Graph Neural Networks

A "full-fledged" GNN layer is a multi-dimensional generalization of the multi-layer graph perceptron given by

$H_{\ell ,k}\in {\mathbb{R}}^{d_{\ell -1}\times d_{\ell }}$ for $1\le \ell \le L$, where

• $X_0=X\in {\mathbb{R}}^{n\times d_0}$
• $Y=X_L\in {\mathbb{R}}^{n\times d_{\ell }}$
• $\sigma :\mathbb{R}\to \mathbb{R}$ is a pointwise nonlinearity
• and the multiplication is a convolutional filter bank

Here, $X_{\ell }\in {\mathbb{R}}^{n\times d_{\ell }}$ is still called an $\ell$-layer embedding.

These GNNs are sometimes called convolutional GNNs because they are based on graph convolution.

$$X_{\ell }=\sigma (U_{\ell })=\sigma \left(\sum _{k=0}^{K-1}{S}^k{X}_{\ell -1}{H}_{\ell ,k}\right)$$

^layer-equation