In this post, I’ll walk you through the process of training a Graph Convolutional Network (GCN) for edge classification, a common task in graph-based machine learning applications. Edge classification involves predicting the type of relationship (or edge) between two nodes in a graph, which is particularly useful in areas like knowledge graph completion, social network analysis, and recommendation systems.

### The Basics of GCNs

Graph Convolutional Networks (GCNs) are a class of neural networks specifically designed to operate on graph-structured data. Instead of assuming that data points are independent and identically distributed (as in traditional neural networks), GCNs take into account the connections (edges) between data points (nodes). The core idea of a GCN is to iteratively update the feature representation of each node by aggregating information from its neighbors.

Mathematically, the update rule for a single layer in a GCN can be expressed as:

\[H^{(l+1)} = \sigma\left( \hat{D}^{-1/2} \hat{A} \hat{D}^{-1/2} H^{(l)} W^{(l)} \right)\]Where:

- \(H^{(l)}\) is the matrix of node features at layer \(l\).
- \(\hat{A} = A + I\) is the adjacency matrix with added self-loops.
- \(\hat{D}\) is the diagonal degree matrix of \(\hat{A}\).
- \(W^{(l)}\) is the layer-specific trainable weight matrix.
- \(\sigma\) is the activation function, such as ReLU.

### Initial Approach: Random Node Features

In the first approach, we create random graphs by generating random node features and edges. Each graph is independent, and the GCN is trained to predict the type of relationship (edge) between nodes based purely on these random features. While this method is simple and provides a good introduction to GCNs, it has limitations. Since the node features and edges are random, there’s no meaningful structure for the GCN to learn, which limits the model’s performance.

When batching multiple graphs together during training, we need to increment the edge indices to ensure that each graph’s nodes are correctly referenced within the combined set of node features. This process ensures that the graphs remain independent within the batch, allowing the GCN to aggregate information correctly without mixing up nodes from different graphs.

### Alternative Approach 1: Learnable Node Embeddings

To address the shortcomings of the initial approach, we introduce learnable node embeddings. Instead of generating random node features for each graph, we use a fixed set of embeddings that are shared across all graphs in the dataset. These embeddings are updated during training, allowing the model to learn a more meaningful representation of each node.

Similar to the initial approach, batching requires incrementing the edge indices. This ensures that nodes from different graphs in a batch are correctly referenced, maintaining the independence of the graphs while leveraging the shared node embeddings.

### Alternative Approach 2: Treating All Graphs as One Large Graph

One might consider treating all graphs in a batch as part of one large, interconnected graph. This method involves using a single set of node embeddings and keeping the edge indices as-is, without any adjustments. The idea here is to have a consistent node representation across all graphs.

However, **this approach doesn’t work!** The loss (cross entropy loss) doesn’t go down at
all. If you decrease the batch size, the loss does start go down, but very slowly.
What’s happening here?

### Code

If you’re interested in exploring this further or experimenting with the code, you can check out the full implementation on GitHub. The repository contains the Jupyter notebook used for this post, along with the data and scripts needed to reproduce the experiments.