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
This is a single graph attention layer. A GAT is made up of multiple such layers.
It takes , where as input and outputs , where .
32class GraphAttentionLayer(nn.Module):
in_features
, , is the number of input features per node out_features
, , is the number of output features per node n_heads
, , 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 activation46 def __init__(self, in_features: int, out_features: int, n_heads: int,
47 is_concat: bool = True,
48 dropout: float = 0.6,
49 leaky_relu_negative_slope: float = 0.2):
58 super().__init__()
59
60 self.is_concat = is_concat
61 self.n_heads = n_heads
Calculate the number of dimensions per head
64 if is_concat:
65 assert out_features % n_heads == 0
If we are concatenating the multiple heads
67 self.n_hidden = out_features // n_heads
68 else:
If we are averaging the multiple heads
70 self.n_hidden = out_features
Linear layer for initial transformation; i.e. to transform the node embeddings before self-attention
74 self.linear = nn.Linear(in_features, self.n_hidden * n_heads, bias=False)
Linear layer to compute attention score
76 self.attn = nn.Linear(self.n_hidden * 2, 1, bias=False)
The activation for attention score
78 self.activation = nn.LeakyReLU(negative_slope=leaky_relu_negative_slope)
Softmax to compute attention
80 self.softmax = nn.Softmax(dim=1)
Dropout layer to be applied for attention
82 self.dropout = nn.Dropout(dropout)
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
.
84 def forward(self, h: torch.Tensor, adj_mat: torch.Tensor):
Number of nodes
95 n_nodes = h.shape[0]
The initial transformation, for each head. We do single linear transformation and then split it up for each head.
100 g = self.linear(h).view(n_nodes, self.n_heads, self.n_hidden)
We calculate these for each head . We have omitted for simplicity.
is the attention score (importance) from node to node . We calculate this for each head.
is the attention mechanism, that calculates the attention score. The paper concatenates , and does a linear transformation with a weight vector followed by a .
First we calculate for all pairs of .
g_repeat
gets where each node embedding is repeated n_nodes
times.
131 g_repeat = g.repeat(n_nodes, 1, 1)
g_repeat_interleave
gets where each node embedding is repeated n_nodes
times.
136 g_repeat_interleave = g.repeat_interleave(n_nodes, dim=0)
Now we concatenate to get
144 g_concat = torch.cat([g_repeat_interleave, g_repeat], dim=-1)
Reshape so that g_concat[i, j]
is
146 g_concat = g_concat.view(n_nodes, n_nodes, self.n_heads, 2 * self.n_hidden)
Calculate e
is of shape [n_nodes, n_nodes, n_heads, 1]
154 e = self.activation(self.attn(g_concat))
Remove the last dimension of size 1
156 e = e.squeeze(-1)
The adjacency matrix should have shape [n_nodes, n_nodes, n_heads]
or[n_nodes, n_nodes, 1]
160 assert adj_mat.shape[0] == 1 or adj_mat.shape[0] == n_nodes
161 assert adj_mat.shape[1] == 1 or adj_mat.shape[1] == n_nodes
162 assert adj_mat.shape[2] == 1 or adj_mat.shape[2] == self.n_heads
Mask based on adjacency matrix. is set to if there is no edge from to .
165 e = e.masked_fill(adj_mat == 0, float('-inf'))
We then normalize attention scores (or coefficients)
where is the set of nodes connected to .
We do this by setting unconnected to which makes for unconnected pairs.
175 a = self.softmax(e)
Apply dropout regularization
178 a = self.dropout(a)
Calculate final output for each head
Note: The paper includes the final activation in 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.
187 attn_res = torch.einsum('ijh,jhf->ihf', a, g)
Concatenate the heads
190 if self.is_concat:
192 return attn_res.reshape(n_nodes, self.n_heads * self.n_hidden)
Take the mean of the heads
194 else:
196 return attn_res.mean(dim=1)