Elementary analysis of graph convolution neural network (GCN)

Multilayer graph convolution network （GCN） The interlayer propagation rule of is ：
$$H^{(l+1)}=\sigma \left({\tilde{D}}^{-\frac{1}{2}}\tilde{A}{\tilde{D}}^{-\frac{1}{2}}{H}^{(l)}{W}^{(l)}\right)$$
among ,$\tilde{A}=A+I_N$,$A$ Is the adjacency matrix of a graph ,$I_N$ Is the unit matrix .${\tilde{D}}_{ii}={\sum }_j{\tilde{A}}_{ij}$.$W^{(l)}$ Is the trainable weight parameter of a specific layer .$\sigma (\cdot )$ It's an activation function .

The following figure is an example to illustrate how these parameters are generated :

Its adjacency matrix A by ：

$$\begin{gathered} A: \\ [[0.1.1.0.] \\ [1.0.1.0.] \\ [1.1.0.1.] \\ [0.0.1.0.]] \end{gathered}$$

$\tilde{A}=A+I_N$ by ：

$$\begin{gathered} \tilde{A}: \\ [[1.1.1.0.] \\ [1.1.1.0.] \\ [1.1.1.1.] \\ [0.0.1.1.]] \end{gathered}$$

${\tilde{D}}_{ii}={\sum }_j{\tilde{A}}_{ij}$ by （ Equivalent to degree matrix ）：

$$\begin{gathered} \tilde{D}: \\ [[3.0.0.0.] \\ [0.3.0.0.] \\ [0.0.4.0.] \\ [0.0.0.2.]] \end{gathered}$$

${\tilde{D}}^{-\frac{1}{2}}$ by ：

$$\begin{gathered} {\tilde{D}}^{-\frac{1}{2}}: \\ [[0.577350270.0.0.] \\ [0.0.577350270.0.] \\ [0.0.0.50.] \\ [0.0.0.0.70710678]] \end{gathered}$$

Take the following simple undirected graph to illustrate what calculation means ：

The formula is ：
$$H^{(l+1)}=\sigma \left({\tilde{D}}^{-\frac{1}{2}}\tilde{A}{\tilde{D}}^{-\frac{1}{2}}{H}^{(l)}{W}^{(l)}\right)$$
Let's not consider ${\tilde{D}}^{-\frac{1}{2}}$, That is, only consider $H^{(l+1)}=\sigma \left(\tilde{A}{H}^{(l)}{W}^{(l)}\right)$

Adjacency matrix $A$ and $\tilde{A}$ by ：
$$\begin{gathered} A= \\ [[011] \\ [100] \\ [100]] \\ \tilde{A}= \\ [[111] \\ [110] \\ [101]] \end{gathered}$$
The eigenvector matrix composed of the features of each node in the figure is ：
$$\begin{gathered} X= \\ [[0.10.4] \\ [0.20.3] \\ [0.10.2]] \end{gathered}$$
When only the adjacency matrix is considered , The calculation formula is ：$H^{(l+1)}=\sigma \left(A{H}^{(l)}{W}^{(l)}\right)$. The calculation process is （ Without multiplying by the weight ）：
$$\begin{gathered} [[011][[0.10.4][[0.2+0.10.3+0.2] \\ [100]\ast [0.20.3]=[0.10.4] \\ [100]][0.10.2]][0.10.4]] \end{gathered}$$
Now the characteristic of each node becomes the sum of the eigenvalues of all its neighbors . For example, the eigenvalue of the first node becomes [0.2+0.1 0.3+0.2], This is i yes 2、3 Sum of nodes .

Let's see what the calculation process means when considering the adjacency matrix plus the identity matrix , Its calculation formula is ：$H^{(l+1)}=\sigma \left(\tilde{A}{H}^{(l)}{W}^{(l)}\right)$, The calculation process is （ Without multiplying by the weight ）：
$$\begin{gathered} [[111][[0.10.4][[0.2+0.1+0.10.4+0.3+0.2] \\ [110]\ast [0.20.3]=[0.1+0.20.4+0.3] \\ [101]][0.10.2]][0.1+0.10.4+0.2]] \end{gathered}$$
After the adjacency matrix plus the identity matrix , The eigenvalue of each node becomes the sum of the eigenvalues of its own node and all neighbor nodes .

If a node has more neighbors , Then the eigenvalue after adding becomes very large , Therefore, it needs to be normalized , and ${\tilde{D}}^{-\frac{1}{2}}$ The function of plays a role of normalization , therefore GCN The inter layer propagation rule of is $H^{(l+1)}=\sigma \left({\tilde{D}}^{-\frac{1}{2}}\tilde{A}{\tilde{D}}^{-\frac{1}{2}}{H}^{(l)}{W}^{(l)}\right)$.

The following figure for GCN A schematic diagram of , Enter a feature dimension as C Graph structure of , after GCN Then get a new graph , The characteristics and dimensions of each node in the graph change （ from C Dimension becomes F dimension ）, But the structure of the graph remains unchanged .

The last layer GCN The resulting graph is used for specific tasks . In the classification task , Generally, each node in the graph is classified , therefore ：

1. The last layer GCN The number of hidden layer features of is equal to the number of categories , Used directly for softmax Output probability .
2. In the output of F After Witt's sign , Another full connection layer , Then turn to dimensions with the same number of categories .

To sum up

With symmetric adjacency matrix A Two levels of semi supervised node classification on the graph GCN For example , First, calculate in the pretreatment step $\hat{A}={\tilde{D}}^{-\frac{1}{2}}\tilde{A}{\tilde{D}}^{-\frac{1}{2}}$, The forward calculation of the model is ：
$$Z=f(X,A)=\mathrm{softmax}\left(\hat{A}\mathrm{ReLU}\left(\hat{A}X{W}^{(0)}\right)W^{(1)}\right)$$
$W^{(0)}\in {\mathbb{R}}^{C\times H}$ Is a person with $H$ Input of hidden layers of dimensional features - Hide the weight matrix .$W^{(1)}\in {\mathbb{R}}^{H\times F}$ Is a weight matrix hidden in the output .softmax Activation function . For classification tasks , Using the cross entropy loss function ：
$$\mathcal{L}=-\sum _{l\in {\mathcal{Y}}_L}\sum _{f=1}^F{Y}_{lf}\ln Z_{lf}$$

Code section

First import the package used .

import scipy.sparse as sp
import numpy as np
import torch
import math
from torch.nn.parameter import Parameter
import torch.nn as nn
import torch.nn.functional as F
import time
import argparse
import torch.optim as optim
from visdom import Visdom

Code address ：https://github.com/tkipf/pygcn

1、 Data processing

cora The data set consists of machine learning papers . These papers fall into one of the following seven categories ：

1. Case_Based
2. Genetic_Algorithms
3. Neural_Networks、
4. Probabilistic_Methods
5. Reinforcement_Learning
6. Rule_Learning
7. Theory

cora The data set includes two files ：

1. cora.content The file is the information of all data , Each line includes Node number , Eigenvector , Category .
2. cora.cites The file is side information , Every act node 1, node 2, It means that these two nodes are connected to form an edge .

The main function of the data processing part is ：

1. Use the information of edges to generate adjacency matrix , Then normalize it , In this paper, the normalization method is ：$\hat{A}={\tilde{D}}^{-\frac{1}{2}}\tilde{A}{\tilde{D}}^{-\frac{1}{2}}$, The normalization method in the code implementation is ：$\hat{A}={\tilde{D}}^{-1}\tilde{A}$.
2. Deal with eigenvectors , And normalize the eigenvector （ There is no normalization in the paper , not essential ）.
3. Divide the training set 、 Validation set and test set .

The specific code is as follows .

def encode_onehot(labels):
    """  Change the label to one_hot code  :param labels: :return: """
    # Get all categories in the tag 
    #set  Create an unordered and unrepeatable collection 
    classes = set(labels)
    # Assign one to each tag one-hot code 
    #identity The function is used for a n*n The identity matrix of ( The main diagonal elements are all 1, The rest are all 0 Matrix ).
    classes_dict = {
    c : np.identity(len(classes))[i,:] for i,c in enumerate(classes)}
    # Change the original label to one-hot code 
    labels_onehot = np.array(list(map(classes_dict.get,labels)),dtype = np.int32)
    return labels_onehot

def normalize(mx):
    '''  Row normalized sparse matrix  :param mx: :return: '''
    # Matrix row summation 
    rowsum = np.array(mx.sum(1))
    # Summative -1 Power 
    r_inv = np.power(rowsum,-1).flatten()
    # If it is inf, convert to 0
    r_inv[np.isinf(r_inv)] = 0.
    # Construct diagonal matrix 
    r_mat_inv = sp.diags(r_inv)
    # structure D^(-1)*A ,  Normalization operation 
    # In the formula is D^(-1/2)*A*D^(-1/2), Here is a simplified normalization operation 
    mx = r_mat_inv.dot(mx)
    return mx


def sparse_mx_to_torch_sparse_tensor(sparse_mx):
    '''  Transform a huge sparse matrix into torch Sparse tensor . :param sparse_mx: :return: '''
    sparse_mx = sparse_mx.tocoo().astype(np.float32)
    indices = torch.from_numpy(
        np.vstack((sparse_mx.row,sparse_mx.col)).astype(np.int64))
    values = torch.from_numpy(sparse_mx.data)
    shape = torch.Size(sparse_mx.shape)
    return torch.sparse.FloatTensor(indices,values,shape)



def load_data(path = "./data/cora/", dataset = "cora"):
    print('Loading {} dataset...'.format(dataset))

    # Load data from a text file 
    #genfromtxt(): Load data from a text file , Processing data at the same time .
    #idx_features_labels The shape of the ：( Number of nodes , Number + features + Category ) = (2708,1435)
    idx_features_labels = np.genfromtxt("{}{}.content".format(path,dataset),dtype = np.dtype(str))

    # Take the eigenvalue of each node , At the same time, it is transformed into a sparse matrix  .A You can take its value 
    features = sp.csr_matrix(idx_features_labels[:,1:-1],dtype = np.float32)
    # Change the tag value to one-hot
    labels = encode_onehot(idx_features_labels[:,-1])

    # Get the id
    idx = np.array(idx_features_labels[:,0],dtype = np.int32)
    # Will all id In order   Construct a   Dictionaries  key Is the node order of the source data ,value For from 0 To 1 Order value of 
    idx_map = {
    j:i for i,j in enumerate(idx)}
    # Read edge information , Each row of data consists of two nodes , It means that these two nodes are connected to form an edge 
    edges_unordered = np.genfromtxt("{}{}.cites".format(path,dataset),dtype = np.int32)
    # For each node   Renumber 
    #flatten() : Reduce the dimension of multidimensional data 
    edges = np.array(list(map(idx_map.get,edges_unordered.flatten())),dtype = np.int32).reshape(edges_unordered.shape)
    # Building adjacency matrix 
    #coo_matrix： Generate a matrix in coordinate format 
    adj = sp.coo_matrix((np.ones(edges.shape[0]),(edges[:,0],edges[:,1])),
                        shape = (labels.shape[0],labels.shape[0]),
                        dtype = np.float32)
    # Calculate transpose matrix 
    adj = adj + adj.T.multiply(adj.T > adj) - adj.multiply(adj.T > adj)

    # Normalize the features , Not absolutely necessary 
    features = normalize(features)
    # Yes A+I Normalize ,
    adj = normalize(adj + sp.eye(adj.shape[0]))

    # Divide training 、 verification 、 Samples tested 
    idx_train = range(140)
    idx_val = range(200,500)
    idx_test = range(500,1500)

    # take numpy The data goes to torch Format 
    features = torch.FloatTensor(np.array(features.todense()))
    labels = torch.LongTensor(np.where(labels)[1])
    adj = sparse_mx_to_torch_sparse_tensor(adj)

    idx_train = torch.LongTensor(idx_train)
    idx_val = torch.LongTensor(idx_val)
    idx_test = torch.LongTensor(idx_test)

    return adj,features,labels,idx_train,idx_val,idx_test

2、GCN Calculation process

The calculation process is to use the processed adjacency matrix, weight and eigenvector to calculate , Its operation formula is ：$\left(\hat{A} X W\right) $. The specific code is as follows ：

class GraphConvolution(nn.Module):
    """ Simple GCN layer, similar to https://arxiv.org/abs/1609.02907 """
    def __init__(self,in_features,out_features,bias = True):
        super(GraphConvolution, self).__init__()
        # Enter the feature dimension 
        self.in_features = in_features
        # Output feature dimensions 
        self.out_features = out_features
        # Weight parameters 
        self.weight = Parameter(torch.FloatTensor(in_features,out_features))
        # bias 
        if bias:
            self.bias = Parameter(torch.FloatTensor(out_features))
        else:
            self.register_parameter('bias',None)
        self.reset_parameters()
    # Initialize parameters 
    def reset_parameters(self):
        stdv = 1. / math.sqrt(self.weight.size(1))
        self.weight.data.uniform_(-stdv,stdv)
        if self.bias is not None:
            self.bias.data.uniform_(-stdv,stdv)


    def forward(self,input,adj):
        # The eigenvector is multiplied by the weight 
        support = torch.mm(input,self.weight)
        # Then multiply with the processed adjacency matrix 
        output = torch.spmm(adj,support)
        # Whether to add offset 
        if self.bias is not None:
            return output + self.bias
        else:
            return output
    # Used to display properties 
    def __repr__(self):
        return self.__class__.__name__ + ' ('\
            + str(self.in_features) + ' -> '\
            + str(self.out_features) + ')'

3、 Build a two-tier GCN The Internet

Use the above GCN The calculation process of can build a two-tier GCN The Internet , It also uses dropout, The formula is ：$\mathrm{softmax}\left(\hat{A}\mathrm{ReLU}\left(\hat{A}X{W}^{(0)}\right)W^{(1)}\right)$. The code is as follows ：

class GCN(nn.Module):
    def __init__(self,nfeat,nhid,nclass,dropout):
        super(GCN, self).__init__()
        self.gc1 = GraphConvolution(nfeat,nhid)
        self.gc2 = GraphConvolution(nhid,nclass)
        self.dropout = dropout

    def forward(self,x,adj):
        x = F.relu(self.gc1(x,adj))
        x = F.dropout(x,self.dropout,training = self.training)
        x = self.gc2(x,adj)
        return F.log_softmax(x,dim = 1)

4、 model training

Define a function of calculation accuracy .

def accuracy(output,labels):
    preds = output.max(1)[1].type_as(labels)
    correct = preds.eq(labels).double()
    correct = correct.sum()
    return correct / len(labels)

The following is the code of the training model

# Set training parameters 
parser = argparse.ArgumentParser()
parser.add_argument('--no-cuda', action='store_true', default=False,
                    help='Disables CUDA training.')
parser.add_argument('--fastmode', action='store_true', default=False,
                    help='Validate during training pass.')
parser.add_argument('--seed', type=int, default=42, help='Random seed.')
parser.add_argument('--epochs', type=int, default=200,
                    help='Number of epochs to train.')
parser.add_argument('--lr', type=float, default=0.01,
                    help='Initial learning rate.')
parser.add_argument('--weight_decay', type=float, default=5e-4,
                    help='Weight decay (L2 loss on parameters).')
parser.add_argument('--hidden', type=int, default=16,
                    help='Number of hidden units.')
parser.add_argument('--dropout', type=float, default=0.5,
                    help='Dropout rate (1 - keep probability).')

#args = parser.parse_args()
args = parser.parse_known_args()[0]
args.cuda = not args.no_cuda and torch.cuda.is_available()

# Set random seeds 
np.random.seed(args.seed)
torch.manual_seed(args.seed)
if args.cuda:
    torch.cuda.manual_seed(args.seed)

# Load data 
adj,features,labels,idx_train,idx_val,idx_test = load_data()

# Load models and optimizers 
model = GCN(nfeat = features.shape[1],
            nhid = args.hidden,
            nclass = labels.max().item() + 1,
            dropout = args.dropout)
optimizer = optim.Adam(model.parameters(),
                       lr = args.lr,weight_decay = args.weight_decay)

# If you use GPU, Send data to GPU
if model.cuda():
    model.cuda()
    features = features.cuda()
    adj = adj.cuda()
    labels = labels.cuda()
    idx_train = idx_train.cuda()
    idx_val = idx_val.cuda()
    idx_test = idx_test.cuda()

# Training functions 
def train(epoch,viz):
    t = time.time()
    model.train()
    optimizer.zero_grad()
    output = model(features,adj)
    # Calculate the training loss and accuracy 
    loss_train = F.nll_loss(output[idx_train],labels[idx_train])
    acc_train = accuracy(output[idx_train],labels[idx_train])
    loss_train.backward()
    optimizer.step()

    if not args.fastmode:
        model.eval()
        output = model(features,adj)

    # Calculate the verification loss and accuracy 
    loss_val = F.nll_loss(output[idx_val],labels[idx_val])
    acc_val = accuracy(output[idx_val],labels[idx_val])
    viz.line(Y=np.column_stack((acc_train.cpu(),acc_val.cpu())),
             X=np.column_stack((epoch, epoch)),
                    win='line1',
                    opts=dict(legend=['acc_train','acc_val'],
                        title='acc',
                        xlabel='epoch',
                        ylabel='acc'),
                    update = None if epoch == 0 else 'append'
                    )
    viz.line(Y=np.column_stack((loss_train.detach().cpu().numpy(),loss_val.detach().cpu().numpy())),
             X=np.column_stack((epoch, epoch)),
                    win='line2',
                    opts=dict(legend=['loss_train','loss_val'],
                        title='loss',
                        xlabel='epoch',
                        ylabel='loss'),
                    update = None if epoch == 0 else 'append'
                    )
	''' print('Epoch: {:04d}'.format(epoch + 1), 'loss_train: {:.4f}'.format(loss_train.item()), 'acc_train: {:.4f}'.format(acc_train.item()), 'loss_val: {:.4f}'.format(loss_val.item()), 'acc_val: {:.4f}'.format(acc_val.item()), 'time: {:.4f}s'.format(time.time() - t)) '''
             
# Test functions 
def test():
    model.eval()
    output = model(features,adj)
    loss_test = F.nll_loss(output[idx_test],labels[idx_test])
    acc_test = accuracy(output[idx_test],labels[idx_test])
    print("Test set results:",
          "loss= {:.4f}".format(loss_test.item()),
          "accuracy= {:.4f}".format(acc_test.item()))

Start training

t_total = time.time()
viz = Visdom(env='test1')
for epoch in range(args.epochs):
    train(epoch,viz)
print('Optimization Finished!')
print("Total time elapsed: {:.4f}s".format(time.time() - t_total))

# Testing
test()

Optimization Finished!
Total time elapsed: 8.1138s
Test set results: loss= 0.7548 accuracy= 0.8260

The loss visualization curve during training is ：

The visualization curve of accuracy during training is ：