Graph Convolutional Network

Spatial-based convolutional graph neural networks, or GCNs for short, are a common type of graph neural network.

• They are convolutional in that they update each node by aggregating information from nearby nodes. As such, they induce a relational inductive bias, prioritizing information from neighbors.
• They are spatial-based because they use the original graph structure. This contrasts with spectral-based methods, which apply convolutions in the Fourier domain.

Each layer of the GCN is a function $\mathbf{F}[\bullet ]$ with parameters $\varphi$ that takes the node embeddings and adjacency matrix and outputs new node embeddings. The network can hence be written as:

$$\begin{array}{rl}{\mathbf{H}}_1& =\mathbf{F}[\mathbf{X},\mathbf{A},{\varphi }_0]\\ {\mathbf{H}}_2& =\mathbf{F}[{\mathbf{H}}_1,\mathbf{A},{\varphi }_1]\\ {\mathbf{H}}_3& =\mathbf{F}[{\mathbf{H}}_1,\mathbf{A},{\varphi }_2]\\ ⋮& =⋮\\ {\mathbf{H}}_K& =\mathbf{F}[{\mathbf{H}}_{K-1},\mathbf{A},{\varphi }_{K-1}]\end{array}$$

where $\mathbf{X}$ is the input, $\mathbf{A}$ is the adjacency matrix, ${\mathbf{H}}_k$ contains the modified node embeddings at the $k$-th layer, and ${\varphi }_k$ denotes the parameters that map from layer $k$ to $k+1$.