Recent studies reveal the connection between GNNs and the diffusion process, which motivates many diffusion-based GNNs to be proposed. However, since these two mechanisms are closely related, one fundamental question naturally arises: Is there a general diffusion framework that can formally unify these GNNs? The answer to this question can not only deepen our understanding of the learning process of GNNs, but also may open a new door to design a broad new class of GNNs. In this paper, we propose a general diffusion equation framework with the fidelity term, which formally establishes the relationship between the diffusion process with more GNNs. Meanwhile, with this framework, we identify one characteristic of graph diffusion networks, i.e., the current neural diffusion process only corresponds to the first-order diffusion equation. However, by an experimental investigation, we show that the labels of high-order neighbors actually exhibit monophily property, which induces the similarity based on labels among high-order neighbors without requiring the similarity among first-order neighbors. This discovery motives to design a new high-order neighbor-aware diffusion equation, and derive a new type of graph diffusion network (HiD-Net) based on the framework. With the high-order diffusion equation, HiD-Net is more robust against attacks and works on both homophily and heterophily graphs. We not only theoretically analyze the relation between HiD-Net with high-order random walk, but also provide a theoretical convergence guarantee. Extensive experimental results well demonstrate the effectiveness of HiD-Net over state-of-the-art graph diffusion networks.

翻译：近期研究揭示了图神经网络（GNNs）与扩散过程之间的关联，这促使大量基于扩散的GNNs被提出。然而，由于这两种机制紧密相关，一个基本问题自然浮现：是否存在一个通用的扩散框架能够正式统一这些GNNs？回答这一问题不仅能深化我们对GNNs学习过程的理解，还可能为设计全新类别的GNNs打开大门。本文提出了一种带有保真项的广义扩散方程框架，正式建立了扩散过程与更多GNNs之间的联系。同时，借助该框架，我们识别出图扩散网络的一个特征：当前神经扩散过程仅对应于一阶扩散方程。但通过实验研究，我们发现高阶邻居的标签实际上表现出"单亲性"（monophily property），即基于标签的高阶邻居间相似性并不要求一阶邻居间存在相似性。这一发现促使我们设计了一种新的高阶邻居感知扩散方程，并基于该框架推导出一种新型图扩散网络（HiD-Net）。凭借高阶扩散方程，HiD-Net对攻击具有更强的鲁棒性，并能在同质性与异质性图上均有效工作。我们不仅从理论上分析了HiD-Net与高阶随机游走之间的关系，还提供了理论收敛性保证。大量实验结果表明，HiD-Net在性能上显著优于当前最先进的图扩散网络。