Overview of graph neural networks and examples of application and implementation in python

Graph Neural Networks

A graph neural network (GNN) is a type of neural network for data with a graph structure. ) to express relationships between elements. Examples of graph-structured data include social networks, road networks, chemical molecule structures, and knowledge graphs.

A GNN generates a feature representation that takes into account the graph structure by learning the features of each node and each edge using a neural network, and this feature representation can be used to solve tasks such as graph classification, graph generation, node classification, and link prediction, for example.

GNNs can be applied to both semi-supervised and unsupervised learning, and can also be combined with Convolutional Neural Networks (CNNs) as described in “Overview of CNNs, Algorithms, and Examples of Implementations” (see “Overview, Algorithms, and Application of Graph Convolutional Neural Networks (GCNs)” for details) and combined with RNN described in “Overview of RNN and examples of algorithms and implementations” (see “Overview, Algorithms, and Examples of Graph Embedding” for details). and recent research has addressed more advanced tasks such as graph generation using GNNs and working with sequence data and graph structures simultaneously.

Algorithms used in graph neural networks

As mentioned above, GNN is a type of neural network for learning and predicting on graph-structured data, which is capable of receiving node and edge features as input and performing information propagation and feature extraction in the graph. The following is a description of the algorithms commonly used in GNNs.

• Graph Convolutional Network (GCN): GCN is a method that applies the idea of convolutional neural networks (CNN) described in “Overview of CNN and examples of algorithms and implementations” to graphs. By aggregating information from neighboring nodes and edges, updating the features of each node, and using the information from neighboring nodes, it is able to combine the local information of nodes with information from the global graph structure. For more information on convolutional neural networks, see “Convolutional Neural Networks (1) Forward and Backward Propagation Algorithms and Mini-Batch.
• Graph Attention Mechanism: The Graph Attention Mechanism is used to learn the importance between nodes in a GNN. The attention mechanism, described in “Attention in Deep Learning” is the most popular deep learning technique in recent years and allows the network to concentrate important information on specific nodes by giving each node a different weight, thereby drawing attention to the important nodes. This allows the network to focus important information on a specific node.
• Graph Pooling: Graph pooling is used to reduce the size of the graph. While common pooling operations reduce the spatial dimension of feature maps, graph pooling reduces the nodes and edges of the graph, thereby reducing computational complexity when processing large graphs. For more details on pooling, see “Deep Learning for Computer Vision with Python and Keras (1) – Convolution and Pooling.
• Graph Generative Models: Graph Generative Models are methods for generating new graphs from existing graph data. Examples include graph autoencoders and graph adversarial networks (GANs), which are used for tasks such as graph generation, completion, and anomaly detection. For more information on autoencoders, see “Autoencoders“; for GANs, see “Overview of GANs and their various applications and implementations“.

Although these algorithms are part of GNNs, research on GNNs is evolving and various derivative methods and applications exist; GNNs are increasingly being used as useful tools in a variety of domains, including social network analysis, molecular structure analysis of chemical substances, and recommendation systems.

Libraries and platforms used for graph neural networks

Below are the various libraries and platforms available for implementing GNNs.

• PyTorch Geometric: PyTorch Geometric is an extension package to PyTorch, a Python deep learning library dedicated to the study and implementation of graph neural networks, including GNNs, supporting tasks such as graph data preprocessing, GNN model building, It supports tasks such as graph data preprocessing, GNN model building, training, and evaluation.
• Deep Graph Library (DGL): DGL will be a library of graph neural networks integrated with Python and other major deep learning frameworks (PyTorch, TensorFlow, etc.). They support tasks such as graph-structured data manipulation, convolution, pooling, and graph generation.
• NetworkX: NetworkX is a powerful library used for network analysis in Python that supports graph data creation, visualization, manipulation, and analysis. libraries.
• StellarGraph: StellarGraph is a Python graph machine learning library and tool that supports machine learning and GNN on graph data. It supports different types of graph structure data and provides a variety of GNN models and training algorithms.

About GNN Application Examples

GNNs have been applied to a variety of problems, including classification of nodes on graphs, clustering of graphs, and graph generation. The following are typical examples of their application.

• Node Classification: GNNs are used to classify the groups of nodes in graphs such as social networks. For example, in social networks such as Facebook and Twitter, GNNs are used to classify new users into groups based on the attributes of the grouped users.
• Graph Generation: Various methods have been proposed to generate nodes and edges using GNNs. This allows, for example, the generation of spatial graphs to represent physical distances, co-occurrence graphs to represent co-occurrence, etc.
• Graph Clustering: GNNs are also used for clustering nodes and edges. For example, by applying GNNs to a graph representing the similarity of web pages, it is possible to cluster similar pages.
• Recommendation: GNNs are also used in recommendation algorithms. For example, by applying a graph neural network to a graph representing a user’s browsing history and rating data, it is possible to recommend appropriate items to the user.

In addition to these, GNNs are used in a wide range of other fields, including modeling of physical phenomena, bioinformatics, and image processing.

Details and specific implementations of each of these examples are described below.

Node classification using GNN

<Overview>

The general flow of the node classification procedure using GNN is as follows

1. Data Preparation:
   • Node features: Prepare a matrix representing the features of each node. Each node has a feature vector, which is represented as a matrix.
   • Edge information: Define the connections and relationships between nodes in the form of graph edge information. Typically, an edge index or adjacency matrix is used to represent the edge connections.
2. Model Construction:
   • Define the architecture of the GNN model. Typically, multiple graph convolutional layers (GCNConv, GraphSAGE described in “GraphSAGE Overview, Algorithm, and Example Implementation“, GIN, etc.) may be stacked or pooling layers (graph pooling or global pooling) may be used. Each layer takes as input information on node features and edges and updates the representation of the node.
   • The final output layer uses the appropriate activation or softmax function as described in “Overview of softmax functions and related algorithms and implementation examples” to determine the prediction class of the node.
3. Training:
   • Initialize the parameters of the model and select a loss function and optimization method. Common loss functions include cross-entropy error described in “Overview of Cross-Entropy and Related Algorithms and Implementation Examples,” and log-likelihood loss.
   • Train the model using a training data set. Optimize the parameters of the model using mini-batch gradient descent or its variants to properly adjust hyper-parameters such as epoch number, learning rate, and regularization.
4. Testing:
   • Evaluate the performance of the model using a test data set. Compare the predicted class against the true class and calculate metrics such as accuracy, goodness of fit, and recall.
5. Model Improvement:
   • If the model’s performance is unsatisfactory, attempt to improve it by adjusting the model’s architecture or hyperparameters. In addition, data preprocessing and feature engineering may be performed.

Examples of python implementations of these techniques are described below.

<Implementation in Python>

The following is a basic implementation of node classification using GNN in Python. This example uses PyTorch and the PyTorch Geometric library. First, import the necessary libraries.

import torch
import torch.nn as nn
import torch.nn.functional as F
from torch_geometric.nn import GCNConv
from torch_geometric.datasets import Planetoid

Next, the GNN model is defined.

class GCN(nn.Module):
    def __init__(self, in_channels, hidden_channels, out_channels):
        super(GCN, self).__init__()
        self.conv1 = GCNConv(in_channels, hidden_channels)
        self.conv2 = GCNConv(hidden_channels, out_channels)
    
    def forward(self, x, edge_index):
        x = self.conv1(x, edge_index)
        x = F.relu(x)
        x = F.dropout(x, training=self.training)
        x = self.conv2(x, edge_index)
        return F.log_softmax(x, dim=1)

The GCNConv used in the model is a PyTorch Geometric class and represents the graph convolution layer. Next, load the dataset. We use the Planetoid dataset here, but the same procedure can be applied to other datasets such as the Karate Club dataset.

dataset = Planetoid(root='/path/to/dataset', name='Cora')
data = dataset[0]

Obtain node features and edge information from the dataset.

x = data.x
edge_index = data.edge_index
y = data.y

Initialize the model and set the optimization method and loss function.

model = GCN(in_channels=dataset.num_features, hidden_channels=16, out_channels=dataset.num_classes)
optimizer = torch.optim.Adam(model.parameters(), lr=0.01)
criterion = nn.NLLLoss()

Define a training loop to iterate through the data and train the model.

def train():
    model.train()
    optimizer.zero_grad()
    out = model(x, edge_index)
    loss = criterion(out[data.train_mask], y[data.train_mask])
    loss.backward()
    optimizer.step()

for epoch in range(200):
    train()

After training the model, test data is used to predict the nodes.

model.eval()
out = model(x, edge_index)
pred = out.argmax(dim=1)

Next, we discuss graph generation using GNNs.

Graph generation using GNN

<Overview>

One of the applications of GNNs in graph generation is the generation of molecules, 3D models, and inference-based modeling of physical phenomena. For example, in the case of molecule generation, a graph neural network can be used to learn the structure of a molecule and generate a new molecule based on that structure.

It can also be used to generate knowledge graphs as described in “Automatic Generation of Knowledge Graphs and Various Implementation Examples” and to automatically generate graph data for question-and-answer systems as described in “Chatbots and Question-and-Answer Technology.

The general procedure for generating graphs using GNNs is as follows

1. Data Preparation:
   • Graph features: Define the features of the graph to be generated. For example, node features and edge features may be considered.
   • Graph Topology: Define the topology of the graph to be generated. This could be, for example, the connection relationships between nodes or the types of edges.
2. Model Construction:
   • Define the architecture of the GNN model. A GNN model called a Generator is commonly used for graph generation. Generators have mechanisms for updating the representation of nodes and edges and generating new nodes and edges.
3. Training:
   • The parameters of the generator model are initialized and a loss function and optimization method are selected. The general loss function will be the loss function for evaluating the similarity between the generated graph and the target graph.
   • A training dataset is used to train the generator model. The generator is trained to generate a graph that is close to the topology of the target graph.
4. Graph Generation:
   • Generate a new graph using the trained generator model. Common methods include generating random nodes, generating connection relations for nodes, and generating edges.
   • The generated graph is expected to reflect the patterns and features learned by the generator model.
5. Model Improvements:
   • If the generated graphs do not meet the target characteristics or constraints, we will attempt to improve them by adjusting the model architecture and hyperparameters. The use of various loss functions and training techniques may also be considered.

Proper execution of these steps will allow us to perform the task of graph generation using GNNs. However, graph generation is a very complex task, and since there are many different approaches and methods, the appropriate model and algorithm must be selected depending on the specific task and requirements.

Below we present two examples of implementation of automatic graph generation using python.

<Example 1 implementation in python>

The code below describes an example implementation using the PyTorch Geometric library. First, import the necessary libraries.

import torch
import torch.nn as nn
from torch_geometric.data import Data
from torch_geometric.nn import GATConv

Next, a graph generator (Generator) model is defined.

class GraphGenerator(nn.Module):
    def __init__(self, num_nodes, input_dim, hidden_dim, output_dim):
        super(GraphGenerator, self).__init__()
        self.num_nodes = num_nodes
        self.input_dim = input_dim
        self.hidden_dim = hidden_dim
        self.output_dim = output_dim

        self.conv1 = GATConv(input_dim, hidden_dim)
        self.conv2 = GATConv(hidden_dim, output_dim)

    def forward(self):
        x = torch.randn(self.num_nodes, self.input_dim)
        edge_index = self.generate_edge_index()
        x = self.conv1(x, edge_index)
        x = torch.relu(x)
        x = self.conv2(x, edge_index)
        return x, edge_index

    def generate_edge_index(self):
        # Implement according to how edges are generated
        # For example, generating random edges or generating edges based on a specific pattern.
        pass

In this example, GATConv is used to update the node features.

The forward method of the graph generator outputs the node’s feature value x and the index of the generated edge, edge_index. As for the edge generation method, it is necessary to implement logic to generate the index of the edge in the generate_edge_index method according to the edge generation method. Specifically, it is possible to generate random edges or use a probabilistic model as described in “About Probabilistic Generation Models.

The following is an example of model training and graph generation.

num_nodes = 100  # Number of nodes in the graph to be generated
input_dim = 16   # Number of dimensions of input features
hidden_dim = 32  # Number of hidden layer dimensions
output_dim = 2   # Number of dimensions of output features

generator = GraphGenerator(num_nodes, input_dim, hidden_dim, output_dim)
optimizer = torch.optim.Adam(generator.parameters(), lr=0.01)

# training loop
for epoch in range(100):
    optimizer.zero_grad()
    output, _ = generator()
    loss = compute_loss(output)  # Calculation of the loss function (to be defined according to the characteristics of the generated graph)
    loss.backward()
    optimizer.step()

# Graph Generation
generated_output, generated_edge_index = generator()

In the above example, the model training and graph generation are repeated. The appropriate loss function should be computed and the gradient updated within the training loop. In the graph generation part, the features and edge indices of the generated graph can be obtained by calling the generator() method.

<Example of Python Implementation 2>

The following is an example of a Python implementation of automatic graph generation using the Deep Graph Library (DGL). First, import the necessary libraries.

import dgl
import torch
from dgl.data import DGLDataset
from dgl.nn import GraphConv

Next, the dataset class of the automatic graph generator (Generator) is defined.

class GraphGeneratorDataset(DGLDataset):
    def __init__(self, num_graphs, num_nodes, input_dim, hidden_dim, output_dim):
        self.num_graphs = num_graphs
        self.num_nodes = num_nodes
        self.input_dim = input_dim
        self.hidden_dim = hidden_dim
        self.output_dim = output_dim

        super(GraphGeneratorDataset, self).__init__(name='graph_generator')

    def process(self):
        for i in range(self.num_graphs):
            graph = self.generate_graph()
            self.add_graph(graph)

    def generate_graph(self):
        g = dgl.DGLGraph()
        g.add_nodes(self.num_nodes)

        # Implement according to how edges are generated
        # For example, generating random edges or generating edges based on a specific pattern.

        return g

    def add_graph(self, graph):
        self.graphs.append(graph)

    def __getitem__(self, idx):
        return self.graphs[idx]

    def __len__(self):
        return self.num_graphs

In this example, the dataset class for the graph generator is created by inheriting from DGLDataset, and the specified number of graphs are generated and added to the dataset in the process method. Specifically, it is possible to generate random edges or to use a probabilistic model as described in “About Probabilistic Generation Models.

Next, we define the graph generator model.

class GraphGeneratorModel(torch.nn.Module):
    def __init__(self, input_dim, hidden_dim, output_dim):
        super(GraphGeneratorModel, self).__init__()
        self.conv1 = GraphConv(input_dim, hidden_dim)
        self.conv2 = GraphConv(hidden_dim, output_dim)

    def forward(self, g):
        x = g.ndata['feat']
        x = self.conv1(g, x)
        x = torch.relu(x)
        x = self.conv2(g, x)
        return x

In this example, GraphConv is used to update the graph features. The following is an example of model training and graph generation.

num_graphs = 100  # Number of graphs to generate
num_nodes = 100   # Number of nodes in graph
input_dim = 16    # Number of dimensions of input features
hidden_dim = 32   # Number of hidden layer dimensions
output_dim = 2    # Number of dimensions of output features

dataset = GraphGeneratorDataset(num_graphs, num_nodes, input_dim, hidden_dim, output_dim)
loader = dgl.dataloading.GraphDataLoader(dataset, batch_size=1, shuffle=True)

model = GraphGeneratorModel(input_dim, hidden_dim, output_dim)
optimizer = torch.optim.Adam(model.parameters(), lr=0.01)

# training loop
for epoch in range(100):
    for batched_graph in loader:
        optimizer.zero_grad()
        output = model(batched_graph)
        loss = compute_loss(output)  # Calculation of the loss function (to be defined according to the characteristics of the generated graph)
        loss.backward()
        optimizer.step()

# Graph Generation
generated_graph = dataset.generate_graph()

In the above example, the dataset class is used to generate the graph and train the model. Within the training loop, the appropriate loss function is computed and the gradient is updated. In the graph generation part, the generate_graph method is called to generate a new graph.

Next, graph clustering using GNN is described.

Graph clustering using GNN

<Overview>

Graph clustering using GNNs is the task of partitioning nodes in graph data into meaningful groups; GNNs utilize node features and graph topology to perform clustering. The following is a general procedure for graph clustering using GNNs.

1. Data Preparation:
   • Create graph data: Create the graph data to be clustered. This includes the connections between nodes (edges) and the features of each node.
2. Model construction: Define the architecture of the GNN model.
   • Define the architecture of the GNN model. Common GNN models include Graph Convolutional Network (GCN), GraphSAGE, and GIN.
3. Training:
   • Train the GNN model using a training dataset. The training dataset may contain the correct answer information for clustering, but for unsupervised learning, the correct answer information is not required.
   • A common approach is to consider the features of a node’s neighbors when updating the node’s features. In some cases, the features of edges are also considered.
4. Performing clustering:
   • Perform clustering on unknown graph data using a trained GNN model.
   • There are various clustering methods. A common method is to extract the features of each node and apply a clustering algorithm (k-means, hierarchical clustering, etc.).
5. Evaluation:
   • Clustering evaluation metrics (e.g. Adjusted Rand Index, Normalized Mutual Information) are used to evaluate clustering results.

Next, we describe a concrete example of a python implementation of the above.

<Implementation in python>

The following code uses the PyTorch Geometric library. First, import the necessary libraries.

import torch
import torch.nn as nn
import torch.nn.functional as F
from torch_geometric.data import Data
from torch_geometric.nn import GCNConv, global_mean_pool
from sklearn.cluster import KMeans

Next, the GNN model is defined.

class GraphClusteringModel(nn.Module):
    def __init__(self, input_dim, hidden_dim, output_dim):
        super(GraphClusteringModel, self).__init__()
        self.conv1 = GCNConv(input_dim, hidden_dim)
        self.conv2 = GCNConv(hidden_dim, output_dim)

    def forward(self, x, edge_index):
        x = self.conv1(x, edge_index)
        x = F.relu(x)
        x = self.conv2(x, edge_index)
        return x

In this example, GCNConv is used to update the node features. Next, the execution part of the graph clustering is shown.

# Creation of graph data (here created manually as an example)
x = torch.tensor([[1], [2], [3], [4], [5]], dtype=torch.float)  # Node features
edge_index = torch.tensor([[0, 1, 2, 2, 3, 4], [1, 0, 2, 3, 4, 3]], dtype=torch.long)  # Index of Edge

# Instantiation of GNN model
model = GraphClusteringModel(input_dim=1, hidden_dim=16, output_dim=8)

# Graph Clustering Training
optimizer = torch.optim.Adam(model.parameters(), lr=0.01)
num_epochs = 100

for epoch in range(num_epochs):
    optimizer.zero_grad()
    output = model(x, edge_index)
    loss = compute_loss(output)  # Calculate the loss function (to be defined for clustering purposes)
    loss.backward()
    optimizer.step()

# Perform clustering
cluster_embeddings = model(x, edge_index)
kmeans = KMeans(n_clusters=2)
cluster_labels = kmeans.fit_predict(cluster_embeddings.detach().numpy())

In the above example, the GNN model is trained by manually creating graphical data. The specific design of the clustering loss function and evaluation metrics should be defined appropriately according to the purpose of the clustering and the data. See “Overview of python Keras and examples of its application to basic deep learning tasks” and “Deep Learning with Python and Keras: A Methodology for Deep Learning” for more details.

In the clustering execution part, the output of the GNN model is used to apply clustering algorithms such as K-means.

Recommendation using GNN

<Overview>

A GNN-based recommendation system will be tasked with recommending the most suitable items to the user using the user’s behavioral history and graph data representing the item’s characteristics; the GNN will learn the relationship between the user and the item and utilize this information to make individual recommendations. The following is a typical procedure for a recommendation system using a GNN.

1. Data Preparation:
   • The user and item data are represented in a graph structure. Nodes represent users and items, and edges represent relationships between users and items. Nodes may contain features (e.g., user attributes, item characteristics, etc.).
2. Model Architecture:
   • Define the architecture of the GNN model. Common GNN models used include Graph Convolutional Network (GCN), GraphSAGE, and GAT.
   • Parameters in the model are used to learn user and item features and to generate an embedding vector representing user and item associations.
3. Training:
   • A training dataset is used to train the GNN model. Training datasets include user behavior history and item ratings.
   • The general approach is to propagate information through the GNN model in order to learn the association between users and items. A loss function is defined and the model is trained to minimize the loss.
4. Running the recommendation:
   • Use the trained GNN model to recommend the best item for a particular user.
   • Using the user’s features as input, the GNN model generates an embedding vector for the user. The embedding vector is then compared to the item’s feature values and a recommendation score is calculated. Items with high recommendation scores are recommended to the user.
5. Evaluation:
   • Evaluation metrics (e.g., Precision, Recall, NDCG, etc.) are used to evaluate the performance of the recommendation system. This allows us to measure and improve the accuracy and effectiveness of the recommendation system.

Next, an example of a concrete implementation in python is shown below.

<Example of python implementation of a recommendation system using GNN>

The following code uses the PyTorch Geometric library. First, import the necessary libraries.

import torch
import torch.nn as nn
import torch.nn.functional as F
from torch_geometric.data import Data
from torch_geometric.nn import GCNConv
from sklearn.metrics import pairwise_distances_argmin_min

Next, the GNN model is defined.

class RecommendationModel(nn.Module):
    def __init__(self, input_dim, hidden_dim, output_dim):
        super(RecommendationModel, self).__init__()
        self.conv1 = GCNConv(input_dim, hidden_dim)
        self.conv2 = GCNConv(hidden_dim, output_dim)

    def forward(self, x, edge_index):
        x = self.conv1(x, edge_index)
        x = F.relu(x)
        x = self.conv2(x, edge_index)
        return x

In this example, GCNConv is used to update the node features. Next, the execution part of the recommendation is shown.

# Create user and item features (manually created here as an example)
user_features = torch.tensor([[1, 0, 0], [0, 1, 0], [0, 0, 1]], dtype=torch.float)  # User Features
item_features = torch.tensor([[1, 0, 0], [0, 1, 0], [0, 0, 1]], dtype=torch.float)  # Item Feature Value
x = torch.cat([user_features, item_features], dim=0)

# Creating Graph Data
num_users = user_features.size(0)
num_items = item_features.size(0)
edge_index = torch.tensor([[i, j] for i in range(num_users) for j in range(num_users, num_users + num_items)], dtype=torch.long).t().contiguous()

# Instantiation of GNN model
model = RecommendationModel(input_dim=3, hidden_dim=16, output_dim=8)

# Executing a recommendation
recommendation_embeddings = model(x, edge_index)
user_embeddings = recommendation_embeddings[:num_users]
item_embeddings = recommendation_embeddings[num_users:]
user_indices = torch.arange(num_users)
item_indices = torch.arange(num_users, num_users + num_items)

recommendations = []
for i in range(num_users):
    user_embedding = user_embeddings[i]
    item_embedding = item_embeddings
    nearest_item_index = pairwise_distances_argmin_min(user_embedding.unsqueeze(0), item_embedding)[0][0]
    recommendations.append(nearest_item_index.item())

print(recommendations)

In the above example, the user and item features are manually created and the GNN model is used to make recommendations. The distance between the user’s embedding vector and the item’s embedding vector (in this case, the nearest neighbor item) is used as the criterion for recommendation.

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

“Hands-On Graph Neural Networks Using Python: Practical techniques and architectures for building powerful graph and deep learning apps with PyTorch“

“Graph Neural Networks: Foundations, Frontiers, and Applications“等がある。

“Introduction to Graph Neural Networks“

“Graph Neural Networks in Action“