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 oversquashing 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.

Graph coarsening aims to reduce the size of a large graph while preserving some of its key properties, which has been used in many applications to reduce computational load and memory footprint. For instance, in graph machine learning, training Graph Neural Networks (GNNs) on coarsened graphs leads to drastic savings in time and memory. However, GNNs rely on the Message-Passing (MP) paradigm, and classical spectral preservation guarantees for graph coarsening do not directly lead to theoretical guarantees when performing naive message-passing on the coarsened graph. In this work, we propose a new message-passing operation specific to coarsened graphs, which exhibit theoretical guarantees on the preservation of the propagated signal. Interestingly, and in a sharp departure from previous proposals, this operation on coarsened graphs is often oriented, even when the original graph is undirected. We conduct node classification tasks on synthetic and real data and observe improved results compared to performing naive message-passing on the coarsened graph.

Interestingly, and in a sharp departure from previous proposals, this operation on coarsened graphs is often oriented, even when the original graph is undirected.

Message Passing Neural Networks (MPNNs) are a class of Graph Neural Networks (GNNs) that leverage the graph topology to propagate messages across increasingly larger neighborhoods. The message-passing scheme leads to two distinct challenges: over-smoothing and over-squashing. DropEdge and its variants - DropNode, DropAgg and DropGNN - have successfully addressed the over-smoothing problem, 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 longrange tasks. In this work, we take the first step towards closing this gap by studying the aforementioned algorithms in the context of over-squashing. We present novel theoretical results that characterize the negative effects of DropEdge on sensitivity between distant nodes, suggesting its unsuitability for long-range tasks. Our findings are easily extended to its variants, allowing us to build a comprehensive understanding of how they affect over-squashing. We evaluate these methods using real-world datasets, demonstrating their detrimental effects. Specifically, we show that while DropEdge-variants improve test-time performance in short-range tasks, they deteriorate performance in long-range ones. Our theory explains these results as follows: random edge-dropping lowers the effective receptive field of GNNs, which although beneficial for short-range tasks, misaligns the models on long-range ones. This forces the models to overfit to short-range artefacts in the training set, resulting in poor generalization. Our conclusions highlight the need to re-evaluate various methods designed for training deep GNNs, with a renewed focus on modelling long-range interactions. Graph-structured data is ubiquitous - it is found in social media platforms, online retail platforms, molecular structures, transportation networks, and even computer systems.

Graph coarsening aims to reduce the size of a large graph while preserving some of its key properties, which has been used in many applications to reduce computational load and memory footprint. For instance, in graph machine learning, training Graph Neural Networks (GNNs) on coarsened graphs leads to drastic savings in time and memory. However, GNNs rely on the Message-Passing (MP) paradigm, and classical spectral preservation guarantees for graph coarsening do not directly lead to theoretical guarantees when performing naive message-passing on the coarsened graph. In this work, we propose a new message-passing operation specific to coarsened graphs, which exhibit theoretical guarantees on the preservation of the propagated signal. Interestingly, and in a sharp departure from previous proposals, this operation on coarsened graphs is oriented, even when the original graph is undirected. We conduct node classification tasks on synthetic and real data and observe improved results compared to performing naive message-passing on the coarsened graph.

The smoothing issue leads to indistinguishable node representations, which poses a significant challenge in the field of graph learning. However, this issue also presents an opportunity to reveal underlying properties behind different types of nodes, which have been overlooked in previous studies. Through empirical and theoretical analysis of real-world node anomaly detection (NAD) datasets, we observe that anomalous and normal nodes show different patterns in the smoothing process, which can be leveraged to enhance NAD tasks. Motivated by these findings, in this paper, we propose a novel unsupervised NAD framework. Specifically, according to our theoretical analysis, we design a Smoothing Learning Component. Subsequently, we introduce a Smoothing-aware Spectral Graph Neural Network, which establishes the connection between the spectral space of graphs and the smoothing process. Additionally, we demonstrate that the Dirichlet Energy, which reflects the smoothness of a graph, can serve as coefficients for node representations across different dimensions of the spectral space. Building upon these observations and analyses, we devise a novel anomaly measure for the NAD task. Extensive experiments on 9 real-world datasets show that SmoothGNN outperforms the best rival by an average of 14.66% in AUC and 7.28% in Precision, with 75x running time speed-up, which validates the effectiveness and efficiency of our framework.

This study addresses the limitations of the traditional analysis of message-passing, central to graph learning, by defining {\em \textbf{generalized propagation}} with directed and weighted graphs. The significance manifest in two ways. \textbf{Firstly}, we propose {\em Generalized Propagation Neural Networks} (\textbf{GPNNs}), a framework that unifies most propagation-based graph neural networks. By generating directed-weighted propagation graphs with adjacency function and connectivity function, GPNNs offer enhanced insights into attention mechanisms across various graph models. We delve into the trade-offs within the design space with empirical experiments and emphasize the crucial role of the adjacency function for model expressivity via theoretical analysis. \textbf{Secondly}, we propose the {\em Continuous Unified Ricci Curvature} (\textbf{CURC}), an extension of celebrated {\em Ollivier-Ricci Curvature} for directed and weighted graphs. Theoretically, we demonstrate that CURC possesses continuity, scale invariance, and a lower bound connection with the Dirichlet isoperimetric constant validating bottleneck analysis for GPNNs. We include a preliminary exploration of learned propagation patterns in datasets, a first in the field. We observe an intriguing ``{\em \textbf{decurve flow}}'' - a curvature reduction during training for models with learnable propagation, revealing the evolution of propagation over time and a deeper connection to over-smoothing and bottleneck trade-off.

Graph Neural Networks (GNNs) have achieved tremendous success in a variety of real-world applications by relying on the fixed graph data as input. However, the initial input graph might not be optimal in terms of specific downstream tasks, because of information scarcity, noise, adversarial attacks, or discrepancies between the distribution in graph topology, features, and groundtruth labels. In this paper, we propose a bi-level optimization approach for learning the optimal graph structure via directly learning the Personalized PageRank propagation matrix as well as the downstream semi-supervised node classification simultaneously. We also explore a low-rank approximation model for further reducing the time complexity. Empirical evaluations show the superior efficacy and robustness of the proposed model over all baseline methods.