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）与扩散过程之间的内在联系，这促使大量基于扩散机制的GNN模型被提出。然而，由于这两种机制密切相关，一个根本问题自然浮现：是否存在一个能够形式化统一这些GNN模型的通用扩散框架？对该问题的解答不仅能深化我们对GNN学习过程的理解，还可能为设计全新的GNN类别开辟新途径。本文提出了一种带保真项（fidelity term）的通用扩散方程框架，该框架正式建立了扩散过程与更多GNN模型之间的关联。同时，基于该框架我们发现图扩散网络的一个特征：当前神经扩散过程仅对应一阶扩散方程。然而，通过实验探究证明，高阶邻居的标签实际上展现出单亲性（monophily property），这种性质使得基于标签的相似性能在高阶邻居间建立，而不要求一阶邻居间存在相似性。这一发现启发我们设计了一种新型高阶邻居感知扩散方程，并基于该框架推导出新一代图扩散网络（HiD-Net）。借助高阶扩散方程，HiD-Net对攻击更具鲁棒性，且能同时处理同配图（homophily graphs）与异配图（heterophily graphs）。我们不仅从理论上分析了HiD-Net与高阶随机游走的关系，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。