sunshinecxm_BJTU

GNN-3-基于图神经网络的节点表征学习

参考开源学习地址：datawhale

1. 引言

在图节点预测或边预测任务中，首先需要生成节点表征（representation）。高质量节点表征应该能用于衡量节点的相似性，然后基于节点表征可以实现高准确性的节点预测或边预测，因此节点表征的生成是图节点预测和边预测任务成功的关键。基于图神经网络的节点表征学习可以理解为对图神经网络进行基于监督学习的训练，使得图神经网络学会产生高质量的节点表征。

在节点预测任务中，我们拥有一个图，图上有很多节点，部分节点的标签已知，剩余节点的标签未知。将节点的属性（x）、边的端点信息（edge_index）、边的属性（edge_attr，如果有的话）输入到多层图神经网络，经过图神经网络每一层的一次节点间信息传递，图神经网络为节点生成节点表征。

我们的任务是根据节点的属性(可以是类别型、也可以是数值型)、边的信息、边的属性（如果有的话）、已知的节点预测标签，对未知标签的节点做预测。

我们将以Cora数据集为例子进行说明，Cora是一个论文引用网络，节点代表论文，如果两篇论文存在引用关系，那么认为对应的两个节点之间存在边，每个节点由一个1433维的词包特征向量描述。我们的任务是推断每个文档的类别（共7类）。

2. 准备工作

获取并分析数据集

from torch_geometric.datasets import Planetoid
from torch_geometric.transforms import NormalizeFeatures

dataset = Planetoid(root='data/Planetoid', name='Cora', transform=NormalizeFeatures())

print()
print(f'Dataset: {dataset}:')
print('======================')
print(f'Number of graphs: {len(dataset)}')
print(f'Number of features: {dataset.num_features}')
print(f'Number of classes: {dataset.num_classes}')

data = dataset[0]  # Get the first graph object.

print()
print(data)
print('======================')

# Gather some statistics about the graph.
print(f'Number of nodes: {data.num_nodes}')
print(f'Number of edges: {data.num_edges}')
print(f'Average node degree: {data.num_edges / data.num_nodes:.2f}')
print(f'Number of training nodes: {data.train_mask.sum()}')
print(f'Training node label rate: {int(data.train_mask.sum()) / data.num_nodes:.2f}')
print(f'Contains isolated nodes: {data.contains_isolated_nodes()}')
print(f'Contains self-loops: {data.contains_self_loops()}')
print(f'Is undirected: {data.is_undirected()}')

3. MLP在图节点分类中的应用

理论上，我们应该能够仅根据文件的内容，即它的词包特征表示来推断文件的类别，而无需考虑文件之间的任何关系信息。让我们通过构建一个简单的MLP来验证这一点，该网络只对输入节点的特征进行操作，它在所有节点之间共享权重。

MLP图节点分类器：

from torch_geometric.datasets import Planetoid
from torch_geometric.transforms import NormalizeFeatures

dataset = Planetoid(root='data/Planetoid', name='Cora', transform=NormalizeFeatures())

data = dataset[0]  # Get the first graph object.

import torch
from torch.nn import Linear
import torch.nn.functional as F

class MLP(torch.nn.Module):
    def __init__(self, hidden_channels):
        super(MLP, self).__init__()
        torch.manual_seed(12345)
        self.lin1 = Linear(dataset.num_features, hidden_channels)
        self.lin2 = Linear(hidden_channels, dataset.num_classes)

    def forward(self, x):
        x = self.lin1(x)
        x = x.relu()
        x = F.dropout(x, p=0.5, training=self.training)
        x = self.lin2(x)
        return x

model = MLP(hidden_channels=16)
print(model)

criterion = torch.nn.CrossEntropyLoss()  # Define loss criterion.
optimizer = torch.optim.Adam(model.parameters(), lr=0.01, weight_decay=5e-4)  # Define optimizer.

def train():
    model.train()
    optimizer.zero_grad()  # Clear gradients.
    out = model(data.x)  # Perform a single forward pass.
    loss = criterion(out[data.train_mask], data.y[data.train_mask])  # Compute the loss solely based on the training nodes.
    loss.backward()  # Derive gradients.
    optimizer.step()  # Update parameters based on gradients.
    return loss

for epoch in range(1, 201):
    loss = train()
    print(f'Epoch: {epoch:03d}, Loss: {loss:.4f}')
    
def test():
    model.eval()
    out = model(data.x)
    pred = out.argmax(dim=1)  # Use the class with highest probability.
    test_correct = pred[data.test_mask] == data.y[data.test_mask]  # Check against ground-truth labels.
    test_acc = int(test_correct.sum()) / int(data.test_mask.sum())  # Derive ratio of correct predictions.
    return test_acc

test_acc = test()
print(f'Test Accuracy: {test_acc:.4f}')  
# Test Accuracy: 0.5900

正如我们所看到的，我们的MLP表现相当糟糕，只有大约59%的测试准确性。
为什么MLP没有表现得更好呢？ 其中一个重要原因是，用于训练此神经网络的有标签节点数量过少，此神经网络被过拟合，它对未见过的节点泛化性很差。

4. GCN及其在图节点分类任务中的应用

GCN的定义

GCN 神经网络层来源于论文“Semi-supervised Classification with Graph Convolutional Network”，其数学定义为，
$\mathbf{X}^{\prime} = \mathbf{\hat{D}}^{-1/2} \mathbf{\hat{A}} \mathbf{\hat{D}}^{-1/2} \mathbf{X} \mathbf{\Theta},$
其中 $\mathbf{\hat{A}} = \mathbf{A} + \mathbf{I}$ 表示插入自环的邻接矩阵， $\hat{D}_{ii} = \sum_{j=0} \hat{A}_{ij}$ 表示其对角线度矩阵。邻接矩阵可以包括不为 $1$ 的值，当邻接矩阵不为{0,1}值时，表示邻接矩阵存储的是边的权重。 $\mathbf{\hat{D}}^{-1/2} \mathbf{\hat{A}} \mathbf{\hat{D}}^{-1/2}$ 对称归一化矩阵。

它的节点式表述为：
$\mathbf{x}^{\prime}_i = \mathbf{\Theta} \sum_{j \in \mathcal{N}(v) \cup \{ i \}} \frac{e_{j,i}}{\sqrt{\hat{d}_j \hat{d}_i}} \mathbf{x}_j$
其中， $\hat{d}_i = 1 + \sum_{j \in \mathcal{N}(i)} e_{j,i}$ ， $e_{j,i}$ 表示从源节点 $j$ 到目标节点 $i$ 的边的对称归一化系数（默认值为1.0）。

PyG中GCNConv 模块说明

GCNConv构造函数接口：

GCNConv(in_channels: int, out_channels: int, improved: bool = False, cached: bool = False, add_self_loops: bool = True, normalize: bool = True, bias: bool = True, **kwargs)

其中：

in_channels：输入数据维度；
out_channels：输出数据维度；
improved：如果为true， $\mathbf{\hat{A}} = \mathbf{A} + 2\mathbf{I}$ ，其目的在于增强中心节点自身信息；
cached：是否存储 $\mathbf{\hat{D}}^{-1/2} \mathbf{\hat{A}} \mathbf{\hat{D}}^{-1/2}$ 的计算结果以便后续使用，这个参数只应在归纳学习（transductive learning）的景中设置为true；
add_self_loops：是否在邻接矩阵中增加自环边；
normalize：是否添加自环边并在运行中计算对称归一化系数；
bias：是否包含偏置项。

详细内容请大家参阅GCNConv官方文档。

基于GCN图神经网络的图节点分类

print('==============obtain data=============')
import torch
from torch_geometric.datasets import Planetoid
from torch_geometric.transforms import NormalizeFeatures

dataset = Planetoid(root='data/Planetoid', name='Cora', transform=NormalizeFeatures())

data = dataset[0]  # Get the first graph object.

# GCNConv
from torch_geometric.nn import GCNConv
import torch.nn.functional as F

class GCN(torch.nn.Module):
    def __init__(self, hidden_channels):
        super(GCN, self).__init__()
        torch.manual_seed(12345)
        self.conv1 = GCNConv(dataset.num_features, hidden_channels)
        self.conv2 = GCNConv(hidden_channels, dataset.num_classes)

    def forward(self, x, edge_index):
        x = self.conv1(x, edge_index)
        x = x.relu()
        x = F.dropout(x, p=0.5, training=self.training)
        x = self.conv2(x, edge_index)
        return x

model = GCN(hidden_channels=16)
print(model)

model.eval()

print('==============visualize=============')
import matplotlib.pyplot as plt
from sklearn.manifold import TSNE

def visualize(h, color):
    z = TSNE(n_components=2).fit_transform(out.detach().cpu().numpy())
    plt.figure(figsize=(10,10))
    plt.xticks([])
    plt.yticks([])

    plt.scatter(z[:, 0], z[:, 1], s=70, c=color, cmap="Set2")
    plt.show()
    
   
out = model(data.x, data.edge_index)
print('before training:')  
visualize(out, color=data.y)

print('==============GCNConv=============')
model = GCN(hidden_channels=16)
optimizer = torch.optim.Adam(model.parameters(), lr=0.01, weight_decay=5e-4)
criterion = torch.nn.CrossEntropyLoss()

def train():
      model.train()
      optimizer.zero_grad()  # Clear gradients.
      out = model(data.x, data.edge_index)  # Perform a single forward pass.
      loss = criterion(out[data.train_mask], data.y[data.train_mask])  # Compute the loss solely based on the training nodes.
      loss.backward()  # Derive gradients.
      optimizer.step()  # Update parameters based on gradients.
      return loss

for epoch in range(1, 201):
    loss = train()
    print(f'Epoch: {epoch:03d}, Loss: {loss:.4f}')

def test():
      model.eval()
      out = model(data.x, data.edge_index)
      pred = out.argmax(dim=1)  # Use the class with highest probability.
      test_correct = pred[data.test_mask] == data.y[data.test_mask]  # Check against ground-truth labels.
      test_acc = int(test_correct.sum()) / int(data.test_mask.sum())  # Derive ratio of correct predictions.
      return test_acc

test_acc = test()
print(f'Test Accuracy: {test_acc:.4f}')
# Test Accuracy: 0.8140

out = model(data.x, data.edge_index)
print('after training:')  
visualize(out, color=data.y)

通过将torch.nn.Linear layers 替换为PyG的GNN Conv Layers，我们可以轻松地将MLP模型转化为GNN模型。在下方的例子中，我们将MLP例子中的linear层替换为GCNConv层。

经过visualize函数的处理，7维特征的节点被嵌入到2维的平面上。

Before training:

通过简单地将线性层替换成GCN层，我们可以达到81.4%的测试准确率！与前面的仅获得59%的测试准确率的MLP分类器相比，现在的分类器准确性要高得多。这表明节点的邻接信息在取得更好的准确率方面起着关键作用。

最后我们还可以通过可视化我们训练过的模型输出的节点表征来再次验证这一点，现在同类节点的聚集在一起的情况更加明显了。

After training:

5. GAT及其在图节点分类任务中的应用

GAT的定义

图注意网络（GAT）来源于论文 Graph Attention Networks。其数学定义为，
$\mathbf{x}^{\prime}_i = \alpha_{i,i}\mathbf{\Theta}\mathbf{x}_{i} + \sum_{j \in \mathcal{N}(i)} \alpha_{i,j}\mathbf{\Theta}\mathbf{x}_{j},$
其中注意力系数 $\alpha_{i,j}$ 的计算方法为，
$\alpha_{i,j} = \frac{ \exp\left(\mathrm{LeakyReLU}\left(\mathbf{a}^{\top} [\mathbf{\Theta}\mathbf{x}_i \, \Vert \, \mathbf{\Theta}\mathbf{x}_j] \right)\right)} {\sum_{k \in \mathcal{N}(i) \cup \{ i \}} \exp\left(\mathrm{LeakyReLU}\left(\mathbf{a}^{\top} [\mathbf{\Theta}\mathbf{x}_i \, \Vert \, \mathbf{\Theta}\mathbf{x}_k] \right)\right)}.$

PyG中`GATConv` 模块说明

GATConv构造函数接口：

GATConv(in_channels: Union[int, Tuple[int, int]], out_channels: int, heads: int = 1, concat: bool = True, negative_slope: float = 0.2, dropout: float = 0.0, add_self_loops: bool = True, bias: bool = True, **kwargs)

其中：

in_channels：输入数据维度；
out_channels：输出数据维度；
heads：在GATConv使用多少个注意力模型（Number of multi-head-attentions）；
concat：如为true，不同注意力模型得到的节点表征被拼接到一起（表征维度翻倍），否则对不同注意力模型得到的节点表征求均值；

详细内容请大家参阅GATConv官方文档

基于GAT图神经网络的图节点分类

这一次，我们将MLP例子中的linear层替换为GATConv层，来实现基于GAT的图节点分类神经网络。

print('==============obtain data=============')
import torch
from torch_geometric.datasets import Planetoid
from torch_geometric.transforms import NormalizeFeatures

dataset = Planetoid(root='data/Planetoid', name='Cora', transform=NormalizeFeatures())

data = dataset[0]  # Get the first graph object.

# GATConv
from torch_geometric.nn import GATConv
import torch.nn.functional as F

class GAT(torch.nn.Module):
    def __init__(self, hidden_channels):
        super(GAT, self).__init__()
        torch.manual_seed(12345)
        self.conv1 = GATConv(dataset.num_features, hidden_channels)
        self.conv2 = GATConv(hidden_channels, dataset.num_classes)

    def forward(self, x, edge_index):
        x = self.conv1(x, edge_index)
        x = x.relu()
        x = F.dropout(x, p=0.5, training=self.training)
        x = self.conv2(x, edge_index)
        return x

model = GAT(hidden_channels=16)
print(model)

model.eval()

print('==============visualize=============')
import matplotlib.pyplot as plt
from sklearn.manifold import TSNE

def visualize(h, color):
    z = TSNE(n_components=2).fit_transform(out.detach().cpu().numpy())
    plt.figure(figsize=(10,10))
    plt.xticks([])
    plt.yticks([])

    plt.scatter(z[:, 0], z[:, 1], s=70, c=color, cmap="Set2")
    plt.show()
    
   
out = model(data.x, data.edge_index)
print('before training:')  
visualize(out, color=data.y)

print('==============GCNConv=============')
model = GAT(hidden_channels=16)
optimizer = torch.optim.Adam(model.parameters(), lr=0.01, weight_decay=5e-4)
criterion = torch.nn.CrossEntropyLoss()

def train():
      model.train()
      optimizer.zero_grad()  # Clear gradients.
      out = model(data.x, data.edge_index)  # Perform a single forward pass.
      loss = criterion(out[data.train_mask], data.y[data.train_mask])  # Compute the loss solely based on the training nodes.
      loss.backward()  # Derive gradients.
      optimizer.step()  # Update parameters based on gradients.
      return loss

for epoch in range(1, 201):
    loss = train()
    print(f'Epoch: {epoch:03d}, Loss: {loss:.4f}')

def test():
      model.eval()
      out = model(data.x, data.edge_index)
      pred = out.argmax(dim=1)  # Use the class with highest probability.
      test_correct = pred[data.test_mask] == data.y[data.test_mask]  # Check against ground-truth labels.
      test_acc = int(test_correct.sum()) / int(data.test_mask.sum())  # Derive ratio of correct predictions.
      return test_acc

test_acc = test()
print(f'Test Accuracy: {test_acc:.4f}')
# Test Accuracy: 0.7380

out = model(data.x, data.edge_index)
print('after training:')  
visualize(out, color=data.y)