While message-passing neural networks (MPNNs) have shown promising results, their real-world impact remains limited. Although various limitations have been identified, their theoretical foundations remain poorly understood, leading to fragmented research efforts. In this thesis, we provide an in-depth theoretical analysis and identify several key properties limiting the performance of MPNNs. Building on these findings, we propose several frameworks that address these shortcomings. We identify two properties exhibited by many MPNNs: shared component amplification (SCA), where each message-passing iteration amplifies the same components across all feature channels, and component dominance (CD), where a single component gets increasingly amplified as more message-passing steps are applied. These properties lead to the observable phenomenon of rank collapse of node representations, which generalizes the established over-smoothing phenomenon. By generalizing and decomposing over-smoothing, we enable a deeper understanding of MPNNs, more targeted solutions, and more precise communication within the field. To avoid SCA, we show that utilizing multiple computational graphs or edge relations is necessary. Our multi-relational split (MRS) framework transforms any existing MPNN into one that leverages multiple edge relations. Additionally, we introduce the spectral graph convolution for multiple feature channels (MIMO-GC), which naturally uses multiple computational graphs. A localized variant, LMGC, approximates the MIMO-GC while inheriting its beneficial properties. To address CD, we demonstrate a close connection between MPNNs and the PageRank algorithm. Based on personalized PageRank, we propose a variant of MPNNs that allows for infinitely many message-passing iterations, while preserving initial node features. Collectively, these results deepen the theoretical understanding of MPNNs.

翻译：尽管消息传递神经网络（MPNNs）已展现出有前景的结果，但其在实际应用中的影响力仍然有限。尽管已识别出多种局限性，其理论基础仍未被充分理解，导致相关研究呈现碎片化。本论文提供了深入的理论分析，并识别出限制MPNNs性能的几个关键特性。基于这些发现，我们提出了多个框架以应对这些不足。我们识别出许多MPNNs表现出的两种特性：共享成分放大（SCA），即每次消息传递迭代在所有特征通道中放大相同的成分；以及成分主导（CD），即随着应用更多消息传递步骤，单一成分被持续放大。这些特性导致了节点表示秩塌缩的可观测现象，该现象推广了已确立的过平滑现象。通过对过平滑现象进行推广和分解，我们得以更深入地理解MPNNs，提出更具针对性的解决方案，并促进领域内更精确的交流。为避免SCA，我们证明利用多个计算图或边关系是必要的。我们的多关系分割（MRS）框架可将任何现有MPNN转换为能够利用多种边关系的模型。此外，我们引入了针对多特征通道的谱图卷积（MIMO-GC），它自然地使用了多个计算图。其局部化变体LMGC在继承MIMO-GC有益特性的同时对其进行了近似。为应对CD，我们揭示了MPNNs与PageRank算法之间的紧密联系。基于个性化PageRank，我们提出了一种MPNNs的变体，该变体允许无限次消息传递迭代，同时保留初始节点特征。总体而言，这些成果深化了对MPNNs的理论理解。