graph convolutional network

[[concept]]

Graph Convolutional Networks

Introduced by Kipf and Willing, 2017, a graph convolutional network layer is given by $(x_{\ell }{)}_i=\sigma \left[{\sum }_{j\in N(i)}\frac{(x_{\ell -1}{)}_j}{|N(i)|}{H}_{\ell }\right]$ Note that $H_{\ell }$ are row vectors. We can think of each layer as a “degree normalized aggregation”

Note

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.

see also GNN

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