  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.
