human
Solve the following python programming problem given the following code:
# GAT implementation adapted from https://github.com/deepmind/jraph/blob/master/jraph/_src/models.py#L442.
def GAT(attention_query_fn: Callable,
        attention_logit_fn: Callable,
        node_update_fn: Optional[Callable] = None,
        add_self_edges: bool = True) -> Callable:
  """Returns a method that applies a Graph Attention Network layer.

  Graph Attention message passing as described in
  https://arxiv.org/pdf/1710.10903.pdf. This model expects node features as a
  jnp.array, may use edge features for computing attention weights, and
  ignore global features. It does not support nests.
  Args:
    attention_query_fn: function that generates attention queries from sender
      node features.
    attention_logit_fn: function that converts attention queries into logits for
      softmax attention.
    node_update_fn: function that updates the aggregated messages. If None, will
      apply leaky relu and concatenate (if using multi-head attention).

  Returns:
    A function that applies a Graph Attention layer.
  """
  # pylint: disable=g-long-lambda
  if node_update_fn is None:
    # By default, apply the leaky relu and then concatenate the heads on the
    # feature axis.
    node_update_fn = lambda x: jnp.reshape(
        jax.nn.leaky_relu(x), (x.shape[0], -1))

  def _ApplyGAT(graph: jraph.GraphsTuple) -> jraph.GraphsTuple:
    """Applies a Graph Attention layer."""
    nodes, edges, receivers, senders, _, _, _ = graph
    # Equivalent to the sum of n_node, but statically known.
    try:
      sum_n_node = nodes.shape[0]
    except IndexError:
      raise IndexError('GAT requires node features')

    # Pass nodes through the attention query function to transform
    # node features, e.g. with an MLP.
    nodes = attention_query_fn(nodes)

    total_num_nodes = tree.tree_leaves(nodes)[0].shape[0]
    if add_self_edges:
      # We add self edges to the senders and receivers so that each node
      # includes itself in aggregation.
      receivers, senders = add_self_edges_fn(receivers, senders,
  ...
Given the above code, generate some markdown instructions for it.
assistant
#Here is the Python program you requested.

Graph Attention (GAT) Layer
While the GCN we covered in the previous section can learn meaningful representations, it also has some shortcomings. Can you think of any?
In the GCN layer, the messages from all its neighbours and the node itself are equally weighted. This may lead to loss of node-specific information. E.g., consider the case when a set of nodes shares the same set of neighbors, and start out with different node features. Then because of averaging, their resulting output features would be the same. Adding self-edges mitigates this issue by a small amount, but this problem is magnified with increasing number of GCN layers and number of edges connecting to a node.
The graph attention (GAT) mechanism, as proposed by Velickovic et al. ( 2017), allows the network to learn how to weigh / assign importance to the node features from the neighbourhood when computing the new node features. This is very similar to the idea of using attention in Transformers, which were introduced in Vaswani et al. (2017).
(One could even argue that Transformers are graph attention networks operating on the special case of fully-connected graphs.)
In the figure below, $\vec{h}$ are the node features and $\vec{\alpha}$ are the learned attention weights.
<image src="https://storage.googleapis.com/dm-educational/assets/graph-nets/gat1.png" width="400px">
Figure Credit: Velickovic et al. ( 2017).
(Detail: This image is showing multi-headed attention with 3 heads, each color corresponding to a different head. At the end, an aggregation function is applied over all the heads.)
To obtain the output node features of the GAT layer, we compute:
$$ \vec{h}'i = \sum {j \in \mathcal{N}(i)}\alpha_{ij} \mathbf{W} \vec{h}_j$$
Here, $\mathbf{W}$ is a weight matrix which performs a linear transformation on the input.
How do we obtain $\alpha$, or in other words, learn what to pay attention to?
Intuitively, the attention coefficient $\alpha_{ij}$ should rely on both the transformed features from nodes $i$ and $j$. So let's first define an...