空域图卷积模型

回顾经典卷积的操作：采样（即构建邻域）+聚合（聚合邻居结点的信息）。将固定数量的邻域结点排序后，与相同数量的卷积核参数相乘求和。

对于图结构数据如何定义卷积操作？

　　1）构建邻域；

　　2）对邻域的点与卷积核参数内积；

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GNN，构建邻域的大小为p，p个固定数量的卷积核参数。GNN使用随机游走的方法，为每个结点选取了最紧密相连的p个结点作为邻域，然后与固定大小的卷积核参数进行内积。

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Comparison Between Spectral and Spatial Models [1]:

Spatial models are preferred over spectral models due to efficiency, generality, and flexibility issues.

 

为什么空域图卷积更受到大家的关注？原因：效率，泛化性，灵活性。

First, spectral models are less efficient than spatial models. Spectral models either need to perform eigenvector computation or handle the whole graph at the same time.  Spatial models are more scalable to large graphs as they directly perform convolutions in the graph domain via information propagation. The computationcan be performed in a batch of nodes instead of the whole graph.

1、谱域图卷积需要处理整张图的Laplace矩阵，同时谱分解是高复杂度的。空余图卷积则更适合于大规模的图，因为其直接再结点域执行卷积操作（信息的传播）。而且是针对每一个节点，具有局部性。

Second, spectral models that rely on a graph Fourier basis generalize poorly to new graphs. They assume a fixed graph. Any perturbations to a graph would result in a change of eigenbasis. Spatial-based models, on the other hand, perform graph convolutions locally on each node, where weights can be easily shared across different locations and structures.

2、谱域图卷积高度依赖于Laplace矩阵的图傅里叶变换正交基，因此，只能针对固定的图。空域图卷积则针对节点的邻域进行卷积操作（信息的加权聚合），同时也可以实现参数共享。

Third, spectral-based models are limited to operate on undirected graphs. Spatial-based models are more flexible to handle multisource graph inputs, such as edge inputs directed graphs  signed graphs  and heterogeneous graphs  because these graph inputs can be incorporated into the aggregation function easily.

3、由于Laplace矩阵的特征值分解的需要，因此，谱域图卷积只能针对于无向图（保证Laplace矩阵为对称半正定矩阵，从而一定可以谱分解），但空域图卷积不存在这样的约束，可以处理更多类型的输入，更加的灵活。

 

[1]: A Comprehensive Survey on Graph Neural Networks