries and offers superior adaptability in data-scarce experimental settings. Finally, we address a critical gap in the literature by conducting a systematic comparison between GNNs and the interpretable paradigm of Probabilistic Graphical Models (PGMs). We show that PGMs are more robust in real-world scenarios characterized by low-dimensional or noisy node features and complex graph connectivity, such as heterophily. GNNs, in contrast, excel when high-dimensional features are available, and we find that their performance on heterophilic graphs significantly improves when features are strategically selected to incorporate information about the graph structure.