The long-term progression of neurodegenerative diseases is commonly conceptualized as a spatiotemporal diffusion process that consists of a graph diffusion process across the structural brain connectome and a localized reaction process within brain regions. However, modeling this progression remains challenging due to 1) the scarcity of longitudinal data obtained through irregular and infrequent subject visits and 2) the complex interplay of pathological mechanisms across brain regions and disease stages, where traditional models assume fixed mechanisms throughout disease progression. To address these limitations, we propose a novel stage-aware Mixture of Experts (MoE) framework that explicitly models how different contributing mechanisms dominate at different disease stages through time-dependent expert weighting. This architecture is a key innovation designed to maximize the utility of small datasets and provide interpretable insights into disease etiology. Data-wise, we utilize an iterative dual optimization method to properly estimate the temporal position of individual observations, constructing a cohort-level progression trajectory from irregular snapshots. Model-wise, we enhance the spatial component with an inhomogeneous graph neural diffusion model (IGND) that allows diffusivity to vary based on node states and time, providing more flexible representations of brain networks. We also introduce a localized neural reaction module to capture complex dynamics beyond standard processes.The resulting IGND-MoE model dynamically integrates these components across temporal states, offering a principled way to understand how stage-specific pathological mechanisms contribute to progression. When used to model tau pathology propagation in human brains, IGND-MoE outperforms purely pathophysiological and purely neural baselines in long-term prediction accuracy. Moreover, its stage-wise weights yield novel clinical insights that align with literature, suggesting that graph-related processes are more influential at early stages, while other unknown physical processes become dominant later on. Our findings highlight the necessity of designing hybrid and expert-constrained models that account for the evolving nature of neurodegenerative processes.

Inferring the graph structure from observed data is a key task in graph machine learning to capture the intrinsic relationship between data entities. While significant advancements have been made in learning the structure of homogeneous graphs, many real-world graphs exhibit heterogeneous patterns where nodes and edges have multiple types. This paper fills this gap by introducing the first approach for heterogeneous graph structure learning (HGSL). To this end, we first propose a novel statistical model for the data-generating process (DGP) of heterogeneous graph data, namely hidden Markov networks for heterogeneous graphs (H2MN). Then we formalize HGSL as a maximum a-posterior estimation problem parameterized by such DGP and derive an alternating optimization method to obtain a solution together with a theoretical justification of the optimization conditions. Finally, we conduct extensive experiments on both synthetic and real-world datasets to demonstrate that our proposed method excels in learning structure on heterogeneous graphs in terms of edge type identification and edge weight recovery.

Hypergraphs are crucial for modelling higher-order interactions in real-world data. Hypergraph neural networks (HNNs) effectively utilise these structures by message passing to generate informative node features for various downstream tasks like node classification. However, the message passing module in existing HNNs typically requires a computationally intensive training process, which limits their practical use. To tackle this challenge, we propose an alternative approach by decoupling the usage of hypergraph structural information from the model learning stage. This leads to a novel training-free message passing module, named TF-MP-Module, which can be precomputed in the data preprocessing stage, thereby reducing the computational burden. We refer to the hypergraph neural network equipped with our TF-MP-Module as TF-HNN. We theoretically support the efficiency and effectiveness of TF-HNN by showing that: 1) It is more training-efficient compared to existing HNNs; 2) It utilises as much information as existing HNNs for node feature generation; and 3) It is robust against the oversmoothing issue while using long-range interactions. Experiments based on seven real-world hypergraph benchmarks in node classification and hyperlink prediction show that, compared to state-of-the-art HNNs, TF-HNN exhibits both competitive performance and superior training efficiency. Specifically, on the large-scale benchmark, Trivago, TF-HNN outperforms the node classification accuracy of the best baseline by 10% with just 1% of the training time of that baseline.

Message Passing Neural Networks (MPNNs) are a class of Graph Neural Networks (GNNs) that propagate information across the graph via local neighborhoods. The scheme gives rise to two key challenges: over-smoothing and over-squashing. While several Dropout-style algorithms, such as DropEdge and DropMessage, have successfully addressed over-smoothing, their impact on over-squashing remains largely unexplored. This represents a critical gap in the literature, as failure to mitigate over-squashing would make these methods unsuitable for long-range tasks – the intended use case of deep MPNNs. In this work, we study the aforementioned algorithms, and closely related edge-dropping algorithms – DropNode, DropAgg and DropGNN – in the context of over-squashing. We present theoretical results showing that DropEdge-variants reduce sensitivity between distant nodes, limiting their suitability for long-range tasks. To address this, we introduce DropSens, a sensitivity-aware variant of DropEdge, which is developed following the message-passing scheme of GCN. DropSens explicitly controls the proportion of information lost due to edge-dropping, thereby increasing sensitivity to distant nodes despite dropping the same number of edges. Our experiments on long-range synthetic and real-world datasets confirm the predicted limitations of existing edge-dropping and feature-dropping methods. Moreover, DropSens with GCN consistently outperforms graph rewiring techniques designed to mitigate over-squashing, suggesting that simple, targeted modifications can substantially improve a model’s ability to capture long-range interactions. Our conclusions highlight the need to re-evaluate and re-design existing methods for training deep GNNs, with a renewed focus on modelling long-range interactions. The code for reproducing the results in our this work is available at https://github.com/ignasa007/Dropout-Effects-GNNs.