Link Prediction using Graph Neural Networks

In the introduction, you have already learned the basic workflow of using GNNs for node classification, i.e. predicting the category of a node in a graph. This tutorial will teach you how to train a GNN for link prediction, i.e. predicting the existence of an edge between two arbitrary nodes in a graph.

By the end of this tutorial you will be able to

• Build a GNN-based link prediction model.

• Train and evaluate the model on a small DGL-provided dataset.

(Time estimate: 28 minutes)

import itertools
import os

os.environ["DGLBACKEND"] = "pytorch"

import dgl
import dgl.data
import numpy as np
import scipy.sparse as sp
import torch
import torch.nn as nn
import torch.nn.functional as F

Overview of Link Prediction with GNN

Many applications such as social recommendation, item recommendation, knowledge graph completion, etc., can be formulated as link prediction, which predicts whether an edge exists between two particular nodes. This tutorial shows an example of predicting whether a citation relationship, either citing or being cited, between two papers exists in a citation network.

This tutorial formulates the link prediction problem as a binary classification problem as follows:

• Treat the edges in the graph as positive examples.

• Sample a number of non-existent edges (i.e. node pairs with no edges between them) as negative examples.

• Divide the positive examples and negative examples into a training set and a test set.

• Evaluate the model with any binary classification metric such as Area Under Curve (AUC).

Note

The practice comes from SEAL, although the model here does not use their idea of node labeling.

In some domains such as large-scale recommender systems or information retrieval, you may favor metrics that emphasize good performance of top-K predictions. In these cases you may want to consider other metrics such as mean average precision, and use other negative sampling methods, which are beyond the scope of this tutorial.

Loading graph and features

Following the introduction, this tutorial first loads the Cora dataset.

dataset = dgl.data.CoraGraphDataset()
g = dataset[0]

Out:

  NumNodes: 2708
  NumEdges: 10556
  NumFeats: 1433
  NumClasses: 7
  NumTrainingSamples: 140
  NumValidationSamples: 500
  NumTestSamples: 1000
Done loading data from cached files.

Prepare training and testing sets

This tutorial randomly picks 10% of the edges for positive examples in the test set, and leave the rest for the training set. It then samples the same number of edges for negative examples in both sets.

# Split edge set for training and testing
u, v = g.edges()

eids = np.arange(g.num_edges())
eids = np.random.permutation(eids)
test_size = int(len(eids) * 0.1)
train_size = g.num_edges() - test_size
test_pos_u, test_pos_v = u[eids[:test_size]], v[eids[:test_size]]
train_pos_u, train_pos_v = u[eids[test_size:]], v[eids[test_size:]]

# Find all negative edges and split them for training and testing
adj = sp.coo_matrix((np.ones(len(u)), (u.numpy(), v.numpy())))
adj_neg = 1 - adj.todense() - np.eye(g.num_nodes())
neg_u, neg_v = np.where(adj_neg != 0)

neg_eids = np.random.choice(len(neg_u), g.num_edges())
test_neg_u, test_neg_v = (
    neg_u[neg_eids[:test_size]],
    neg_v[neg_eids[:test_size]],
)
train_neg_u, train_neg_v = (
    neg_u[neg_eids[test_size:]],
    neg_v[neg_eids[test_size:]],
)

When training, you will need to remove the edges in the test set from the original graph. You can do this via dgl.remove_edges.

Note

dgl.remove_edges works by creating a subgraph from the original graph, resulting in a copy and therefore could be slow for large graphs. If so, you could save the training and test graph to disk, as you would do for preprocessing.

train_g = dgl.remove_edges(g, eids[:test_size])

Define a GraphSAGE model

This tutorial builds a model consisting of two GraphSAGE layers, each computes new node representations by averaging neighbor information. DGL provides dgl.nn.SAGEConv that conveniently creates a GraphSAGE layer.

from dgl.nn import SAGEConv


# ----------- 2. create model -------------- #
# build a two-layer GraphSAGE model
class GraphSAGE(nn.Module):
    def __init__(self, in_feats, h_feats):
        super(GraphSAGE, self).__init__()
        self.conv1 = SAGEConv(in_feats, h_feats, "mean")
        self.conv2 = SAGEConv(h_feats, h_feats, "mean")

    def forward(self, g, in_feat):
        h = self.conv1(g, in_feat)
        h = F.relu(h)
        h = self.conv2(g, h)
        return h

The model then predicts the probability of existence of an edge by computing a score between the representations of both incident nodes with a function (e.g. an MLP or a dot product), which you will see in the next section.

\[\hat{y}_{u\sim v} = f(h_u, h_v)\]

Positive graph, negative graph, and apply_edges

In previous tutorials you have learned how to compute node representations with a GNN. However, link prediction requires you to compute representation of pairs of nodes.

DGL recommends you to treat the pairs of nodes as another graph, since you can describe a pair of nodes with an edge. In link prediction, you will have a positive graph consisting of all the positive examples as edges, and a negative graph consisting of all the negative examples. The positive graph and the negative graph will contain the same set of nodes as the original graph. This makes it easier to pass node features among multiple graphs for computation. As you will see later, you can directly feed the node representations computed on the entire graph to the positive and the negative graphs for computing pair-wise scores.

The following code constructs the positive graph and the negative graph for the training set and the test set respectively.

train_pos_g = dgl.graph((train_pos_u, train_pos_v), num_nodes=g.num_nodes())
train_neg_g = dgl.graph((train_neg_u, train_neg_v), num_nodes=g.num_nodes())

test_pos_g = dgl.graph((test_pos_u, test_pos_v), num_nodes=g.num_nodes())
test_neg_g = dgl.graph((test_neg_u, test_neg_v), num_nodes=g.num_nodes())

The benefit of treating the pairs of nodes as a graph is that you can use the DGLGraph.apply_edges method, which conveniently computes new edge features based on the incident nodes’ features and the original edge features (if applicable).

DGL provides a set of optimized builtin functions to compute new edge features based on the original node/edge features. For example, dgl.function.u_dot_v computes a dot product of the incident nodes’ representations for each edge.

import dgl.function as fn


class DotPredictor(nn.Module):
    def forward(self, g, h):
        with g.local_scope():
            g.ndata["h"] = h
            # Compute a new edge feature named 'score' by a dot-product between the
            # source node feature 'h' and destination node feature 'h'.
            g.apply_edges(fn.u_dot_v("h", "h", "score"))
            # u_dot_v returns a 1-element vector for each edge so you need to squeeze it.
            return g.edata["score"][:, 0]

You can also write your own function if it is complex. For instance, the following module produces a scalar score on each edge by concatenating the incident nodes’ features and passing it to an MLP.

class MLPPredictor(nn.Module):
    def __init__(self, h_feats):
        super().__init__()
        self.W1 = nn.Linear(h_feats * 2, h_feats)
        self.W2 = nn.Linear(h_feats, 1)

    def apply_edges(self, edges):
        """
        Computes a scalar score for each edge of the given graph.

        Parameters
        ----------
        edges :
            Has three members ``src``, ``dst`` and ``data``, each of
            which is a dictionary representing the features of the
            source nodes, the destination nodes, and the edges
            themselves.

        Returns
        -------
        dict
            A dictionary of new edge features.
        """
        h = torch.cat([edges.src["h"], edges.dst["h"]], 1)
        return {"score": self.W2(F.relu(self.W1(h))).squeeze(1)}

    def forward(self, g, h):
        with g.local_scope():
            g.ndata["h"] = h
            g.apply_edges(self.apply_edges)
            return g.edata["score"]

Note

The builtin functions are optimized for both speed and memory. We recommend using builtin functions whenever possible.

Note

If you have read the message passing tutorial, you will notice that the argument apply_edges takes has exactly the same form as a message function in update_all.

Training loop

After you defined the node representation computation and the edge score computation, you can go ahead and define the overall model, loss function, and evaluation metric.

The loss function is simply binary cross entropy loss.

\[\mathcal{L} = -\sum_{u\sim v\in \mathcal{D}}\left( y_{u\sim v}\log(\hat{y}_{u\sim v}) + (1-y_{u\sim v})\log(1-\hat{y}_{u\sim v})) \right)\]

The evaluation metric in this tutorial is AUC.

model = GraphSAGE(train_g.ndata["feat"].shape[1], 16)
# You can replace DotPredictor with MLPPredictor.
# pred = MLPPredictor(16)
pred = DotPredictor()


def compute_loss(pos_score, neg_score):
    scores = torch.cat([pos_score, neg_score])
    labels = torch.cat(
        [torch.ones(pos_score.shape[0]), torch.zeros(neg_score.shape[0])]
    )
    return F.binary_cross_entropy_with_logits(scores, labels)


def compute_auc(pos_score, neg_score):
    scores = torch.cat([pos_score, neg_score]).numpy()
    labels = torch.cat(
        [torch.ones(pos_score.shape[0]), torch.zeros(neg_score.shape[0])]
    ).numpy()
    return roc_auc_score(labels, scores)

The training loop goes as follows:

Note

This tutorial does not include evaluation on a validation set. In practice you should save and evaluate the best model based on performance on the validation set.

# ----------- 3. set up loss and optimizer -------------- #
# in this case, loss will in training loop
optimizer = torch.optim.Adam(
    itertools.chain(model.parameters(), pred.parameters()), lr=0.01
)

# ----------- 4. training -------------------------------- #
all_logits = []
for e in range(100):
    # forward
    h = model(train_g, train_g.ndata["feat"])
    pos_score = pred(train_pos_g, h)
    neg_score = pred(train_neg_g, h)
    loss = compute_loss(pos_score, neg_score)

    # backward
    optimizer.zero_grad()
    loss.backward()
    optimizer.step()

    if e % 5 == 0:
        print("In epoch {}, loss: {}".format(e, loss))

# ----------- 5. check results ------------------------ #
from sklearn.metrics import roc_auc_score

with torch.no_grad():
    pos_score = pred(test_pos_g, h)
    neg_score = pred(test_neg_g, h)
    print("AUC", compute_auc(pos_score, neg_score))


# Thumbnail credits: Link Prediction with Neo4j, Mark Needham
# sphinx_gallery_thumbnail_path = '_static/blitz_4_link_predict.png'

Out:

In epoch 0, loss: 0.7089807987213135
In epoch 5, loss: 0.6925361156463623
In epoch 10, loss: 0.6763476729393005
In epoch 15, loss: 0.6368499398231506
In epoch 20, loss: 0.5828432440757751
In epoch 25, loss: 0.5465787053108215
In epoch 30, loss: 0.518890380859375
In epoch 35, loss: 0.495853066444397
In epoch 40, loss: 0.4819425940513611
In epoch 45, loss: 0.4622977077960968
In epoch 50, loss: 0.4424958825111389
In epoch 55, loss: 0.42348989844322205
In epoch 60, loss: 0.4055698812007904
In epoch 65, loss: 0.38643723726272583
In epoch 70, loss: 0.3656672537326813
In epoch 75, loss: 0.3464341163635254
In epoch 80, loss: 0.32723113894462585
In epoch 85, loss: 0.3090077042579651
In epoch 90, loss: 0.29131653904914856
In epoch 95, loss: 0.2741481065750122
AUC 0.8682850789515061