Note

Go to the end to download the full example code

# 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]
```

```
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.

## 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.

## 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.

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'
```

```
In epoch 0, loss: 0.7021162509918213
In epoch 5, loss: 0.689262330532074
In epoch 10, loss: 0.6682888865470886
In epoch 15, loss: 0.6222915053367615
In epoch 20, loss: 0.5650363564491272
In epoch 25, loss: 0.5266429781913757
In epoch 30, loss: 0.4944186508655548
In epoch 35, loss: 0.4720936417579651
In epoch 40, loss: 0.4443744719028473
In epoch 45, loss: 0.4237181842327118
In epoch 50, loss: 0.40078991651535034
In epoch 55, loss: 0.3775203824043274
In epoch 60, loss: 0.3544590473175049
In epoch 65, loss: 0.33114877343177795
In epoch 70, loss: 0.30858904123306274
In epoch 75, loss: 0.285332590341568
In epoch 80, loss: 0.2629302740097046
In epoch 85, loss: 0.2413136214017868
In epoch 90, loss: 0.2202303111553192
In epoch 95, loss: 0.19989247620105743
AUC 0.8439091664607714
```

**Total running time of the script:** (0 minutes 4.066 seconds)