Graph Convolutional Network
This is a gentle introduction of using DGL to implement Graph Convolutional Networks (Kipf & Welling et al., Semi-Supervised Classification with Graph Convolutional Networks). We build upon the earlier tutorial on DGLGraph and demonstrate how DGL combines graph with deep neural network and learn structural representations.
Model Overview
Graph Convolutional Network
GCN from the perspective of message passing
We describe a layer of graph convolutional neural network from a message passing perspective; the math can be found here. It boils down to the following step, for each node u:
-
Aggregate neighbors' representations
to produce an intermediate representation
.
-
Transform the aggregated representation
non-linearity:.
We will implement step 1 with DGL message passing, and step 2 with the apply_nodes method, whose node UDF will be a PyTorch nn.Module.
GCN implementation with DGL
We first define the message and reduce function as usual. Since the aggregation on a node only involves summing over the neighbors’ representations
, we can simply
use builtin functions:
import dgl
import dgl.function as fn
import torch as th
import torch.nn as nn
import torch.nn.functional as F
from dgl import DGLGraph
gcn_msg = fn.copy_src(src='h', out='m')
gcn_reduce = fn.sum(msg='m', out='h')
We then define the node UDF for apply_nodes, which is a fully-connected layer:
class NodeApplyModule(nn.Module):
def __init__(self, in_feats, out_feats, activation):
super(NodeApplyModule, self).__init__()
self.linear = nn.Linear(in_feats, out_feats)
self.activation = activation
def forward(self, node):
h = self.linear(node.data['h'])
if self.activation is not None:
h = self.activation(h)
return {'h' : h}
We then proceed to define the GCN module. A GCN layer essentially performs message passing on all the nodes then applies the NodeApplyModule. Note that we omitted the dropout in the paper for simplicity.
class GCN(nn.Module):
def __init__(self, in_feats, out_feats, activation):
super(GCN, self).__init__()
self.apply_mod = NodeApplyModule(in_feats, out_feats, activation)
def forward(self, g, feature):
g.ndata['h'] = feature
g.update_all(gcn_msg, gcn_reduce)
g.apply_nodes(func=self.apply_mod)
return g.ndata.pop('h')
The forward function is essentially the same as any other commonly seen NNs model in PyTorch. We can initialize GCN like any nn.Module. For example, let’s define a simple neural network consisting of two GCN layers. Suppose we are training the classifier for the cora dataset(the input feature size is 1433 and the number of classes is 7). The last GCN layer computes node embeddings, so the last layer in general doesn’t apply activation.
class Net(nn.Module):
def __init__(self):
super(Net, self).__init__()
self.gcn1 = GCN(1433, 16, F.relu)
self.gcn2 = GCN(16, 7, None)
def forward(self, g, features):
x = self.gcn1(g, features)
x = self.gcn2(g, x)
return x
net = Net()
print(net)
# Results:
Net(
(gcn1): GCN(
(apply_mod): NodeApplyModule(
(linear): Linear(in_features=1433, out_features=16, bias=True)
)
)
(gcn2): GCN(
(apply_mod): NodeApplyModule(
(linear): Linear(in_features=16, out_features=7, bias=True)
)
)
)
We load the cora dataset using DGL’s built-in data module.
from dgl.data import citation_graph as citegrh
import networkx as nx
def load_cora_data():
data = citegrh.load_cora()
features = th.FloatTensor(data.features)
labels = th.LongTensor(data.labels)
train_mask = th.BoolTensor(data.train_mask)
test_mask = th.BoolTensor(data.test_mask)
g = data.graph
# add self loop
g.remove_edges_from(nx.selfloop_edges(g))
g = DGLGraph(g)
g.add_edges(g.nodes(), g.nodes())
return g, features, labels, train_mask, test_mask
When a model is trained, we can use the following method to evaluate the performance of the model on the test dataset:
def evaluate(model, g, features, labels, mask):
model.eval()
with th.no_grad():
logits = model(g, features)
logits = logits[mask]
labels = labels[mask]
_, indices = th.max(logits, dim=1)
correct = th.sum(indices == labels)
return correct.item() * 1.0 / len(labels)
We then train the network as follows:
import time
import numpy as np
g, features, labels, train_mask, test_mask = load_cora_data()
optimizer = th.optim.Adam(net.parameters(), lr=1e-3)
dur = []
for epoch in range(50):
if epoch >=3:
t0 = time.time()
net.train()
logits = net(g, features)
logp = F.log_softmax(logits, 1)
loss = F.nll_loss(logp[train_mask], labels[train_mask])
optimizer.zero_grad()
loss.backward()
optimizer.step()
if epoch >=3:
dur.append(time.time() - t0)
acc = evaluate(net, g, features, labels, test_mask)
print("Epoch {:05d} | Loss {:.4f} | Test Acc {:.4f} | Time(s) {:.4f}".format(
epoch, loss.item(), acc, np.mean(dur)))
# Results:
Epoch 00000 | Loss 1.9180 | Test Acc 0.3090 | Time(s) nan
Epoch 00001 | Loss 1.8872 | Test Acc 0.3100 | Time(s) nan
Epoch 00002 | Loss 1.8609 | Test Acc 0.3200 | Time(s) nan
Epoch 00003 | Loss 1.8365 | Test Acc 0.3310 | Time(s) 0.1041
Epoch 00004 | Loss 1.8127 | Test Acc 0.3420 | Time(s) 0.1042
Epoch 00005 | Loss 1.7886 | Test Acc 0.3520 | Time(s) 0.1038
Epoch 00006 | Loss 1.7641 | Test Acc 0.3880 | Time(s) 0.1066
Epoch 00007 | Loss 1.7396 | Test Acc 0.4090 | Time(s) 0.1093
Epoch 00008 | Loss 1.7151 | Test Acc 0.4110 | Time(s) 0.1094
Epoch 00009 | Loss 1.6901 | Test Acc 0.4210 | Time(s) 0.1088
Epoch 00010 | Loss 1.6649 | Test Acc 0.4300 | Time(s) 0.1081
Epoch 00011 | Loss 1.6396 | Test Acc 0.4370 | Time(s) 0.1089
......
......
......
Epoch 00044 | Loss 0.9971 | Test Acc 0.6620 | Time(s) 0.1101
Epoch 00045 | Loss 0.9845 | Test Acc 0.6690 | Time(s) 0.1100
Epoch 00046 | Loss 0.9721 | Test Acc 0.6710 | Time(s) 0.1099
Epoch 00047 | Loss 0.9599 | Test Acc 0.6710 | Time(s) 0.1099
Epoch 00048 | Loss 0.9480 | Test Acc 0.6740 | Time(s) 0.1097
Epoch 00049 | Loss 0.9364 | Test Acc 0.6780 | Time(s) 0.1114
GCN in one formula
Mathematically, the GCN model follows this formula:
Here, denotes the
layer in the network,
is the
non-linearity, and is the
weight matrix for this layer. and
, as commonly seen, represent
degree matrix and adjacency matrix, respectively. The ~ is a renormalization trick in which we add a self-connection to each node of the graph, and build the corresponding degree and adjacency matrix. The shape of the input is
, where
is
the number of nodes and is
the number of input features. We can chain up multiple layers as such to produce a node-level representation output with shape :mathN times F, where is the dimension of the output node feature vector.
The equation can be efficiently implemented using sparse matrix multiplication kernels (such as Kipf’s pygcn code). The above DGL implementation in fact has already used this trick due to the use of builtin functions. To understand what is under the hood, please read our tutorial on PageRank.
