Generalized GCN Layer (Hamilton 2020) — layer_gcn

Implements a single Graph Convolutional Network (GCN) layer following Hamilton 2020:

$$\mathbf{H}^{(k)} = \sigma\left(\mathbf{A}\mathbf{H}^{(k-1)}\mathbf{W}^{(k)}_{\text{neigh}} + \mathbf{H}^{(k-1)}\mathbf{W}^{(k)}_{\text{self}}\right)$$

This can also be written as (Guo et al. 2025):

$$\mathbf{X}^{(l)} = \sigma\left(\mathbf{D}^{-1}\mathbf{A}\mathbf{X}^{(l-1)}\boldsymbol{\Theta}^{(l)} + \mathbf{X}^{(l-1)}\boldsymbol{\Phi}^{(l)} + \boldsymbol{\Psi}^{(l)}\right)$$

This layer combines:

Neighbor aggregation: $D^{-1}AX^{(l-1)}\Theta^{(l)}$
Self transformation: $X^{(l-1)}\Phi^{(l)}$ focal node transformation
Global bias: $\Psi^{(l)}$ additive bias term

Parameters:

$\Theta$ (theta): in_features x out_features transforms aggregated neighbor features
$\Phi$ (phi): in_features x out_features transforms node's own features
$\Psi$ (psi): out_features global bias term (shared across all nodes)

Usage

layer_gcn_general(in_features, out_features, bias = TRUE, normalize = FALSE)

Arguments

in_features: Integer. Number of input features per node
out_features: Integer. Number of output features per node
bias: Logical. Add learnable bias term ($\Psi$). Default: TRUE
normalize: Logical. Whether to add self-loops and row-normalize the adjacency matrix on-the-fly. Default: FALSE

Value

Tensor n_nodes x out_features. Transformed node features (before activation)

Details

The adjacency matrix is expected to be row-normalized $D^{-1}A$ where $D$ is the degree matrix. This layer does NOT perform normalization internally.

Forward pass

layer(x, adj)

x: Tensor n_nodes x in_features. Node feature matrix.
adj: Sparse COO tensor n_nodes x n_nodes. Adjacency matrix. Unless normalize = TRUE, it is expected to be row-normalized $D^{-1}A$, where $D$ is the degree matrix. Can be binary or weighted.

References

Hamilton, W. L. (2020). Graph Representation Learning. In Synthesis Lectures on Artificial Intelligence and Machine Learning. Springer International Publishing. doi:10.1007/978-3-031-01588-5

Guo, H., Wang, H., Zhu, D., Wu, L., Fotheringham, A. S., & Liu, Y. (2025). RegionGCN: Spatial-Heterogeneity-Aware Graph Convolutional Networks. Annals of the American Association of Geographers, 1–17. doi:10.1080/24694452.2025.2558661

Examples

if (FALSE) { # torch::torch_is_installed()
adj <- adj_from_edgelist(from = c(1, 2, 3, 4), to = c(2, 3, 4, 1))
x <- torch::torch_randn(4, 8)

# This layer expects a row-normalized adjacency matrix
adj_norm <- adj_row_normalize(add_graph_self_loops(adj))

layer <- layer_gcn_general(8, 4)
layer(x, adj_norm)
}