graph convolutional network

Introduced by Kipf and Willing, 2017, a graph convolutional network layer is given by

Note that $H_{\ell }$ are row vectors. We can think of each layer as a "degree normalized aggregation"
^def

Advantages of GCNs:

• One local diffusion step per layer
• Simple and interpretable
• Neighborhood size normalization prevents vanishing or exploding gradient problem - only one diffusion step

Disadvantages

• no edge weights
• only supports the adjacency $S=A$
• No self loops unless they are present in $A$ (embedding at layer $\ell$ not informed by the embedding at layer $\ell -1$). Even if self-loops are in the graph, there is no ability to weight $(x_{\ell }{)}_i$ and $(x_{\ell -1}{)}_j,j\in N(i)$ differently.