理解图卷积：图神经网络的基础构建与设计选择

中文摘要

本文探讨了图神经网络（GNN）的核心组件——图卷积的构建原理与设计方法。图卷积是处理非欧几里得数据结构（如社交网络、分子图）的关键技术，通过将传统卷积操作扩展到图域，实现对节点特征的高效聚合与传播。文章系统分析了图卷积的数学基础，包括邻接矩阵的归一化处理、消息传递机制以及不同聚合函数（如均值、最大值、求和）的适用场景。同时，对比了谱域方法与空间域方法的优劣，并讨论了现代GNN架构（如GCN、GAT、GraphSAGE）的设计选择。最后，文章总结了图卷积在节点分类、链接预测、图分类等任务中的应用挑战与未来研究方向，为深度学习在复杂关系数据上的应用提供了重要理论支撑。

English Summary

Understanding Convolutions on Graphs: Building Blocks and Design Choices of Graph Neural Networks

This article delves into the foundational principles and design methodologies of graph convolutions, the core component of graph neural networks (GNNs). As a key technique for processing non-Euclidean data structures (e.g., social networks, molecular graphs), graph convolution extends traditional convolution operations to graph domains, enabling efficient aggregation and propagation of node features. The paper systematically examines the mathematical foundations of graph convolutions, covering adjacency matrix normalization, message-passing mechanisms, and the applicability of different aggregation functions (e.g., mean, max, sum). It compares spectral-domain and spatial-domain approaches while analyzing design choices in modern GNN architectures (e.g., GCN, GAT, GraphSAGE). The study concludes with challenges in node classification, link prediction, and graph classification tasks, along with future research directions, providing crucial theoretical insights for deep learning applications on complex relational data.