$\tilde{D}^{-\frac{1}{2}} \tilde{A} \tilde{D}^{-\frac{1}{2}}=\left\{\begin{array}{cccccc}\frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & 0 & 0 & \frac{1}{\sqrt{3}} & 0 \\\frac{1}{\sqrt{4}} & \frac{1}{\sqrt{4}} & \frac{1}{\sqrt{4}} & 0 & \frac{1}{\sqrt{4}} & 0 \\0 & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & 0 & 0 \\0 & 0 & \frac{1}{\sqrt{4}} & \frac{1}{\sqrt{4}} & \frac{1}{\sqrt{4}} & \frac{1}{\sqrt{4}} \\\frac{1}{\sqrt{4}} & \frac{1}{\sqrt{4}} & 0 & \frac{1}{\sqrt{4}} & \frac{1}{\sqrt{4}} & 0 \\0 & 0 & 0 & \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}}\end{array}\right\}\left\{\begin{array}{cccccc}\frac{1}{\sqrt{3}} & 0 & 0 & 0 & 0 & 0 \\0 & \frac{1}{\sqrt{4}} & 0 & 0 & 0 & 0 \\0 & 0 & \frac{1}{\sqrt{3}} & 0 & 0 & 0 \\0 & 0 & 0 & \frac{1}{\sqrt{4}} & 0 & 0 \\0 & 0 & 0 & 0 & \frac{1}{\sqrt{4}} & 0 \\0 & 0 & 0 & 0 & 0 & \frac{1}{\sqrt{2}}\end{array}\right\}$
$\tilde{D}^{-\frac{1}{2}} \tilde{A} \tilde{D}^{-\frac{1}{2}}=\left\{\begin{array}{cccccc}\frac{1}{3} & \frac{1}{\sqrt{12}} & 0 & 0 & \frac{1}{\sqrt{12}} & 0 \\\frac{1}{\sqrt{12}} & \frac{1}{4} & \frac{1}{\sqrt{12}} & 0 & \frac{1}{4} & 0 \\0 & \frac{1}{\sqrt{12}} & \frac{1}{3} & \frac{1}{\sqrt{12}} & 0 & 0 \\0 & 0 & \frac{1}{\sqrt{12}} & \frac{1}{4} & \frac{1}{4} & \frac{1}{\sqrt{8}} \\\frac{1}{\sqrt{12}} & \frac{1}{4} & 0 & \frac{1}{4} & \frac{1}{4} & 0 \\0& 0& 0& \frac{1}{\sqrt{8}} & 0 & \frac{1}{2}\end{array}\right\}$
2.1 Spectral graph convolutions 2.1.1 foundation of Spectral graph convolutionsWe consider spectral convolutions on graphs：
假设 $x$ 是特征函数，$g$ 是卷积核，则图卷积为：
$(g * x)=F^{-1}[F[g] \odot F[x]]$
$\left(g * x\right)=U\left(U^{T} x \odot U^{T} g\right)=U\left(U^{T} g \odot U^{T} x\right)$
把 $U^{T} g$ 整体看作可学习的卷积核 :
$U^{T} g=\left[\begin{array}{c}\hat{g}_{\theta}\left(\lambda_{0}\right) \\\hat{g}_{\theta}\left(\lambda_{1}\right) \\\ldots \\\hat{g}_{\theta}\left(\lambda_{n-1}\right)\end{array}\right]$
其中 $\theta$ 为 $g$ 的参数。
则可得：
$\begin{array}{l}\left(U^{T} g\right) \odot\left(U^{T} x\right)&=\left[\begin{array}{c}\hat{g}_{\theta}\left(\lambda_{0}\right) \\\hat{g_{\theta}}\left(\lambda_{1}\right) \\\cdots \\\hat{g_{\theta}}\left(\lambda_{n-1}\right)\end{array}\right] \odot\left[\begin{array}{c}\hat{x}\left(\lambda_{0}\right) \\\hat{x}\left(\lambda_{1}\right) \\\cdots \\\hat{x}\left(\lambda_{n-1}\right)\end{array}\right]\\&=\left[\begin{array}{c}\hat{g}_{\theta}\left(\lambda_{0}\right) \cdot \hat{x}\left(\lambda_{0}\right) \\\hat{g}_{\theta}\left(\lambda_{1}\right) \cdot \hat{x}\left(\lambda_{1}\right) \\\cdots \\\hat{g}_{\theta}\left(\lambda_{n-1}\right) \cdot \hat{x}\left(\lambda_{n-1}\right)\end{array}\right]\\&=\left[\begin{array}{cccc}\hat{g}_{\theta}\left(\lambda_{0}\right) & 0 & \cdots & 0 \\0 & \hat{g}_{\theta}\left(\lambda_{1}\right) & \cdots & 0 \\\vdots & \vdots & \ddots & \vdots \\0 & 0 & \cdots & \hat{g}_{\theta}\left(\lambda_{n-1}\right)\end{array}\right] \cdot\left[\begin{array}{c}\hat{x}\left(\lambda_{0}\right) \\\hat{x}\left(\lambda_{1}\right) \\\cdots \\\hat{x}\left(\lambda_{n-1}\right)\end{array}\right]\\&=g_{\theta}(\Lambda)U^{T} x\end{array}$
最终图上的卷积公式是: