#

Graph Attention Networks (GAT)

This is a PyTorch implementation of the paper Graph Attention Networks.

GATs work on graph data. A graph consists of nodes and edges connecting nodes. For example, in Cora dataset the nodes are research papers and the edges are citations that connect the papers.

GAT uses masked self-attention, kind of similar to transformers. GAT consists of graph attention layers stacked on top of each other. Each graph attention layer gets node embeddings as inputs and outputs transformed embeddings. The node embeddings pay attention to the embeddings of other nodes it's connected to. The details of graph attention layers are included alongside the implementation.

Here is the training code for training a two-layer GAT on Cora dataset.

28import torch
29from torch import nn
30
31from labml_helpers.module import Module

#

Graph attention layer

This is a single graph attention layer. A GAT is made up of multiple such layers.

It takes $h = {h_{1}, h_{2}, \dots, h_{N}}$ , where $h_{i} \in R^{F}$ as input and outputs $h^{'} = {h_{1}^{'}, h_{2}^{'}, \dots, h_{N}^{'}}$ , where $h_{i}^{'} \in R^{F^{'}}$ .

34class GraphAttentionLayer(Module):

#

in_features , $F$ , is the number of input features per node
out_features , $F^{'}$ , is the number of output features per node
n_heads , $K$ , is the number of attention heads
is_concat whether the multi-head results should be concatenated or averaged
dropout is the dropout probability
leaky_relu_negative_slope is the negative slope for leaky relu activation

48    def __init__(self, in_features: int, out_features: int, n_heads: int,
49                 is_concat: bool = True,
50                 dropout: float = 0.6,
51                 leaky_relu_negative_slope: float = 0.2):

#

60        super().__init__()
61
62        self.is_concat = is_concat
63        self.n_heads = n_heads

#

Calculate the number of dimensions per head

66        if is_concat:
67            assert out_features % n_heads == 0

#

If we are concatenating the multiple heads

69            self.n_hidden = out_features // n_heads
70        else:

#

If we are averaging the multiple heads

72            self.n_hidden = out_features

#

Linear layer for initial transformation; i.e. to transform the node embeddings before self-attention

76        self.linear = nn.Linear(in_features, self.n_hidden * n_heads, bias=False)

#

Linear layer to compute attention score $e_{i j}$

78        self.attn = nn.Linear(self.n_hidden * 2, 1, bias=False)

#

The activation for attention score $e_{i j}$

80        self.activation = nn.LeakyReLU(negative_slope=leaky_relu_negative_slope)

#

Softmax to compute attention $α_{i j}$

82        self.softmax = nn.Softmax(dim=1)

#

Dropout layer to be applied for attention

84        self.dropout = nn.Dropout(dropout)

#

h , $h$ is the input node embeddings of shape [n_nodes, in_features] .
adj_mat is the adjacency matrix of shape [n_nodes, n_nodes, n_heads] . We use shape [n_nodes, n_nodes, 1] since the adjacency is the same for each head.

Adjacency matrix represent the edges (or connections) among nodes. adj_mat[i][j] is True if there is an edge from node i to node j .

86    def forward(self, h: torch.Tensor, adj_mat: torch.Tensor):

#

Number of nodes

97        n_nodes = h.shape[0]

#

The initial transformation, $g_{i}^{k} = W^{k} h_{i}$ for each head. We do single linear transformation and then split it up for each head.

102        g = self.linear(h).view(n_nodes, self.n_heads, self.n_hidden)

#

Calculate attention score

We calculate these for each head $k$ . We have omitted $\cdot^{k}$ for simplicity.

$e_{i j} = a (W h_{i}, W h_{j}) = a (g_{i}, g_{j})$

$e_{i j}$ is the attention score (importance) from node $j$ to node $i$ . We calculate this for each head.

$a$ is the attention mechanism, that calculates the attention score. The paper concatenates $g_{i}$ , $g_{j}$ and does a linear transformation with a weight vector $a \in R^{2 F^{'}}$ followed by a $Le a k yReLU$ .

$e_{i j} = Le a k yReLU (a^{⊤} [g_{i} ∥ g_{j}])$

#

First we calculate $[g_{i} ∥ g_{j}]$ for all pairs of $i, j$ .

g_repeat gets ${g_{1}, g_{2}, \dots, g_{N}, g_{1}, g_{2}, \dots, g_{N}, ...}$ where each node embedding is repeated n_nodes times.

133        g_repeat = g.repeat(n_nodes, 1, 1)

#

g_repeat_interleave gets ${g_{1}, g_{1}, \dots, g_{1}, g_{2}, g_{2}, \dots, g_{2}, ...}$ where each node embedding is repeated n_nodes times.

138        g_repeat_interleave = g.repeat_interleave(n_nodes, dim=0)

#

Now we concatenate to get ${g_{1} ∥ g_{1}, g_{1} ∥ g_{2}, \dots, g_{1} ∥ g_{N}, g_{2} ∥ g_{1}, g_{2} ∥ g_{2}, \dots, g_{2} ∥ g_{N}, ...}$

146        g_concat = torch.cat([g_repeat_interleave, g_repeat], dim=-1)

#

Reshape so that g_concat[i, j] is $g_{i} ∥ g_{j}$

148        g_concat = g_concat.view(n_nodes, n_nodes, self.n_heads, 2 * self.n_hidden)

#

Calculate $e_{i j} = Le a k yReLU (a^{⊤} [g_{i} ∥ g_{j}])$ e is of shape [n_nodes, n_nodes, n_heads, 1]

156        e = self.activation(self.attn(g_concat))

#

Remove the last dimension of size 1

158        e = e.squeeze(-1)

#

The adjacency matrix should have shape [n_nodes, n_nodes, n_heads] or[n_nodes, n_nodes, 1]

162        assert adj_mat.shape[0] == 1 or adj_mat.shape[0] == n_nodes
163        assert adj_mat.shape[1] == 1 or adj_mat.shape[1] == n_nodes
164        assert adj_mat.shape[2] == 1 or adj_mat.shape[2] == self.n_heads

#

Mask $e_{i j}$ based on adjacency matrix. $e_{i j}$ is set to $- \infty$ if there is no edge from $i$ to $j$ .

167        e = e.masked_fill(adj_mat == 0, float('-inf'))

#

We then normalize attention scores (or coefficients) $α_{i j} = softm a x_{j} (e_{i j}) = \frac{exp ( e _{i j} )}{\sum _{k \in N_{i}} exp ( e _{i k} )}$

where $N_{i}$ is the set of nodes connected to $i$ .

We do this by setting unconnected $e_{i j}$ to $- \infty$ which makes $exp (e_{i j}) \sim 0$ for unconnected pairs.

177        a = self.softmax(e)

#

Apply dropout regularization

180        a = self.dropout(a)

#

Calculate final output for each head $h_{i}^{' k} = j \in N_{i} \sum α_{i j}^{k} g_{j}^{k}$

Note: The paper includes the final activation $σ$ in $h_{i}$ We have omitted this from the Graph Attention Layer implementation and use it on the GAT model to match with how other PyTorch modules are defined - activation as a separate layer.

189        attn_res = torch.einsum('ijh,jhf->ihf', a, g)

#

Concatenate the heads

192        if self.is_concat:

#

$h_{i}^{'} = ∥ ∥_{k = 1}^{K} h_{i}^{' k}$

194            return attn_res.reshape(n_nodes, self.n_heads * self.n_hidden)

#

Take the mean of the heads

196        else:

#

$h_{i}^{'} = \frac{1}{K} k = 1 \sum K h_{i}^{' k}$

198            return attn_res.mean(dim=1)