GCN layer (message passing)

Background

A Graph Convolutional Network (GCN) layer updates each node's features by mixing in its neighbors'. Kipf & Welling's formulation uses the symmetrically normalized adjacency with self-loops: $\hat A = \tilde D^{-1/2}\tilde A\,\tilde D^{-1/2}$ where $\tilde A = A + I$. One layer is then $H' = \sigma(\hat A H W)$ — propagate (aggregate neighbors), transform (linear), activate.

Problem statement

Implement gcn_layer(A, H, W) for one GCN propagation step (no activation):

$$\tilde A = A + I, \quad \tilde D_{ii} = \sum_j \tilde A_{ij}, \quad \hat A = \tilde D^{-1/2}\tilde A\,\tilde D^{-1/2}, \quad H' = \hat A H W$$

Input

• A — adjacency matrix (n, n) (0/1, symmetric, no self-loops).
• H — node features (n, f_in).
• W — weight matrix (f_in, f_out).

Output

An np.ndarray (n, f_out): the propagated, transformed node features.

Examples

Example 1

Input:  A = [[0,1],[1,0]] (two connected nodes), H = [[1],[3]], W = [[1]]
Output: [[2.0], [2.0]]

Explanation: with self-loops each node has degree 2, so $\hat A = [[0.5,0.5],[0.5,0.5]]$. Each node's new feature is the average of both, $(1+3)/2 = 2$, then multiplied by $W=1$.

Constraints

• Add self-loops ($\tilde A = A + I$) before normalizing.
• Use symmetric normalization $\tilde D^{-1/2}\tilde A\,\tilde D^{-1/2}$, not row normalization.
• Apply the linear transform W after propagation (no activation here).

Notes

• Self-loops ensure a node keeps its own features; without them a node would see only its neighbors.
• Symmetric normalization keeps the scale of features stable across nodes of very different degree, avoiding exploding/vanishing signals in deep GCNs.