Graph Convolutional Network

The problem, from first principles

A Graph Convolutional Network (GCN) applies convolution-like operations on graphs. It aggregates neighbor information via message passing: each node updates its representation by aggregating its neighbors' representations.

Where N(i) is the neighborhood of node i, d_i is the degree, W is a learnable weight matrix, and sigma is an activation.

From theory to code

Implement:

gcn_layer(features, adj, weight)

Applies one GCN layer: aggregate neighbors, apply weight matrix, return node embeddings.

Constraints

• features shape: (num_nodes, input_dim).
• adj shape: (num_nodes, num_nodes), adjacency matrix.
• weight shape: (input_dim, output_dim).
• Return shape: (num_nodes, output_dim).