Overview of Graph Convolutional Neural Networks (GCN)
Graph Convolutional Neural Networks (GCN) is a type of neural network that enables convolutional operations on data with a graph structure. While regular convolutional neural networks (CNNs) described in “Overview of CNN and examples of algorithms and implementations” are effective for lattice-like data such as image data, GCNs have been developed as a deep learning method for non-lattice-like data with very complex structures, such as graph data and network data.
Convolution in general image recognition involves scanning a small-sized rectangle called a filter over the input image while performing a sum-of-products operation to detect where in the input image a pattern similar to the pattern represented by the filter is located, and by combining such convolutions, the localization of the input image consisting of individual images By combining such convolutions, local and global features of the input image consisting of individual images are hierarchically recognized.
To apply this to graphs, it is necessary to use data structures and algorithms that apply to graph data that is not localized and that increases or decreases in the number of vertices, rather than (pixel) data that exists regularly in a lattice-like pattern.
There are two approaches to this: (1) Spectral graph convolution and (2) Spacial graph convolution.
Spectral graph convolution is a transformation and inverse transformation of the graph Laplacian into space by eigenvectors, which is then transformed into frequency components by Fourier transform in signal processing. Typical approaches for these include ChebNet, ChebyNet, and GCN (Graph Convolutional Network).
Spacial graph convolution is a method of learning representations by considering the operation of collecting attribute information of surrounding nodes connected by edges to individual nodes as a convolution operation, which is also called graph convolution of message passing described in “Overview of Message Passing in Machine Learning with Examples of Algorithms and Implementations“. Typical approaches include PATCHY-SAN described in “Overview of PATCHY-SAN and examples of algorithms and implementations” DCNN (Diffusion-Convolutional Neural Network) described in “Overview, Algorithm and Implementation of DCNN (Diffusion-Convolutional Neural Networks)“and GraphSAGE.
GCN is a spectral graph convolutional network proposed by Kipf and Wlling in “Semi-Supervised Classification with Graph Convolutional Networks“. The network is based on ChebNet, which is described in “Overview of ChebNet, Algorithm, and Examples of Implementation”.
The basic concepts of GCN are described below.
1. Graph Structure: GCN takes as input a graph structure consisting of nodes (vertices) and edges (edges). Nodes represent data points and edges represent relationships between nodes. For example, in a social network, users are nodes and friendship relationships are represented by edges.
2. Adjacency Matrix: Graph structures are usually represented as adjacency matrices. An adjacency matrix is a matrix representation of the connection relationship between nodes, which is used to obtain information about the neighborhood of a node.
3. convolutional operations: Similar to ordinary convolutional operations, GCN generates a new representation by combining the features of a node with those of its neighbors. A convolutional filter is used to compute a new representation by combining the information of a node and its neighbors.
4. aggregation: In GCN, aggregation is used to combine the information of neighboring nodes. Typical aggregation methods include averaging and weighted sum of features of neighboring nodes.
5. activation functions and pooling: Similar to ordinary convolutional neural networks, GCNs use activation functions (e.g., ReLU) and perform pooling operations.
The key points of the extension of GCN’s algorithm to ChebNet are as follows.
- First-order Chebyshev approximation: As the filter \(g_{\theta}\star \mathbf{x}=\mathbf{U}g_{\theta}(\Lambda)\mathbf{U}^T\mathbf{x}\) of the expression\(g_{\theta}(\Lambda)\mathbf{U}^T\mathbf{x}\) in the Spectral convolution, the Chebyshev polynomial\(g_{\theta}(\Lambda)\) is used as the filter of up to Kth order. Lambda)\)\)\)\)Chebyshev polynomial up to Kth order (g_{\theta}(\Lambda)\approximately\displaystyle\sum_{k=0}^{K-1}\theta_kT_k(\tilde{\Lambda})\)}) with K=1 to avoid overfitting in graphs with wide degree distribution. avoiding overfitting in graphs. For more information on Chebyshev polynomial, please refer to “ChebNet Overview, Algorithm and Implementation Examples“.
- Simplification of the equation by \(\lambda_{max}=2\): This is an approximation of \(\tilde{\mathbf{L}}=\frac{2}{\lambda_{max}}\mathbf{L}-\mathbf{I}\) in ChebNet.
- Parameter reduction: Avoid overfitting by using the single-parameter expression\(x’=\theta(\mathbf{I}+\mathbf{D}^{-\frac{1}{2}}\mathbf{A}\mathbf{D}^{-\frac{1}{2}})x\) in the approximate expression.
- renormalization trick: add self-loops to each node in the above parameter reduction equation to get a \(\tilde{\mathbf{A}}=\mathbf{A}+\mathbf{I},\tilde{\mathbf{D}}_{ii}=\sum_j\tilde{\mathbf{A}}_{ii}\) is introduced to avoid instability of gradient disappearance.
The GCN paper compares the accuracy of the GCN with ManiReg, SemiEmb, LP, DeepWalk described in “DeepWalk Overview, Algorithms, and Example Implementations,”, ICA, and Planetoid for classification problems using citation networks (Citreseer, Cora, Pubmed) between papers and knowledge graphs (NELL) representing relationships among concepts, and shows superiority. The GCN is available in the following git page.
The code of GCN is available on the following git page.
Algorithms related to Graph Convolutional Neural Networks (GCN)
Algorithms and methods related to graph convolutional neural networks (GCNs) are described.
1. Graph Convolutional Networks (GCN): This is the first GCN algorithm proposed by Thomas N. Kipf and Max Welling. This algorithm performs a convolution based on a Laplacian matrix using a first-order approximation, which can be interpreted in terms of spectral filtering.
2. ChebNet (Chebyshev network): An algorithm proposed by Michaël Defferrard et al. ChebNet uses the Chebyshev polynomial to approximate the Laplacian. This method improves computational efficiency by using low order polynomials. For more information, see “ChebNet Overview, Algorithms, and Example Implementations.
3 GraphSAGE (Graph Sample and Aggregated): An algorithm proposed by William L. Hamilton et al. that uses the neighborhood information of randomly sampled nodes for convolution operations. This improves computational efficiency and scalability for large graphs. For details, see “GraphSAGE Overview, Algorithm, and Example Implementation“.
4. GAT (Graph Attention Network): An algorithm proposed by Petar Veličković et al. that uses an attention mechanism in the convolution operation. Each node can consider neighboring nodes of different importance and can learn a more flexible graph representation. For details, please refer to “Overview of GAT (Graph Attention Network), Algorithm and Examples of Implementation.
5. Graph Isomorphism Network (GIN): Algorithm proposed by Xavier Bresson et al. that allows learning graph isomorphism by combining each node’s own feature with the average of its neighbors in the convolution operation. For details, see “Graph Isomorphism Network (GIN) Overview, Algorithm and Example Implementation.
Application Examples of Graph Convolutional Neural Networks (GCN)
Graph convolutional neural networks (GCNs) have been successfully applied in various domains. The following are examples of such applications.
1. Social Network Analysis: In social networks, GCNs are used to learn relationships among nodes (users) and predict friendship relationships and information propagation considering the characteristics of the nodes.
2. Chemistry (molecular structure analysis): In chemistry, molecular structures are represented as graphs, and GCN is used to predict molecular properties and interactions, and to design new molecules.
3. Recommender systems: In the field of recommendation systems, relationships between users and items (e.g., products) are represented as graphs, and GCN is used to recommend items that are suitable for individual users.
4. Bioinformatics: GCN is used to analyze protein-protein interaction networks and gene-gene interaction networks to understand diseases and develop treatments.
5. graph database query: Graph database queries are used for flexible query and pattern detection on graph data with different node and edge attributes.
6. Traffic Flow Prediction: Road network and public transportation data are modeled as graphs, and GCN is used to predict traffic flow and identify optimal routes.
7. Linguistic modeling: Relationships between words in a language are represented as graphs, and GCN is used to analyze the meaning of sentences and to generate word embeddings.
Example implementation of Graph Convolutional Neural Networks (GCN)
Example implementations of GCN have been done primarily using deep learning frameworks (e.g., TensorFlow, PyTorch). Below is an example of a simple GCN implementation using PyTorch.
import torch
import torch.nn as nn
import torch.nn.functional as F
import torch.optim as optim
class GraphConvolutionLayer(nn.Module):
def __init__(self, in_features, out_features):
super(GraphConvolutionLayer, self).__init__()
self.linear = nn.Linear(in_features, out_features)
def forward(self, x, adjacency_matrix):
support = self.linear(x)
output = torch.matmul(adjacency_matrix, support)
return output
class GCN(nn.Module):
def __init__(self, input_size, hidden_size, output_size):
super(GCN, self).__init__()
self.gc1 = GraphConvolutionLayer(input_size, hidden_size)
self.gc2 = GraphConvolutionLayer(hidden_size, output_size)
def forward(self, x, adjacency_matrix):
x = F.relu(self.gc1(x, adjacency_matrix))
x = self.gc2(x, adjacency_matrix)
return F.log_softmax(x, dim=1)
# Example of dummy data and adjacency matrix
input_features = 5
hidden_features = 10
output_classes = 2
# dummy data
x = torch.randn(10, input_features)
# Dummy adjacency matrix (provisional, appropriate data must be used in the actual application)
adjacency_matrix = torch.randn(10, 10)
# Model Instantiation
model = GCN(input_features, hidden_features, output_classes)
# forward pass
output = model(x, adjacency_matrix)
# Definition of loss functions and optimization methods
criterion = nn.CrossEntropyLoss()
optimizer = optim.Adam(model.parameters(), lr=0.01)
# Backward and parameter updates
target = torch.randint(0, output_classes, (10,))
loss = criterion(output, target)
optimizer.zero_grad()
loss.backward()
optimizer.step()In this example, the GraphConvolutionLayer represents the convolution layer of the GCN, and the GCN class combines these to build the model. The data, adjacency matrix, loss function, and optimization methods should be modified appropriately for the actual task and data.
Challenges of Graph Convolutional Neural Networks (GCN) and their Countermeasures
While graph convolutional neural networks (GCNs) are a promising method, several challenges exist. These challenges and their countermeasures are described below.
1. non-locality and missing information:
Challenges: While ordinary convolutional neural networks (CNNs) are suitable for grid-like data such as images, GCNs are specialized for non-grid-like graph structures. However, GCNs may have difficulty adequately capturing relationships with distant nodes.
Solution: Methods have been proposed to account for non-local information by improving the model, such as extending layers and adjacency matrices, and introducing an attention mechanism.
2. computational complexity depending on the graph size:
Challenge: When the graph size is large, the computational complexity of GCN increases, making training and inference time-consuming.
Solution: Methods to improve computational efficiency using sampling, graph pooling, approximation methods, etc. have been proposed, and mini-batch learning described in “Overview of mini-batch learning and examples of algorithms and implementations” and parallel processing have also been applied.
3. improving robustness to graph variation:
Challenge: GCN is sensitive to graph variation, requiring model re-training when nodes are added or deleted.
Solution: Research is underway to improve robustness to node additions and deletions by modifying the graph convolution layer and using methods such as GraphSAGE.
4. problem of missing labels:
Challenge: Insufficient labels make it difficult to train models in supervised learning.
Solution: Semi-supervised learning, weakly supervised learning, and transfer learning described in “Overview of Transfer Learning and Examples of Algorithms and Implementations“have been proposed to deal with the problem of missing labels.
Reference Information and Reference Books
For more information on graph data, see “Graph Data Processing Algorithms and Applications to Machine Learning/Artificial Intelligence Tasks. Also see “Knowledge Information Processing Techniques” for details specific to knowledge graphs. For more information on deep learning in general, see “About Deep Learning.
Reference book is
“Graph Neural Networks: Foundations, Frontiers, and Applications“等がある。
“Introduction to Graph Neural Networks“
“Graph Neural Networks in Action“
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