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A brief introduction to Graph Neural Networks
Graph Neural Network (GNN) is a deep learning method for processing graph-structured data. This tutorial will introduce the basic concepts, main models, application scenarios and code implementation of graph neural network in detail.
What is a graph neural network
Graph Neural Networks (GNN) is a type of neural network model for processing graph-structured data. It is different from traditional neural networks (such as convolutional neural networks, recurrent neural networks, etc.) in processing regular data structures (such as images, time sequence), graph neural networks specialize in processing irregular graph-structured data, such as social networks, knowledge graphs, etc. Graph structure data is a complex relational network composed of nodes and edges, where nodes represent entities and edges represent relationships between entities. Different from traditional neural networks, graph neural networks need to consider the relationship between nodes, so a new way to represent nodes and edges is needed.
The core idea of graph neural network is to aggregate the features of each node with the features of its surrounding nodes to form a new node representation. This process can be implemented through message passing, where each node receives messages from its neighbor nodes and aggregates these messages into a new node representation. This method can be iterated many times to obtain more comprehensive graph structure information.
The structure of a graph neural network usually consists of multiple layers, and each layer contains operations such as node embedding, message passing, and pooling. In the node embedding operation, the features of each node are converted into a low-dimensional vector representation for easy learning and processing by the neural network. In a message passing operation, each node receives information about its neighbor nodes and aggregates these information into a new node representation. In the pooling operation, node representations are merged into a representation of the entire graph to facilitate prediction for graph-level tasks.
At present, graph neural network has been widely used in many fields, such as social network analysis, drug discovery, recommendation system, etc. At the same time, many variants of graph neural network models have emerged, such as Graph Convolutional Network (GCN), Graph Attention Network (GAT), etc., to adapt to different tasks and data types.
Basic Concepts of Graph Neural Networks
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1. Graph
A graph is a mathematical structure that uses vertices (Vertex) and edges (Edge) to represent entities and their relationships. A graph can be represented as G = (V, E), where V is the set of vertices and E is the set of edges. In a graph, vertices represent entities and edges represent relationships between entities. For example, in a social network, vertices may represent users, and edges may represent follow or friendship relationships between users.
There are two types of graphs: directed graphs and undirected graphs. In an undirected graph, the edges have no direction; in a directed graph, the edges have a direction. In addition, graphs can also carry weights, which represent the strength of relationships between entities. Graphs with weights are often called weighted graphs.
2. Adjacency Matrix
The adjacency matrix A is a matrix that represents the relationship between vertices in the graph. Let V be the set of vertices, and the size of A is |V|×|V|. For an undirected graph, element A(i, j) = 1 of A means that vertex i and vertex j are adjacent, that is, there is an edge; A(i, j) = 0 means that vertex i and vertex j are not adjacent. The adjacency matrix of a directed graph represents directed edges.
The adjacency matrix can be used to represent the graph structure, and it can also be used to implement the graph algorithm. Through the adjacency matrix, we can quickly query whether there is an edge between two vertices, as well as information such as the type and weight of the edge.
In addition, the adjacency matrix can be regarded as a representation of a graph. Compared with other representations (such as adjacency list, edge list, etc.), the adjacency matrix can be used to describe dense graphs, with high space utilization and high query efficiency. advantage.
3. Graph Signal
A graph signal is a signal defined on a graph vertex, which can be regarded as a feature of the vertex. For a set of vertices V, we can represent the graph signal as a |V|×d matrix X, where d is the feature dimension.
Graph signals can represent attributes or features of vertices. For example, in social networks, each vertex can be represented as a vector containing user attributes, such as gender, age, occupation, etc. In chemical molecules, each atom can be represented as a vector containing chemical properties such as electron affinity, charge, etc.
Graph signals can be used to analyze and process graph-structured data, such as classifying, clustering, and predicting graphs. Usually, we need to combine the graph signal with the topological structure on the graph, so as to make better use of the characteristics of the graph structure data. For example, in graph convolutional neural networks, we can extract feature representations of nodes through the convolution operation of graph signals and adjacency matrices.
4. Graph Convolution
Graph convolution is an operation that extends the concept of traditional convolution to graph-structured data. Graph convolution is usually expressed as a function between the adjacency matrix A and the graph signal X: f(A, X). Through this function, information transfer and feature extraction can be realized on the graph. Different from traditional convolutions, graph convolutions take into account the topology of nodes on the graph, thus capturing the relationships and dependencies between nodes.
There are several ways to implement graph convolution, the most common of which are spectral domain-based methods and spatial domain-based methods. Spectral-domain based methods exploit the eigenvalues and eigenvectors of the Laplacian matrix of the graph to define the convolution operation. Spatial domain-based methods utilize the adjacency matrix and node features for convolution operations.
Graph convolution can be used for many tasks on graph-structured data, such as node classification, graph classification, link prediction, etc. In graph neural networks, graph convolution is a core operation that can help extract feature representations in graph-structured data, enabling more efficient and accurate graph-structured data analysis and processing.
Main Graph Neural Network Model
1. GCN（Graph Convolutional Networks）
GCN is a graph convolution method based on Spectral Domain. It uses the Laplacian matrix of the graph to perform information transfer and feature extraction by performing convolution operations on the feature vectors. The core idea of GCN is to realize information transfer through the product of adjacency matrix A and graph signal X.
In GCN, the eigenvectors of each node are weighted and averaged with the eigenvectors of its neighbor nodes. The weights are determined by the values in the adjacency matrix A. Specifically, the convolution operation of GCN can be expressed as:
Z = f(A, X) = D⁻¹ A X W
where D is the degree matrix of A and W is the learnable weight matrix.
GCN has been widely used in node classification, graph classification, link prediction and other tasks, and achieved good results. However, the limitation of GCN is that its convolution operation only considers first-order neighbor nodes, and cannot capture longer-range relationships and global information. Therefore, subsequent research proposed many improved versions of graph convolutional networks, such as GAT and GraphSAGE introduced below.
2. GAT（Graph Attention Networks）
GAT is a graph convolution method based on Spatial Domain. GAT introduces an attention mechanism, so that the model can assign different weights to different edges, so as to better capture the relationship between nodes. In GAT, the feature vector of each node is weighted and averaged with the feature vectors of its neighbor nodes, and the weight is calculated by the attention mechanism.
Specifically, the convolution operation of GAT can be expressed as:
Z = f(A, X) = CONCAT(ATTENTION(A, X)W)
Among them, ATTENTION is a function to calculate the attention weight, and CONCAT is a connection operation. The ATTENTION function usually includes two steps: first calculate the similarity between each neighbor node and the current node, and then use the softmax function to convert the similarity into weight. In this way, the weight of each neighbor node can be different, so as to better express the relationship between nodes.
The attention mechanism of GAT enables the model to better adapt to different graph structures and tasks, and has achieved good results in many tasks such as graph classification, node classification, and link prediction.
3. GraphSAGE（Graph Sample and AggregatE）
GraphSAGE is a spatial domain-based graph convolution method, which proposes a neighbor sampling and aggregation strategy, enabling the model to handle large-scale graph data. In GraphSAGE, the eigenvectors of each node are aggregated with the eigenvectors of its neighbors. Different from GCN and GAT, GraphSAGE does not average or weighted average all neighbor nodes, but adopts a certain neighbor sampling strategy, and only considers a node's first-order or k-order neighbor nodes for aggregation, thereby reducing computational complexity.
Specifically, the convolution operation of GraphSAGE can be expressed as:
Z = f(A, X) = AGGREGATE(NEIGHBORS(A, X)) W
Among them, NEIGHBORS is a neighbor sampling function, which is used to randomly sample some nodes from the neighbors of a node as the input of aggregation; AGGREGATE is an aggregation function, which is used to aggregate the feature vectors of neighbor nodes, such as mean value, maximum value, etc.; W is the learnable The weight matrix of .
GraphSAGE can handle large-scale graph data, and has achieved good results in tasks such as node classification, graph classification, and link prediction. Its neighbor sampling and aggregation strategies are also borrowed and improved by many subsequent graph convolutional networks.
Application Scenarios of Graph Neural Networks
Graph neural networks are widely used in many fields, mainly including:
- Node Classification: Predict the category of nodes in the graph, such as predicting user interest tags in social networks.
- Link Prediction: Predict whether there are edges between nodes in the graph, such as predicting the relationship between entities in the knowledge graph.
- Graph Classification: Predict the category of the entire graph, such as predicting the activity of molecules in biomolecular networks.
- Graph Generation: Generate graphs with certain properties, such as generating networks that satisfy specific topological characteristics.
Code
The following is an example code for training a graph convolutional neural network using the Cora dataset, implemented using the PyTorch Geometric library:
import torch
import torch.nn as nn
import torch.nn.functional as F
import torch.optim as optim
from torch_geometric.datasets import Planetoid
from torch_geometric.nn import GCNConv
# 判断是否有可用的GPU，如果有就使用GPU，否则使用CPU
device = torch.device('cuda' if torch.cuda.is_available() else 'cpu')
# 加载Cora数据集
dataset = Planetoid(root='data/Cora', name='Cora')
# 定义一个GCN模型
class GCN(nn.Module):
def __init__(self, input_dim, hidden_dim, output_dim):
super(GCN, self).__init__()
# 第一个图卷积层
self.conv1 = GCNConv(input_dim, hidden_dim)
# 第二个图卷积层
self.conv2 = GCNConv(hidden_dim, output_dim)
# 全连接层
self.fc = nn.Linear(output_dim, dataset.num_classes)
def forward(self, x, edge_index):
# 第一次图卷积，使用ReLU激活函数
x = F.relu(self.conv1(x, edge_index))
# 第二次图卷积
x = self.conv2(x, edge_index))
# Dropout操作，防止过拟合
x = F.dropout(x, training=self.training)
# 全连接层
x = self.fc(x)
# 使用log_softmax进行分类
return F.log_softmax(x, dim=1)
# 创建GCN模型实例，并将模型移动到设备上（GPU或CPU）
model = GCN(dataset.num_features, 16, dataset.num_classes).to(device)
# 定义优化器和损失函数
optimizer = optim.Adam(model.parameters(), lr=0.01)
criterion = nn.NLLLoss()
# 定义训练函数
def train(model, optimizer, criterion, data):
model.train()
optimizer.zero_grad()
out = model(data.x.to(device), data.edge_index.to(device))
loss = criterion(out[data.train_mask], data.y[data.train_mask].to(device))
loss.backward()
optimizer.step()
# 定义测试函数
def test(model, data):
model.eval()
out = model(data.x.to(device), data.edge_index.to(device))
pred = out.argmax(dim=1)
acc = pred[data.test_mask].eq(data.y[data.test_mask].to(device)).sum().item() / data.test_mask.sum().item()
return acc
# 进行模型训练和测试，并输出测试集准确率
for epoch in range(200):
train(model, optimizer, criterion, dataset[0])
test_acc = test(model, dataset[0])
print('Epoch: {:03d}, Test Acc: {:.4f}'.format(epoch, test_acc))
In the above code, first import PyTorch-related libraries and Planetoiddataset classes and GCNConvgraph convolution layer classes. Then, determine whether there is a GPU available, if so, use the GPU, otherwise use the CPU. Next, load the Cora dataset and define a GCNclass for creating a graph convolutional neural network model. In GCNthe class, two graph convolutional layers and a fully connected layer are defined, and forwardthe forward propagation of the model is completed in the method. Next, an instanced model is created and moved to the device (GPU or CPU). Then, the optimizer and loss functions are defined, as well as the training and testing functions. Finally, the model is trained and tested in a loop, and the test set accuracy is output.
Summarize
This tutorial introduces the basic concepts, main models and application scenarios of Graph Neural Network (GNN), as well as sample codes for implementing GCN using PyTorch and PyTorch Geometric. Among them, a graph is a mathematical structure that uses vertices and edges to represent entities and their relationships, an adjacency matrix is a matrix that represents the relationship between vertices in a graph, a graph signal is a signal defined on a graph vertex, and graph convolution is a traditional convolution Operations extended to graph structures. The main GNN models include GCN based on spectral domain, GAT and GraphSAGE based on spatial domain. GNNs are widely used in fields such as node classification, link prediction, graph classification, and graph generation. Through the study of this tutorial, readers can better understand GNN and apply GNN to solve problems in practical problems.