In today’s world, many critical datasets are represented as diagrams or networks—social networks, knowledge graphs, protein interaction networks, the World Wide Web, and more. For a long time, these structured data types were difficult to handle using traditional machine learning techniques. However, in recent years, researchers have started exploring ways to generalize neural network models to work effectively with such graph-structured data.
Currently, several approaches have shown impressive results in specialized domains. Before the rise of graph neural networks, the best performance was achieved using kernel-based methods, graph theory-based regularization, or other similar strategies. These early methods laid the groundwork for what is now known as Graph Convolutional Networks (GCNs).
Outline:
- A brief introduction to neural network diagram models
- Spectral convolution and Graph Convolutional Networks (GCNs)
- Demo: Graph embedding using a simple first-order GCN model
- Viewing GCNs as a micro-generalization of the Weisfeiler-Lehman algorithm
- If you're already familiar with GCNs, skip to "GCNs Part III: Embedding Karate Club Network"
- How powerful are graph convolution networks?
Recent research has focused on adapting well-established neural architectures like RNNs or CNNs to process arbitrary graph structures. Some papers introduced custom architectures tailored for specific tasks, while others built convolution operations based on spectral graph theory.
To define filters in multi-layer neural networks, researchers drew inspiration from classic CNNs. More recently, efforts have aimed to bridge the gap between fast heuristics and slow, theoretically grounded methods. Defferrard et al. (NIPS 2016) used Chebyshev polynomials with learnable parameters in a neural network framework, resulting in an approximate smoothing filter in the spectral domain. Their approach showed promising results on standard datasets like MNIST, performing nearly as well as traditional 2D CNNs.
Kipf and Welling later proposed a simplified version of the spectral convolution framework. This model significantly improved training speed while maintaining accuracy, achieving state-of-the-art results on various benchmark datasets.
GCNs Part I: Definitions
Most current graph neural network models follow a common structure, collectively referred to as Graph Convolutional Networks (GCNs). The term "convolution" comes from the fact that filter parameters are typically shared across all nodes in the graph.
The goal of these models is to learn a function that maps node features into meaningful representations. They take as input:
- A feature matrix **X** of size **N × D**, where **N** is the number of nodes and **D** is the number of input features per node.
- An adjacency matrix **A** representing the graph structure.
From this, they produce a node-level output **Z** of size **N × F**, where **F** is the number of output features per node. Graph-level outputs can be obtained through pooling operations.
Each layer in the network can be expressed as a non-linear function:
$$ H^{(l+1)} = \sigma\left(H^{(l)}W^{(l)}\right) $$
Where $ H^{(0)} = X $, $ H^{(L)} = Z $, and $ L $ is the number of layers. The model's behavior depends on the choice and parameterization of the function $ f(\cdot) $.
GCNs Part II: A Simple Example
Let’s consider a simple propagation rule:
$$ H^{(l+1)} = \sigma\left(D^{-1}A H^{(l)} W^{(l)}\right) $$
Here, $ W^{(l)} $ is the weight matrix for the l-th layer, and $ \sigma $ is a non-linear activation function like ReLU.
Although this model is simple, it is quite powerful. However, there are two important limitations:
1. Multiplying by **A** sums the features of neighboring nodes but excludes the node itself unless a self-loop is added.
2. **A** is usually not normalized, which can distort the feature distribution. Normalizing **A** ensures that each row sums to 1, leading to more stable learning.
By adding a self-loop and normalizing the adjacency matrix, we arrive at the propagation rule used in Kipf & Welling’s work:
$$ H^{(l+1)} = \sigma\left(D^{-1/2}(A + I)D^{-1/2}H^{(l)}W^{(l)}\right) $$
This forms the basis of modern GCN implementations.
GCNs Part III: Embedding the Karate Club Network
To see how GCNs work, let’s look at the famous Zachary’s Karate Club network. We applied a 3-layer GCN with random weights and identity matrix as input features. Even without training, the model produced embeddings that closely matched the community structure of the graph.
This result is intriguing because it shows that even untrained GCNs can capture structural information. Similar results were achieved by DeepWalk, a method that uses unsupervised learning to generate embeddings.
Inspired by this, we can think of GCNs as a differentiable and parameterized version of the Weisfeiler-Lehman algorithm. In that algorithm, node features are updated based on the aggregated features of their neighbors. By replacing the hash function with a linear transformation and applying normalization, we get a similar update rule.
This insight helps explain why GCNs produce meaningful embeddings—nodes that are structurally similar tend to have similar representations.
GCNs Part IV: Semi-Supervised Learning
Since everything in the model is differentiable and parameterizable, we can use semi-supervised learning. By labeling just one node per class, the model can learn to classify all nodes effectively.
Using the semi-supervised approach described in Kipf & Welling’s paper, we observed that the model could separate classes in the hidden space, even with minimal supervision. This demonstrates the power of GCNs in capturing both local and global graph structure.
Conclusion
Research in this field is still in its early stages. While progress has been encouraging, we’ve only scratched the surface of what graph neural networks can achieve. Future work may explore tasks like learning on directed graphs, dynamic graphs, or using graph embeddings for downstream tasks.
This article covers only a part of the broader landscape. I hope to see more innovative applications and extensions in the near future.
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