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Regional GCN Convolutional Layer (Guo et al. 2025)

Source: R/layer-regconv.R
layer_regconv.Rd

Implements the Regional Graph Convolutional Network (RegConv) layer from Guo et al. (2025):

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

This layer extends the standard GCN layer by introducing region-specific parameters to handle spatial heterogeneity (spatial regimes). The computation has two stages:

  1. Base GCN transformation: \(\mathbf{D}^{-1}\mathbf{A}\mathbf{X}^{(l-1)}\boldsymbol{\Theta}^{(l)} + \mathbf{X}^{(l-1)}\boldsymbol{\Phi}^{(l)}\)

  2. Region-specific modulation: Element-wise multiplication by \(\boldsymbol{\Omega}_{reg}\) and addition of \(\boldsymbol{\Psi}_{reg}\)

Parameters:

  • \(\Theta\) (theta): in_features x out_features transforms aggregated neighbor features (global)

  • \(\Phi\) (phi): in_features x out_features transforms node's own features (global)

  • \(\Omega_{reg}\) (omega_reg): n_regions x out_features region-specific weight modulation

  • \(\Psi_{reg}\) (psi_reg): n_regions x out_features region-specific bias terms

Usage

layer_regconv(in_features, out_features, n_regions)

Arguments

in_features

Integer. Number of input features per node

out_features

Integer. Number of output features per node

n_regions

Integer. Number of spatial regions/regimes

Value

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

Details

The RegConv layer is designed for two-stage training:

Stage 1: Train a global GCN to learn \(\Theta\) and \(\Phi\), then freeze these parameters.

Stage 2: Initialize \(\Omega_{reg}\) to all 1s and train region-specific parameters (\(\Omega_{reg}\), \(\Psi_{reg}\)) while keeping \(\Theta\) and \(\Phi\) fixed.

The region-specific parameters allow the model to adjust predictions differently across spatial regimes, enabling the model to capture spatial heterogeneity.

Forward pass

layer(x, adj, region_assignments, edge_weight = NULL)

  • x: Tensor n_nodes x in_features. Node feature matrix.

  • adj: Sparse COO 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.

  • region_assignments: Tensor n_nodes. Integer vector with values in 1:n_regions giving the region each node belongs to. Multiple nodes can belong to the same region.

  • edge_weight: Tensor n_nodes x n_nodes or NULL. Optional edge weights applied to the adjacency matrix. If NULL, the values of adj are used.

References

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))

# Four nodes split across two spatial regimes
regions <- torch::torch_tensor(c(1, 1, 2, 2), dtype = torch::torch_long())

layer <- layer_regconv(8, 4, n_regions = 2)
layer(x, adj_norm, regions)
}