Note

Click here to download the full example code

# Relational Graph Convolutional Network¶

**Author:** Lingfan Yu, Mufei Li, Zheng Zhang

Warning

The tutorial aims at gaining insights into the paper, with code as a mean of explanation. The implementation thus is NOT optimized for running efficiency. For recommended implementation, please refer to the official examples.

In this tutorial, you learn how to implement a relational graph convolutional network (R-GCN). This type of network is one effort to generalize GCN to handle different relationships between entities in a knowledge base. To learn more about the research behind R-GCN, see Modeling Relational Data with Graph Convolutional Networks

The straightforward graph convolutional network (GCN) exploits structural information of a dataset (that is, the graph connectivity) in order to improve the extraction of node representations. Graph edges are left as untyped.

A knowledge graph is made up of a collection of triples in the form subject, relation, object. Edges thus encode important information and have their own embeddings to be learned. Furthermore, there may exist multiple edges among any given pair.

## A brief introduction to R-GCN¶

In *statistical relational learning* (SRL), there are two fundamental
tasks:

**Entity classification**- Where you assign types and categorical properties to entities.**Link prediction**- Where you recover missing triples.

In both cases, missing information is expected to be recovered from the neighborhood structure of the graph. For example, the R-GCN paper cited earlier provides the following example. Knowing that Mikhail Baryshnikov was educated at the Vaganova Academy implies both that Mikhail Baryshnikov should have the label person, and that the triple (Mikhail Baryshnikov, lived in, Russia) must belong to the knowledge graph.

R-GCN solves these two problems using a common graph convolutional network. It’s extended with multi-edge encoding to compute embedding of the entities, but with different downstream processing.

Entity classification is done by attaching a softmax classifier at the final embedding of an entity (node). Training is through loss of standard cross-entropy.

Link prediction is done by reconstructing an edge with an autoencoder architecture, using a parameterized score function. Training uses negative sampling.

This tutorial focuses on the first task, entity classification, to show how to generate entity representation. Complete code for both tasks is found in the DGL Github repository.

## Key ideas of R-GCN¶

Recall that in GCN, the hidden representation for each node \(i\) at \((l+1)^{th}\) layer is computed by:

where \(c_i\) is a normalization constant.

The key difference between R-GCN and GCN is that in R-GCN, edges can represent different relations. In GCN, weight \(W^{(l)}\) in equation \((1)\) is shared by all edges in layer \(l\). In contrast, in R-GCN, different edge types use different weights and only edges of the same relation type \(r\) are associated with the same projection weight \(W_r^{(l)}\).

So the hidden representation of entities in \((l+1)^{th}\) layer in R-GCN can be formulated as the following equation:

where \(N_i^r\) denotes the set of neighbor indices of node \(i\) under relation \(r\in R\) and \(c_{i,r}\) is a normalization constant. In entity classification, the R-GCN paper uses \(c_{i,r}=|N_i^r|\).

The problem of applying the above equation directly is the rapid growth of the number of parameters, especially with highly multi-relational data. In order to reduce model parameter size and prevent overfitting, the original paper proposes to use basis decomposition.

Therefore, the weight \(W_r^{(l)}\) is a linear combination of basis transformation \(V_b^{(l)}\) with coefficients \(a_{rb}^{(l)}\). The number of bases \(B\) is much smaller than the number of relations in the knowledge base.

Note

Another weight regularization, block-decomposition, is implemented in the link prediction.

## Implement R-GCN in DGL¶

An R-GCN model is composed of several R-GCN layers. The first R-GCN layer also serves as input layer and takes in features (for example, description texts) that are associated with node entity and project to hidden space. In this tutorial, we only use the entity ID as an entity feature.

### R-GCN layers¶

For each node, an R-GCN layer performs the following steps:

Compute outgoing message using node representation and weight matrix associated with the edge type (message function)

Aggregate incoming messages and generate new node representations (reduce and apply function)

The following code is the definition of an R-GCN hidden layer.

Note

Each relation type is associated with a different weight. Therefore, the full weight matrix has three dimensions: relation, input_feature, output_feature.

Note

This is showing how to implement an R-GCN from scratch. DGL provides a more
efficient `builtin R-GCN layer module`

.

```
import os
os.environ["DGLBACKEND"] = "pytorch"
from functools import partial
import dgl
import dgl.function as fn
import torch
import torch.nn as nn
import torch.nn.functional as F
from dgl import DGLGraph
class RGCNLayer(nn.Module):
def __init__(
self,
in_feat,
out_feat,
num_rels,
num_bases=-1,
bias=None,
activation=None,
is_input_layer=False,
):
super(RGCNLayer, self).__init__()
self.in_feat = in_feat
self.out_feat = out_feat
self.num_rels = num_rels
self.num_bases = num_bases
self.bias = bias
self.activation = activation
self.is_input_layer = is_input_layer
# sanity check
if self.num_bases <= 0 or self.num_bases > self.num_rels:
self.num_bases = self.num_rels
# weight bases in equation (3)
self.weight = nn.Parameter(
torch.Tensor(self.num_bases, self.in_feat, self.out_feat)
)
if self.num_bases < self.num_rels:
# linear combination coefficients in equation (3)
self.w_comp = nn.Parameter(
torch.Tensor(self.num_rels, self.num_bases)
)
# add bias
if self.bias:
self.bias = nn.Parameter(torch.Tensor(out_feat))
# init trainable parameters
nn.init.xavier_uniform_(
self.weight, gain=nn.init.calculate_gain("relu")
)
if self.num_bases < self.num_rels:
nn.init.xavier_uniform_(
self.w_comp, gain=nn.init.calculate_gain("relu")
)
if self.bias:
nn.init.xavier_uniform_(
self.bias, gain=nn.init.calculate_gain("relu")
)
def forward(self, g):
if self.num_bases < self.num_rels:
# generate all weights from bases (equation (3))
weight = self.weight.view(
self.in_feat, self.num_bases, self.out_feat
)
weight = torch.matmul(self.w_comp, weight).view(
self.num_rels, self.in_feat, self.out_feat
)
else:
weight = self.weight
if self.is_input_layer:
def message_func(edges):
# for input layer, matrix multiply can be converted to be
# an embedding lookup using source node id
embed = weight.view(-1, self.out_feat)
index = edges.data[dgl.ETYPE] * self.in_feat + edges.src["id"]
return {"msg": embed[index] * edges.data["norm"]}
else:
def message_func(edges):
w = weight[edges.data[dgl.ETYPE]]
msg = torch.bmm(edges.src["h"].unsqueeze(1), w).squeeze()
msg = msg * edges.data["norm"]
return {"msg": msg}
def apply_func(nodes):
h = nodes.data["h"]
if self.bias:
h = h + self.bias
if self.activation:
h = self.activation(h)
return {"h": h}
g.update_all(message_func, fn.sum(msg="msg", out="h"), apply_func)
```

### Full R-GCN model defined¶

```
class Model(nn.Module):
def __init__(
self,
num_nodes,
h_dim,
out_dim,
num_rels,
num_bases=-1,
num_hidden_layers=1,
):
super(Model, self).__init__()
self.num_nodes = num_nodes
self.h_dim = h_dim
self.out_dim = out_dim
self.num_rels = num_rels
self.num_bases = num_bases
self.num_hidden_layers = num_hidden_layers
# create rgcn layers
self.build_model()
# create initial features
self.features = self.create_features()
def build_model(self):
self.layers = nn.ModuleList()
# input to hidden
i2h = self.build_input_layer()
self.layers.append(i2h)
# hidden to hidden
for _ in range(self.num_hidden_layers):
h2h = self.build_hidden_layer()
self.layers.append(h2h)
# hidden to output
h2o = self.build_output_layer()
self.layers.append(h2o)
# initialize feature for each node
def create_features(self):
features = torch.arange(self.num_nodes)
return features
def build_input_layer(self):
return RGCNLayer(
self.num_nodes,
self.h_dim,
self.num_rels,
self.num_bases,
activation=F.relu,
is_input_layer=True,
)
def build_hidden_layer(self):
return RGCNLayer(
self.h_dim,
self.h_dim,
self.num_rels,
self.num_bases,
activation=F.relu,
)
def build_output_layer(self):
return RGCNLayer(
self.h_dim,
self.out_dim,
self.num_rels,
self.num_bases,
activation=partial(F.softmax, dim=1),
)
def forward(self, g):
if self.features is not None:
g.ndata["id"] = self.features
for layer in self.layers:
layer(g)
return g.ndata.pop("h")
```

### Handle dataset¶

This tutorial uses Institute for Applied Informatics and Formal Description Methods (AIFB) dataset from R-GCN paper.

```
# load graph data
dataset = dgl.data.rdf.AIFBDataset()
g = dataset[0]
category = dataset.predict_category
train_mask = g.nodes[category].data.pop("train_mask")
test_mask = g.nodes[category].data.pop("test_mask")
train_idx = torch.nonzero(train_mask, as_tuple=False).squeeze()
test_idx = torch.nonzero(test_mask, as_tuple=False).squeeze()
labels = g.nodes[category].data.pop("label")
num_rels = len(g.canonical_etypes)
num_classes = dataset.num_classes
# normalization factor
for cetype in g.canonical_etypes:
g.edges[cetype].data["norm"] = dgl.norm_by_dst(g, cetype).unsqueeze(1)
category_id = g.ntypes.index(category)
```

Out:

```
Done loading data from cached files.
```

### Create graph and model¶

```
# configurations
n_hidden = 16 # number of hidden units
n_bases = -1 # use number of relations as number of bases
n_hidden_layers = 0 # use 1 input layer, 1 output layer, no hidden layer
n_epochs = 25 # epochs to train
lr = 0.01 # learning rate
l2norm = 0 # L2 norm coefficient
# create graph
g = dgl.to_homogeneous(g, edata=["norm"])
node_ids = torch.arange(g.num_nodes())
target_idx = node_ids[g.ndata[dgl.NTYPE] == category_id]
# create model
model = Model(
g.num_nodes(),
n_hidden,
num_classes,
num_rels,
num_bases=n_bases,
num_hidden_layers=n_hidden_layers,
)
```

### Training loop¶

```
# optimizer
optimizer = torch.optim.Adam(model.parameters(), lr=lr, weight_decay=l2norm)
print("start training...")
model.train()
for epoch in range(n_epochs):
optimizer.zero_grad()
logits = model.forward(g)
logits = logits[target_idx]
loss = F.cross_entropy(logits[train_idx], labels[train_idx])
loss.backward()
optimizer.step()
train_acc = torch.sum(logits[train_idx].argmax(dim=1) == labels[train_idx])
train_acc = train_acc.item() / len(train_idx)
val_loss = F.cross_entropy(logits[test_idx], labels[test_idx])
val_acc = torch.sum(logits[test_idx].argmax(dim=1) == labels[test_idx])
val_acc = val_acc.item() / len(test_idx)
print(
"Epoch {:05d} | ".format(epoch)
+ "Train Accuracy: {:.4f} | Train Loss: {:.4f} | ".format(
train_acc, loss.item()
)
+ "Validation Accuracy: {:.4f} | Validation loss: {:.4f}".format(
val_acc, val_loss.item()
)
)
```

Out:

```
start training...
Epoch 00000 | Train Accuracy: 0.1357 | Train Loss: 1.3868 | Validation Accuracy: 0.1944 | Validation loss: 1.3868
Epoch 00001 | Train Accuracy: 0.9571 | Train Loss: 1.3576 | Validation Accuracy: 0.9444 | Validation loss: 1.3644
Epoch 00002 | Train Accuracy: 0.9571 | Train Loss: 1.3148 | Validation Accuracy: 0.9444 | Validation loss: 1.3300
Epoch 00003 | Train Accuracy: 0.9571 | Train Loss: 1.2569 | Validation Accuracy: 0.9444 | Validation loss: 1.2829
Epoch 00004 | Train Accuracy: 0.9571 | Train Loss: 1.1880 | Validation Accuracy: 0.9444 | Validation loss: 1.2250
Epoch 00005 | Train Accuracy: 0.9571 | Train Loss: 1.1164 | Validation Accuracy: 0.9444 | Validation loss: 1.1618
Epoch 00006 | Train Accuracy: 0.9571 | Train Loss: 1.0513 | Validation Accuracy: 0.9444 | Validation loss: 1.1011
Epoch 00007 | Train Accuracy: 0.9571 | Train Loss: 0.9981 | Validation Accuracy: 0.9444 | Validation loss: 1.0497
Epoch 00008 | Train Accuracy: 0.9571 | Train Loss: 0.9566 | Validation Accuracy: 0.9722 | Validation loss: 1.0091
Epoch 00009 | Train Accuracy: 0.9571 | Train Loss: 0.9247 | Validation Accuracy: 0.9722 | Validation loss: 0.9772
Epoch 00010 | Train Accuracy: 0.9571 | Train Loss: 0.8996 | Validation Accuracy: 0.9722 | Validation loss: 0.9515
Epoch 00011 | Train Accuracy: 0.9643 | Train Loss: 0.8794 | Validation Accuracy: 0.9722 | Validation loss: 0.9304
Epoch 00012 | Train Accuracy: 0.9643 | Train Loss: 0.8627 | Validation Accuracy: 0.9722 | Validation loss: 0.9127
Epoch 00013 | Train Accuracy: 0.9643 | Train Loss: 0.8485 | Validation Accuracy: 0.9722 | Validation loss: 0.8977
Epoch 00014 | Train Accuracy: 0.9643 | Train Loss: 0.8367 | Validation Accuracy: 0.9722 | Validation loss: 0.8850
Epoch 00015 | Train Accuracy: 0.9643 | Train Loss: 0.8269 | Validation Accuracy: 0.9722 | Validation loss: 0.8742
Epoch 00016 | Train Accuracy: 0.9643 | Train Loss: 0.8192 | Validation Accuracy: 0.9722 | Validation loss: 0.8650
Epoch 00017 | Train Accuracy: 0.9643 | Train Loss: 0.8132 | Validation Accuracy: 0.9722 | Validation loss: 0.8573
Epoch 00018 | Train Accuracy: 0.9643 | Train Loss: 0.8086 | Validation Accuracy: 0.9722 | Validation loss: 0.8510
Epoch 00019 | Train Accuracy: 0.9643 | Train Loss: 0.8051 | Validation Accuracy: 0.9722 | Validation loss: 0.8459
Epoch 00020 | Train Accuracy: 0.9643 | Train Loss: 0.8023 | Validation Accuracy: 0.9722 | Validation loss: 0.8417
Epoch 00021 | Train Accuracy: 0.9643 | Train Loss: 0.8000 | Validation Accuracy: 0.9722 | Validation loss: 0.8383
Epoch 00022 | Train Accuracy: 0.9643 | Train Loss: 0.7980 | Validation Accuracy: 0.9722 | Validation loss: 0.8354
Epoch 00023 | Train Accuracy: 0.9643 | Train Loss: 0.7961 | Validation Accuracy: 0.9722 | Validation loss: 0.8330
Epoch 00024 | Train Accuracy: 0.9643 | Train Loss: 0.7943 | Validation Accuracy: 0.9722 | Validation loss: 0.8310
```

## The second task, link prediction¶

So far, you have seen how to use DGL to implement entity classification with an R-GCN model. In the knowledge base setting, representation generated by R-GCN can be used to uncover potential relationships between nodes. In the R-GCN paper, the authors feed the entity representations generated by R-GCN into the DistMult prediction model to predict possible relationships.

The implementation is similar to that presented here, but with an extra DistMult layer stacked on top of the R-GCN layers. You can find the complete implementation of link prediction with R-GCN in our Github Python code example.

**Total running time of the script:** ( 0 minutes 2.977 seconds)