Note

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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:

\[\begin{split}h_i^{l+1} = \sigma\left(\sum_{j\in N_i}\frac{1}{c_i} W^{(l)} h_j^{(l)}\right)~~~~~~~~~~(1)\\\end{split}\]

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:

\[\begin{split}h_i^{l+1} = \sigma\left(W_0^{(l)}h_i^{(l)}+\sum_{r\in R}\sum_{j\in N_i^r}\frac{1}{c_{i,r}}W_r^{(l)}h_j^{(l)}\right)~~~~~~~~~~(2)\\\end{split}\]

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.

\[\begin{split}W_r^{(l)}=\sum\limits_{b=1}^B a_{rb}^{(l)}V_b^{(l)}~~~~~~~~~~(3)\\\end{split}\]

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)

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()
        )
    )

start training...
Epoch 00000 | Train Accuracy: 0.2143 | Train Loss: 1.3869 | Validation Accuracy: 0.2778 | Validation loss: 1.3867
Epoch 00001 | Train Accuracy: 0.9429 | Train Loss: 1.3595 | Validation Accuracy: 0.9444 | Validation loss: 1.3667
Epoch 00002 | Train Accuracy: 0.9429 | Train Loss: 1.3176 | Validation Accuracy: 0.9722 | Validation loss: 1.3352
Epoch 00003 | Train Accuracy: 0.9500 | Train Loss: 1.2587 | Validation Accuracy: 0.9722 | Validation loss: 1.2901
Epoch 00004 | Train Accuracy: 0.9429 | Train Loss: 1.1875 | Validation Accuracy: 0.9722 | Validation loss: 1.2333
Epoch 00005 | Train Accuracy: 0.9429 | Train Loss: 1.1142 | Validation Accuracy: 0.9722 | Validation loss: 1.1714
Epoch 00006 | Train Accuracy: 0.9500 | Train Loss: 1.0488 | Validation Accuracy: 0.9722 | Validation loss: 1.1131
Epoch 00007 | Train Accuracy: 0.9500 | Train Loss: 0.9953 | Validation Accuracy: 0.9722 | Validation loss: 1.0631
Epoch 00008 | Train Accuracy: 0.9500 | Train Loss: 0.9535 | Validation Accuracy: 0.9722 | Validation loss: 1.0220
Epoch 00009 | Train Accuracy: 0.9500 | Train Loss: 0.9208 | Validation Accuracy: 0.9722 | Validation loss: 0.9884
Epoch 00010 | Train Accuracy: 0.9571 | Train Loss: 0.8947 | Validation Accuracy: 0.9722 | Validation loss: 0.9602
Epoch 00011 | Train Accuracy: 0.9571 | Train Loss: 0.8734 | Validation Accuracy: 0.9722 | Validation loss: 0.9359
Epoch 00012 | Train Accuracy: 0.9571 | Train Loss: 0.8558 | Validation Accuracy: 0.9722 | Validation loss: 0.9146
Epoch 00013 | Train Accuracy: 0.9571 | Train Loss: 0.8414 | Validation Accuracy: 0.9722 | Validation loss: 0.8963
Epoch 00014 | Train Accuracy: 0.9571 | Train Loss: 0.8298 | Validation Accuracy: 0.9722 | Validation loss: 0.8808
Epoch 00015 | Train Accuracy: 0.9571 | Train Loss: 0.8208 | Validation Accuracy: 0.9722 | Validation loss: 0.8683
Epoch 00016 | Train Accuracy: 0.9571 | Train Loss: 0.8139 | Validation Accuracy: 0.9722 | Validation loss: 0.8582
Epoch 00017 | Train Accuracy: 0.9571 | Train Loss: 0.8088 | Validation Accuracy: 0.9722 | Validation loss: 0.8504
Epoch 00018 | Train Accuracy: 0.9571 | Train Loss: 0.8050 | Validation Accuracy: 0.9722 | Validation loss: 0.8444
Epoch 00019 | Train Accuracy: 0.9571 | Train Loss: 0.8021 | Validation Accuracy: 0.9722 | Validation loss: 0.8398
Epoch 00020 | Train Accuracy: 0.9571 | Train Loss: 0.7997 | Validation Accuracy: 0.9722 | Validation loss: 0.8362
Epoch 00021 | Train Accuracy: 0.9571 | Train Loss: 0.7977 | Validation Accuracy: 0.9722 | Validation loss: 0.8334
Epoch 00022 | Train Accuracy: 0.9571 | Train Loss: 0.7958 | Validation Accuracy: 0.9722 | Validation loss: 0.8311
Epoch 00023 | Train Accuracy: 0.9571 | Train Loss: 0.7939 | Validation Accuracy: 0.9722 | Validation loss: 0.8293
Epoch 00024 | Train Accuracy: 0.9571 | Train Loss: 0.7918 | Validation Accuracy: 0.9722 | Validation loss: 0.8277