Recent advancements in graph learning have revolutionized the way to understand and analyze data with complex structures. Notably, Graph Neural Networks (GNNs), i.e. neural network architectures designed for learning graph representations, have become a popular paradigm. With these models being usually characterized by intuition-driven design or highly intricate components, placing them within the theoretical analysis framework to distill the core concepts, helps understand the key principles that drive the functionality better and guide further development. Given this surge in interest, this article provides a comprehensive summary of the theoretical foundations and breakthroughs concerning the approximation and learning behaviors intrinsic to prevalent graph learning models. Encompassing discussions on fundamental aspects such as expressiveness power, generalization, optimization, and unique phenomena such as over-smoothing and over-squashing, this piece delves into the theoretical foundations and frontier driving the evolution of graph learning. In addition, this article also presents several challenges and further initiates discussions on possible solutions.

翻译：图学习的最新进展彻底改变了理解和分析复杂结构数据的方式。值得注意的是，图神经网络（GNNs），即为学习图表示而设计的神经网络架构，已成为一种主流范式。由于这些模型通常具有直觉驱动的设计或高度复杂的组件，将其置于理论分析框架中以提炼核心概念，有助于更好地理解驱动其功能的关键原理并指导进一步发展。鉴于这一研究热潮，本文全面总结了关于主流图学习模型内在近似与学习行为的理论基础和突破性进展。涵盖表达能力、泛化性、优化等基本方面，以及过度平滑和过度挤压等独特现象的讨论，本文深入探讨了推动图学习演进的理论基础与前沿。此外，本文还提出了若干挑战，并进一步引发了对可能解决方案的讨论。