[Complex Network Modeling]——Link prediction based on Graph Convolutional Networks (GCN)

Table of contents

1. Complex network modeling

2. Graph embedding method (Graph Convolutional Networks (GCN))

1. Picture representation:

2. Adjacency Matrix:

3. Graph Convolutional Layer:

4. Activation function and loss function:

5. Training process:

6. Application:

3. Sample code of GCN implementation based on PyTorch

1. Complex network modeling

A complex network is a network structure composed of a large number of interconnected elements (nodes or vertices), and these connections are usually very complex and dynamic. These networks can be found in a variety of fields, including social networks, biological systems, information technology, and transportation systems, among others.

The study of complex networks mainly focuses on the topological structure, dynamic behavior and functional properties of the network. Some of the common complex network models include small-world networks, scale-free networks, and random networks. These models help us understand the characteristics of information propagation, stability, and robustness in the network.

In practical applications, complex network theory is used to solve many problems, such as social network analysis, disease propagation models, power system optimization, etc. Research in this area is important for understanding the behavior of complex systems in the real world.

There are many methods for modeling complex networks, depending on the nature of the network, the application area and the research question. Here are some common methods for modeling complex networks:

Random Graph Models: This includes Erdős-Rényi models, Gilbert models, etc., where connections between nodes are randomly generated. These models are often used to study the fundamental properties of networks.

Small-World Models: The Watts-Strogatz model is a well-known small-world network model that simulates short paths in reality by adding random connections. and high aggregation.

Scale-Free Models: The Barabási-Albert model is a common scale-free network model, in which the degree distribution of the network follows a power law distribution. This model better captures the reality that some nodes have more connections, which is the so-called "the rich get richer".

Evolutionary Models: These models consider the evolution of the network over time, including the addition and departure of nodes, and the formation and breakage of connections. Leskovec's Preferential Attachment with Fitness Model is an example of an evolving network.

Spacetime Networks Models: Consider the changes in the network over time and space, especially in traffic flow, mobile networks and other fields.

Complex system dynamics model: Use differential equations, difference equations, or algebraic equations to describe the dynamic behavior of nodes and connections in the network. This type of model is usually used to study network stability, synchronization phenomena, etc.

Social network modeling methods: In social networks, methods such as Agent-Based Modeling can be used to consider individual behaviors and interactions to better understand Realistically simulate the evolution of social networks.

Application models of complex networks in specific fields: For example, biological network models, transportation network models, brain network models, etc. These models are adapted and extended to the characteristics of specific domains.

2. Graph embedding method (Graph Convolutional Networks (GCN))

Graph Embedding:
- Definition: Graph embedding is the process of mapping nodes or edges in a graph into a low-dimensional vector space. The goal is to preserve the structural information of the graph in a low-dimensional space so that adjacent nodes or edges are closer in the vector space.
- Applications: Graph embeddings can be used for various tasks such as node classification, node clustering, graph classification, link prediction, etc. Common graph embedding methods include DeepWalk, node2vec, GraphSAGE, etc.
Graph Autoencoder:
- Definition: A graph autoencoder is a neural network structure used to learn representations of graphs. It includes an encoder and a decoder, and by training the network, the input graph is mapped to a low-dimensional representation while trying to retain the structural information of the graph.
- Applications: Graph autoencoders are often used for unsupervised learning tasks, such as graph reconstruction, dimensionality reduction, anomaly detection, etc. They can learn efficient graph representations by minimizing reconstruction errors. Variational Graph Autoencoder (VGAE) is a variant of graph autoencoder that introduces the idea of variational inference.

Summary of differences:

Graph embedding is a broader concept that describes the process of mapping elements in a graph to a low-dimensional space and is not limited to specific learning methods.
Graph autoencoders are a specific type of neural network structure used to learn representations of graphs, usually implemented through the structure of an encoder and a decoder.

Although graph autoencoders can be used for graph embedding, graph embedding methods are not necessarily based on the autoencoder structure, and other techniques and models may also be used.

With graph embedding, you can obtain low-dimensional representations of each node or edge that preserve the structural information of the network in vector space. Such representations can be used for a variety of tasks, such as:

Node classification: After nodes are mapped to a low-dimensional space, node classification tasks can be performed in this space, such as determining the category or label to which the node belongs.

Link prediction: Through low-dimensional representation between nodes, it is possible to predict possible links or edges in the network.

Graph classification: Map the entire graph to a low-dimensional space so that the structural information of the graph is preserved and can be used for graph classification tasks, such as determining the type or attributes of the graph.

Visualization: By representing the network in a low-dimensional space, the network structure can be visualized and help understand the topology of the network.

Graph embedding methods include methods such as DeepWalk, node2vec, GraphSAGE, Graph Convolutional Networks (GCN), etc. These methods can be applied to different types of network data, including social networks, biological networks, knowledge graphs, etc.

Graph Convolutional Networks (GCN) is a deep learning model for graph-structured data, first proposed by Kipf and Welling in 2017. The goal of GCN is to learn the representation of nodes in the graph so that the node representation can capture the information of its neighbor nodes, thereby effectively processing graph-structured tasks, such as node classification, graph classification, link prediction, etc.

The following are the basic principles and key concepts of GCN:

1. Picture representation:

Node representation: Each node in the graph represents an entity, which can be a user, item, paper, etc.
Edge representation: The edges in the graph represent the relationship between nodes, which can be directed edges or undirected edges.

2. Adjacency Matrix:

GCN uses an adjacency matrix to represent the topology of the graph. For graph (G), element (A_{ij}) of adjacency matrix (A) represents whether an edge exists between node (i) and node (j).

3. Graph Convolutional Layer:

Graph convolution operation: GCN updates the representation of nodes through graph convolution operations. The update rule of a single-layer GCN can be expressed as (H' = f(\hat{D}^{-\frac{1}{2}}\hat{A}\hat{D}^{-\frac {1}{2}}XW)), where (H') is the updated node representation, (\hat{A}) is the symmetric normalized adjacency matrix, (\hat{D}) is the Angle matrix, (X) is the node feature matrix, (W) is the weight matrix, (f) is the activation function.
Multi-layer GCN: The multi-layer GCN model gradually aggregates more contextual information and improves the expressive ability of node representation by stacking multiple graph convolution layers.

4. Activation function and loss function:

Activation function: Usually activation functions such as ReLU are used.
Loss function: For supervised learning tasks such as node classification, the cross-entropy loss function is usually used.

5. Training process:

Supervised learning with known node labels. The parameters of the model are updated iteratively through optimization algorithms such as backpropagation and gradient descent.

6. Application:

GCN is widely used in tasks involving graph-structured data, such as social network analysis, bioinformatics, knowledge graphs, etc.

The proposal of GCN fills the shortcomings of traditional convolutional neural networks (CNN) in processing graph structure data, enabling deep learning models to better understand and utilize the structural information of graph data. Although GCN is a successful model, some improved versions have subsequently appeared, such as GraphSAGE, GAT (Graph Attention Network), etc., to deal with different types of graph data and tasks.

3. Sample code of GCN implementation based on 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, input_dim, output_dim):
        super(GraphConvolutionLayer, self).__init__()
        self.linear = nn.Linear(input_dim, output_dim)

    def forward(self, adjacency_matrix, node_features):
        # 对邻接矩阵进行对称归一化
        row_sum = adjacency_matrix.sum(1, keepdim=True)
        normalized_adjacency = adjacency_matrix / row_sum
        # 执行图卷积操作
        result = torch.matmul(normalized_adjacency, node_features)
        result = self.linear(result)
        result = F.relu(result)
        return result

class GCN(nn.Module):
    def __init__(self, input_dim, hidden_dim, output_dim):
        super(GCN, self).__init__()
        self.gc1 = GraphConvolutionLayer(input_dim, hidden_dim)
        self.gc2 = GraphConvolutionLayer(hidden_dim, output_dim)

    def forward(self, adjacency_matrix, node_features):
        h1 = self.gc1(adjacency_matrix, node_features)
        h2 = self.gc2(adjacency_matrix, h1)
        return h2

# 示例数据
# 假设有一个简单的无向图，邻接矩阵表示为：
# adjacency_matrix = [[0, 1, 1],
#                    [1, 0, 1],
#                    [1, 1, 0]]
adjacency_matrix = torch.tensor([[0, 1, 1],
                                [1, 0, 1],
                                [1, 1, 0]], dtype=torch.float32)

# 假设每个节点有一个特征，表示为：
# node_features = [[1, 2],
#                  [3, 4],
#                  [5, 6]]
node_features = torch.tensor([[1, 2],
                              [3, 4],
                              [5, 6]], dtype=torch.float32)

# 创建GCN模型
input_dim = node_features.size(1)
hidden_dim = 16
output_dim = 2
gcn_model = GCN(input_dim, hidden_dim, output_dim)

# 模型前向传播
output = gcn_model(adjacency_matrix, node_features)
print("GCN Output:\n", output)

This is just a simple example. In practice, a more complex model design and training process may be required depending on the task and data. In addition, for larger-scale graph data, it may be necessary to use techniques such as graph sampling to improve training efficiency.

In GCN link prediction, this can be achieved through the following steps:

Generate positive samples and negative samples: Randomly select a part of the existing edges in the graph as positive samples, and then randomly select the same number of positive samples from the edges that do not exist in the graph. edges as negative samples.
Define the loss function: Use the binary cross-entropy loss function to calculate the loss for the probability of the model output.
Training model: Use existing positive and negative samples for supervised learning, and iteratively update the parameters of the model.

import torch
import torch.nn as nn
import dgl
import dgl.function as fn
import torch.optim as optim
from sklearn.metrics import roc_auc_score
from sklearn.model_selection import train_test_split

# 构建一个简单的GCN模型
class GCN(nn.Module):
    def __init__(self, in_feats, hidden_size, out_feats):
        super(GCN, self).__init__()
        self.layer1 = GraphConvolution(in_feats, hidden_size)
        self.layer2 = GraphConvolution(hidden_size, out_feats)

    def forward(self, g, features):
        x = torch.relu(self.layer1(g, features))
        x = self.layer2(g, x)
        return x

# 图卷积层的定义
class GraphConvolution(nn.Module):
    def __init__(self, in_feats, out_feats):
        super(GraphConvolution, self).__init__()
        self.linear = nn.Linear(in_feats, out_feats)

    def forward(self, g, features):
        with g.local_scope():
            g.ndata['h'] = features
            g.update_all(fn.copy_src(src='h', out='m'),
                         fn.sum(msg='m', out='h_neigh'))
            h_neigh = g.ndata['h_neigh']
            return self.linear(h_neigh)

# 数据加载和预处理
# 这里假设你有一个邻接矩阵`adjacency_matrix`和节点特征矩阵`features_matrix`，以及一个标签向量`labels`
# 请根据你的数据格式进行调整

# 构建图
graph = dgl.graph(adjacency_matrix)
features = torch.tensor(features_matrix, dtype=torch.float32)
labels = torch.tensor(labels, dtype=torch.float32)

# 划分训练集和测试集
train_mask, test_mask = train_test_split(range(len(labels)), test_size=0.2, random_state=42)

# 初始化模型、损失函数和优化器
model = GCN(features.shape[1], 16, 1)
criterion = nn.BCEWithLogitsLoss()
optimizer = optim.Adam(model.parameters(), lr=0.01)

# 训练模型
for epoch in range(100):
    model.train()
    logits = model(graph, features)
    loss = criterion(logits[train_mask], labels[train_mask].view(-1, 1))
    
    optimizer.zero_grad()
    loss.backward()
    optimizer.step()

    # 计算AUC
    with torch.no_grad():
        model.eval()
        pred_probs = torch.sigmoid(logits[test_mask]).numpy()
        auc = roc_auc_score(labels[test_mask].numpy(), pred_probs)

    print(f'Epoch {epoch + 1}, Loss: {loss.item():.4f}, AUC: {auc:.4f}')

# 链路预测结果
model.eval()
logits = model(graph, features)
pred_probs = torch.sigmoid(logits).detach().numpy()