text: from typing import Optional, Tuple
  from torch_geometric.typing import Adj, OptTensor, PairTensor
  
  import torch
  from torch import Tensor
  from torch.nn import Parameter
  from torch_scatter import scatter_add
  from torch_sparse import SparseTensor, matmul, fill_diag, sum as sparsesum, mul
  from torch_geometric.nn.inits import zeros
  from torch_geometric.nn.dense.linear import Linear
  from torch_geometric.nn.conv import MessagePassing
  from torch_geometric.utils import add_remaining_self_loops
  from torch_geometric.utils.num_nodes import maybe_num_nodes
  
  
  @torch.jit._overload
  def gcn_norm(edge_index, edge_weight=None, num_nodes=None, improved=False,
               add_self_loops=True, dtype=None):
      # type: (Tensor, OptTensor, Optional[int], bool, bool, Optional[int]) -> PairTensor  # noqa
      pass
  
  
  @torch.jit._overload
  def gcn_norm(edge_index, edge_weight=None, num_nodes=None, improved=False,
               add_self_loops=True, dtype=None):
      # type: (SparseTensor, OptTensor, Optional[int], bool, bool, Optional[int]) -> SparseTensor  # noqa
      pass
  
  
  def gcn_norm(edge_index, edge_weight=None, num_nodes=None, improved=False,
               add_self_loops=True, dtype=None):
  
      fill_value = 2. if improved else 1.
  
      if isinstance(edge_index, SparseTensor):
          adj_t = edge_index
          if not adj_t.has_value():
              adj_t = adj_t.fill_value(1., dtype=dtype)
          if add_self_loops:
              adj_t = fill_diag(adj_t, fill_value)
          deg = sparsesum(adj_t, dim=1)
          deg_inv_sqrt = deg.pow_(-0.5)
          deg_inv_sqrt.masked_fill_(deg_inv_sqrt == float('inf'), 0.)
          adj_t = mul(adj_t, deg_inv_sqrt.view(-1, 1))
          adj_t = mul(adj_t, deg_inv_sqrt.view(1, -1))
          return adj_t
  
      else:
          num_nodes = maybe_num_nodes(edge_index, num_nodes)
  
          if edge_weight is None:
              edge_weight = torch.ones((edge_index.size(1), ), dtype=dtype,
                                       device=edge_index.device)
  
          if add_self_loops:
              edge_index, tmp_edge_weight = add_remaining_self_loops(
                  edge_index, edge_weight, fill_value, num_nodes)
              assert tmp_edge_weight is not None
              edge_weight = tmp_edge_weight
  
          row, col = edge_index[0], edge_index[1]
          deg = scatter_add(edge_weight, col, dim=0, dim_size=num_nodes)
          deg_inv_sqrt = deg.pow_(-0.5)
          deg_inv_sqrt.masked_fill_(deg_inv_sqrt == float('inf'), 0)
          return edge_index, deg_inv_sqrt[row] * edge_weight * deg_inv_sqrt[col]
  
  
  class GCNConv(MessagePassing):
      r"""The graph convolutional operator from the `"Semi-supervised
      Classification with Graph Convolutional Networks"
      <https://arxiv.org/abs/1609.02907>`_ paper
  
      .. math::
          \mathbf{X}^{\prime} = \mathbf{\hat{D}}^{-1/2} \mathbf{\hat{A}}
          \mathbf{\hat{D}}^{-1/2} \mathbf{X} \mathbf{\Theta},
  
      where :math:`\mathbf{\hat{A}} = \mathbf{A} + \mathbf{I}` denotes the
      adjacency matrix with inserted self-loops and
      :math:`\hat{D}_{ii} = \sum_{j=0} \hat{A}_{ij}` its diagonal degree matrix.
      The adjacency matrix can include other values than :obj:`1` representing
      edge weights via the optional :obj:`edge_weight` tensor.
  
      Its node-wise formulation is given by:
  
      .. math::
          \mathbf{x}^{\prime}_i = \mathbf{\Theta}^{\top} \sum_{j \in
          \mathcal{N}(v) \cup \{ i \}} \frac{e_{j,i}}{\sqrt{\hat{d}_j
          \hat{d}_i}} \mathbf{x}_j
  
      with :math:`\hat{d}_i = 1 + \sum_{j \in \mathcal{N}(i)} e_{j,i}`, where
      :math:`e_{j,i}` denotes the edge weight from source node :obj:`j` to target
      node :obj:`i` (default: :obj:`1.0`)
  
      Args:
          in_channels (int): Size of each input sample, or :obj:`-1` to derive
              the size from the first input(s) to the forward method.
          out_channels (int): Size of each output sample.
          improved (bool, optional): If set to :obj:`True`, the layer computes
              :math:`\mathbf{\hat{A}}` as :math:`\mathbf{A} + 2\mathbf{I}`.
              (default: :obj:`False`)
          cached (bool, optional): If set to :obj:`True`, the layer will cache
              the computation of :math:`\mathbf{\hat{D}}^{-1/2} \mathbf{\hat{A}}
              \mathbf{\hat{D}}^{-1/2}` on first execution, and will use the
              cached version for further executions.
              This parameter should only be set to :obj:`True` in transductive
              learning scenarios. (default: :obj:`False`)
          add_self_loops (bool, optional): If set to :obj:`False`, will not add
              self-loops to the input graph. (default: :obj:`True`)
          normalize (bool, optional): Whether to add self-loops and compute
              symmetric normalization coefficients on the fly.
              (default: :obj:`True`)
          bias (bool, optional): If set to :obj:`False`, the layer will not learn
              an additive bias. (default: :obj:`True`)
          **kwargs (optional): Additional arguments of
              :class:`torch_geometric.nn.conv.MessagePassing`.
      """
  
      _cached_edge_index: Optional[Tuple[Tensor, Tensor]]
      _cached_adj_t: Optional[SparseTensor]
  
      def __init__(self, in_channels: int, out_channels: int,
                   improved: bool = False, cached: bool = False,
                   add_self_loops: bool = True, normalize: bool = True,
                   bias: bool = True, **kwargs):
  
          kwargs.setdefault('aggr', 'add')
          super().__init__(**kwargs)
  
          self.in_channels = in_channels
          self.out_channels = out_channels
          self.improved = improved
          self.cached = cached
          self.add_self_loops = add_self_loops
          self.normalize = normalize
  
          self._cached_edge_index = None
          self._cached_adj_t = None
  
          self.lin = Linear(in_channels, out_channels, bias=False,
                            weight_initializer='glorot')
  
          if bias:
              self.bias = Parameter(torch.Tensor(out_channels))
          else:
              self.register_parameter('bias', None)
  
          self.reset_parameters()
  
      def reset_parameters(self):
          self.lin.reset_parameters()
          zeros(self.bias)
          self._cached_edge_index = None
          self._cached_adj_t = None
  
      def forward(self, x: Tensor, edge_index: Adj,
                  edge_weight: OptTensor = None) -> Tensor:
          """"""
  
          if self.normalize:
              if isinstance(edge_index, Tensor):
                  cache = self._cached_edge_index
                  if cache is None:
                      edge_index, edge_weight = gcn_norm(  # yapf: disable
                          edge_index, edge_weight, x.size(self.node_dim),
                          self.improved, self.add_self_loops)
                      if self.cached:
                          self._cached_edge_index = (edge_index, edge_weight)
                  else:
                      edge_index, edge_weight = cache[0], cache[1]
  
              elif isinstance(edge_index, SparseTensor):
                  cache = self._cached_adj_t
                  if cache is None:
                      edge_index = gcn_norm(  # yapf: disable
                          edge_index, edge_weight, x.size(self.node_dim),
                          self.improved, self.add_self_loops)
                      if self.cached:
                          self._cached_adj_t = edge_index
                  else:
                      edge_index = cache
  
          x = self.lin(x)
  
          # propagate_type: (x: Tensor, edge_weight: OptTensor)
          out = self.propagate(edge_index, x=x, edge_weight=edge_weight,
                               size=None)
  
          if self.bias is not None:
              out += self.bias
  
          return out
  
      def message(self, x_j: Tensor, edge_weight: OptTensor) -> Tensor:
          return x_j if edge_weight is None else edge_weight.view(-1, 1) * x_j
  
      def message_and_aggregate(self, adj_t: SparseTensor, x: Tensor) -> Tensor:
          return matmul(adj_t, x, reduce=self.aggr)
  