Skip to contents

Generalized GCN Layer (Hamilton 2020)

layer_gcn_general.Rd

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

x

Tensor n_nodes x in_features. Node feature matrix

adj

Tensor n_nodes x n_nodes. Adjacency matrix. Expected to be row-normalized \(D^{-1}A\) where \(D\) is the degree matrix. Can be binary or weighted. This layer does NOT perform normalization internally

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

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