Graph Convolution Layer

Graph data—molecules, social networks, citations—has no fixed grid. A graph convolution layer averages each node's features with its neighbours, weighted by degree, to build representations that respect graph structure. The operation is matrix multiplication with a renormalised adjacency matrix.

The idea

The core operation is H^(l+1) = σ(D̃^(-1/2) Ã D̃^(-1/2) H^(l) W^(l)), where à = A + I adds self-loops (a node is its own neighbour), D̃ is the diagonal degree matrix of Ã, and the D̃^(-1/2) terms normalise by degree. The result: each node's new features are a weighted sum of its old features and its neighbours', passed through a learned weight matrix and non-linearity.

Without normalisation, high-degree nodes dominate. With symmetric normalisation, each neighbour contributes proportionally to its degree. Stacking layers lets information propagate: a 3-layer GCN sees 3-hop neighbours.

The maths

Try it

Step: 0 / 2

Input

Node 0

Step 0: Input

Interactive demonstration of graph neural networks concepts

Where I used it

Easy to get wrong

Sources

COMP90051 (2023). GCN architectures and applications for node classification on graphs.