torch_geometric.nn

Convolutional Layers

class MessagePassing(aggr='add', flow='source_to_target', node_dim=0)

Base class for creating message passing layers of the form

\[\mathbf{x}_i^{\prime} = \gamma_{\mathbf{\Theta}} \left( \mathbf{x}_i, \square_{j \in \mathcal{N}(i)} \, \phi_{\mathbf{\Theta}} \left(\mathbf{x}_i, \mathbf{x}_j,\mathbf{e}_{i,j}\right) \right),\]

where \(\square\) denotes a differentiable, permutation invariant function, e.g., sum, mean or max, and \(\gamma_{\mathbf{\Theta}}\) and \(\phi_{\mathbf{\Theta}}\) denote differentiable functions such as MLPs. See here for the accompanying tutorial.

Parameters

• aggr (string, optional) – The aggregation scheme to use ("add", "mean", "max" or None). (default: "add")

• flow (string, optional) – The flow direction of message passing ("source_to_target" or "target_to_source"). (default: "source_to_target")

• node_dim (int, optional) – The axis along which to propagate. (default: 0)

aggregate(inputs, index, ptr=None, dim_size=None)

Aggregates messages from neighbors as \(\square_{j \in \mathcal{N}(i)}\).

Takes in the output of message computation as first argument and any argument which was initially passed to propagate().

By default, this function will delegate its call to scatter functions that support “add”, “mean” and “max” operations as specified in __init__() by the aggr argument.

message(x_j)

Constructs messages from node \(j\) to node \(i\) in analogy to \(\phi_{\mathbf{\Theta}}\) for each edge in edge_index. This function can take any argument as input which was initially passed to propagate(). Furthermore, tensors passed to propagate() can be mapped to the respective nodes \(i\) and \(j\) by appending _i or _j to the variable name, .e.g. x_i and x_j.

message_and_aggregate(adj_t)

Fuses computations of message() and aggregate() into a single function. If applicable, this saves both time and memory since messages do not explicitly need to be materialized. This function will only gets called in case it is implemented and propagation takes place based on a torch_sparse.SparseTensor.

propagate(edge_index, size=None, **kwargs)

The initial call to start propagating messages.

Parameters

• adj (Tensor or SparseTensor) – A torch.LongTensor or a torch_sparse.SparseTensor that defines the underlying message propagation. edge_index holds the indices of a general (sparse) assignment matrix of shape [N, M]. If edge_index is of type torch.LongTensor, its shape must be defined as [2, num_messages], where messages from nodes in edge_index[0] are sent to nodes in edge_index[1] (in case flow="source_to_target"). If edge_index is of type torch_sparse.SparseTensor, its sparse indices (row, col) should relate to row = edge_index[1] and col = edge_index[0]. Hence, the only difference between those formats is that we need to input the transposed sparse adjacency matrix into propagate().

• size (list or tuple, optional) – The size [N, M] of the assignment matrix in case edge_index is a LongTensor. If set to None, the size will be automatically inferred and assumed to be quadratic. This argument is ignored in case edge_index is a torch_sparse.SparseTensor. (default: None)

• **kwargs – Any additional data which is needed to construct and aggregate messages, and to update node embeddings.

update(inputs)

Updates node embeddings in analogy to \(\gamma_{\mathbf{\Theta}}\) for each node \(i \in \mathcal{V}\). Takes in the output of aggregation as first argument and any argument which was initially passed to propagate().

class GCNConv(in_channels, out_channels, improved=False, cached=False, bias=True, normalize=True, **kwargs)

The graph convolutional operator from the “Semi-supervised Classification with Graph Convolutional Networks” paper

\[\mathbf{X}^{\prime} = \mathbf{\hat{D}}^{-1/2} \mathbf{\hat{A}} \mathbf{\hat{D}}^{-1/2} \mathbf{X} \mathbf{\Theta},\]

where \(\mathbf{\hat{A}} = \mathbf{A} + \mathbf{I}\) denotes the adjacency matrix with inserted self-loops and \(\hat{D}_{ii} = \sum_{j=0} \hat{A}_{ij}\) its diagonal degree matrix.

Parameters

• in_channels (int) – Size of each input sample.

• out_channels (int) – Size of each output sample.

• improved (bool, optional) – If set to True, the layer computes \(\mathbf{\hat{A}}\) as \(\mathbf{A} + 2\mathbf{I}\). (default: False)

• cached (bool, optional) – If set to True, the layer will cache the computation of \(\mathbf{\hat{D}}^{-1/2} \mathbf{\hat{A}} \mathbf{\hat{D}}^{-1/2}\) on first execution, and will use the cached version for further executions. This parameter should only be set to True in transductive learning scenarios. (default: False)

• bias (bool, optional) – If set to False, the layer will not learn an additive bias. (default: True)

• normalize (bool, optional) – Whether to add self-loops and apply symmetric normalization. (default: True)

• **kwargs (optional) – Additional arguments of torch_geometric.nn.conv.MessagePassing.

forward(x, edge_index, edge_weight=None)

static norm(edge_index, num_nodes, edge_weight=None, improved=False, dtype=None)

reset_parameters()

class ChebConv(in_channels, out_channels, K, normalization='sym', bias=True, **kwargs)

The chebyshev spectral graph convolutional operator from the “Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering” paper

\[\mathbf{X}^{\prime} = \sum_{k=1}^{K} \mathbf{Z}^{(k)} \cdot \mathbf{\Theta}^{(k)}\]

where \(\mathbf{Z}^{(k)}\) is computed recursively by

\[ \begin{align}\begin{aligned}\mathbf{Z}^{(1)} &= \mathbf{X}\\\mathbf{Z}^{(2)} &= \mathbf{\hat{L}} \cdot \mathbf{X}\\\mathbf{Z}^{(k)} &= 2 \cdot \mathbf{\hat{L}} \cdot \mathbf{Z}^{(k-1)} - \mathbf{Z}^{(k-2)}\end{aligned}\end{align} \]

and \(\mathbf{\hat{L}}\) denotes the scaled and normalized Laplacian \(\frac{2\mathbf{L}}{\lambda_{\max}} - \mathbf{I}\).

Parameters

• in_channels (int) – Size of each input sample.

• out_channels (int) – Size of each output sample.

• K (int) – Chebyshev filter size \(K\).

• normalization (str, optional) –

  The normalization scheme for the graph Laplacian (default: "sym"):

  1. None: No normalization \(\mathbf{L} = \mathbf{D} - \mathbf{A}\)

  2. "sym": Symmetric normalization \(\mathbf{L} = \mathbf{I} - \mathbf{D}^{-1/2} \mathbf{A} \mathbf{D}^{-1/2}\)

  3. "rw": Random-walk normalization \(\mathbf{L} = \mathbf{I} - \mathbf{D}^{-1} \mathbf{A}\)

  You need to pass lambda_max to the forward() method of this operator in case the normalization is non-symmetric. lambda_max should be a torch.Tensor of size [num_graphs] in a mini-batch scenario and a scalar when operating on single graphs. You can pre-compute lambda_max via the torch_geometric.transforms.LaplacianLambdaMax transform.

• bias (bool, optional) – If set to False, the layer will not learn an additive bias. (default: True)

• **kwargs (optional) – Additional arguments of torch_geometric.nn.conv.MessagePassing.

forward(x, edge_index, edge_weight=None, batch=None, lambda_max=None)

static norm(edge_index, num_nodes, edge_weight, normalization, lambda_max, dtype=None, batch=None)

reset_parameters()

class SAGEConv(in_channels, out_channels, normalize=False, concat=False, bias=True, **kwargs)

The GraphSAGE operator from the “Inductive Representation Learning on Large Graphs” paper

\[ \begin{align}\begin{aligned}\mathbf{\hat{x}}_i &= \mathbf{\Theta} \cdot \mathrm{mean}_{j \in \mathcal{N(i) \cup \{ i \}}}(\mathbf{x}_j)\\\mathbf{x}^{\prime}_i &= \frac{\mathbf{\hat{x}}_i} {\| \mathbf{\hat{x}}_i \|_2}.\end{aligned}\end{align} \]

Parameters

• in_channels (int) – Size of each input sample.

• out_channels (int) – Size of each output sample.

• normalize (bool, optional) – If set to True, output features will be \(\ell_2\)-normalized. (default: False)

• concat (bool, optional) – If set to True, will concatenate current node features with aggregated ones. (default: False)

• bias (bool, optional) – If set to False, the layer will not learn an additive bias. (default: True)

• **kwargs (optional) – Additional arguments of torch_geometric.nn.conv.MessagePassing.

forward(x, edge_index, edge_weight=None, size=None, res_n_id=None)

Parameters

res_n_id (Tensor, optional) – Residual node indices coming from DataFlow generated by NeighborSampler are used to select central node features in x. Required if operating in a bipartite graph and concat is True. (default: None)

reset_parameters()

class GraphConv(in_channels, out_channels, aggr='add', bias=True, **kwargs)

The graph neural network operator from the “Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks” paper

\[\mathbf{x}^{\prime}_i = \mathbf{\Theta}_1 \mathbf{x}_i + \sum_{j \in \mathcal{N}(i)} \mathbf{\Theta}_2 \mathbf{x}_j.\]

Parameters

• in_channels (int) – Size of each input sample.

• out_channels (int) – Size of each output sample.

• aggr (string, optional) – The aggregation scheme to use ("add", "mean", "max"). (default: "add")

• bias (bool, optional) – If set to False, the layer will not learn an additive bias. (default: True)

• **kwargs (optional) – Additional arguments of torch_geometric.nn.conv.MessagePassing.

forward(x, edge_index, edge_weight=None, size=None)

reset_parameters()

class GatedGraphConv(out_channels, num_layers, aggr='add', bias=True, **kwargs)

The gated graph convolution operator from the “Gated Graph Sequence Neural Networks” paper

\[ \begin{align}\begin{aligned}\mathbf{h}_i^{(0)} &= \mathbf{x}_i \, \Vert \, \mathbf{0}\\\mathbf{m}_i^{(l+1)} &= \sum_{j \in \mathcal{N}(i)} \mathbf{\Theta} \cdot \mathbf{h}_j^{(l)}\\\mathbf{h}_i^{(l+1)} &= \textrm{GRU} (\mathbf{m}_i^{(l+1)}, \mathbf{h}_i^{(l)})\end{aligned}\end{align} \]

up to representation \(\mathbf{h}_i^{(L)}\). The number of input channels of \(\mathbf{x}_i\) needs to be less or equal than out_channels.

Parameters

• out_channels (int) – Size of each input sample.

• num_layers (int) – The sequence length \(L\).

• aggr (string, optional) – The aggregation scheme to use ("add", "mean", "max"). (default: "add")

• bias (bool, optional) – If set to False, the layer will not learn an additive bias. (default: True)

• **kwargs (optional) – Additional arguments of torch_geometric.nn.conv.MessagePassing.

forward(x, edge_index, edge_weight=None)

reset_parameters()

class GATConv(in_channels, out_channels, heads=1, concat=True, negative_slope=0.2, dropout=0, bias=True, **kwargs)

The graph attentional operator from the “Graph Attention Networks” paper

\[\mathbf{x}^{\prime}_i = \alpha_{i,i}\mathbf{\Theta}\mathbf{x}_{i} + \sum_{j \in \mathcal{N}(i)} \alpha_{i,j}\mathbf{\Theta}\mathbf{x}_{j},\]

where the attention coefficients \(\alpha_{i,j}\) are computed as

\[\alpha_{i,j} = \frac{ \exp\left(\mathrm{LeakyReLU}\left(\mathbf{a}^{\top} [\mathbf{\Theta}\mathbf{x}_i \, \Vert \, \mathbf{\Theta}\mathbf{x}_j] \right)\right)} {\sum_{k \in \mathcal{N}(i) \cup \{ i \}} \exp\left(\mathrm{LeakyReLU}\left(\mathbf{a}^{\top} [\mathbf{\Theta}\mathbf{x}_i \, \Vert \, \mathbf{\Theta}\mathbf{x}_k] \right)\right)}.\]

Parameters

• in_channels (int) – Size of each input sample.

• out_channels (int) – Size of each output sample.

• heads (int, optional) – Number of multi-head-attentions. (default: 1)

• concat (bool, optional) – If set to False, the multi-head attentions are averaged instead of concatenated. (default: True)

• negative_slope (float, optional) – LeakyReLU angle of the negative slope. (default: 0.2)

• dropout (float, optional) – Dropout probability of the normalized attention coefficients which exposes each node to a stochastically sampled neighborhood during training. (default: 0)

• bias (bool, optional) – If set to False, the layer will not learn an additive bias. (default: True)

• **kwargs (optional) – Additional arguments of torch_geometric.nn.conv.MessagePassing.

forward(x, edge_index, size=None)

reset_parameters()

class AGNNConv(requires_grad=True, **kwargs)

Graph attentional propagation layer from the “Attention-based Graph Neural Network for Semi-Supervised Learning” paper

\[\mathbf{X}^{\prime} = \mathbf{P} \mathbf{X},\]

where the propagation matrix \(\mathbf{P}\) is computed as

\[P_{i,j} = \frac{\exp( \beta \cdot \cos(\mathbf{x}_i, \mathbf{x}_j))} {\sum_{k \in \mathcal{N}(i)\cup \{ i \}} \exp( \beta \cdot \cos(\mathbf{x}_i, \mathbf{x}_k))}\]

with trainable parameter \(\beta\).

Parameters

• requires_grad (bool, optional) – If set to False, \(\beta\) will not be trainable. (default: True)

• **kwargs (optional) – Additional arguments of torch_geometric.nn.conv.MessagePassing.

forward(x, edge_index)

reset_parameters()

class TAGConv(in_channels, out_channels, K=3, bias=True, **kwargs)

The topology adaptive graph convolutional networks operator from the

“Topology Adaptive Graph Convolutional Networks” paper

\[\mathbf{X}^{\prime} = \sum_{k=0}^K \mathbf{D}^{-1/2} \mathbf{A}^k \mathbf{D}^{-1/2}\mathbf{X} \mathbf{\Theta}_{k},\]

where \(\mathbf{A}\) denotes the adjacency matrix and \(D_{ii} = \sum_{j=0} A_{ij}\) its diagonal degree matrix.

Parameters

• in_channels (int) – Size of each input sample.

• out_channels (int) – Size of each output sample.

• K (int, optional) – Number of hops \(K\). (default: 3)

• bias (bool, optional) – If set to False, the layer will not learn an additive bias. (default: True)

• **kwargs (optional) – Additional arguments of torch_geometric.nn.conv.MessagePassing.

forward(x, edge_index, edge_weight=None)

static norm(edge_index, num_nodes, edge_weight=None, dtype=None)

reset_parameters()

class GINConv(nn, eps=0, train_eps=False, **kwargs)

The graph isomorphism operator from the “How Powerful are Graph Neural Networks?” paper

\[\mathbf{x}^{\prime}_i = h_{\mathbf{\Theta}} \left( (1 + \epsilon) \cdot \mathbf{x}_i + \sum_{j \in \mathcal{N}(i)} \mathbf{x}_j \right),\]

here \(h_{\mathbf{\Theta}}\) denotes a neural network, .i.e. a MLP.

Parameters

• nn (torch.nn.Module) – A neural network \(h_{\mathbf{\Theta}}\) that maps node features x of shape [-1, in_channels] to shape [-1, out_channels], e.g., defined by torch.nn.Sequential.

• eps (float, optional) – (Initial) \(\epsilon\) value. (default: 0)

• train_eps (bool, optional) – If set to True, \(\epsilon\) will be a trainable parameter. (default: False)

• **kwargs (optional) – Additional arguments of torch_geometric.nn.conv.MessagePassing.

forward(x, edge_index)

reset_parameters()

class ARMAConv(in_channels, out_channels, num_stacks=1, num_layers=1, shared_weights=False, act=<function relu>, dropout=0, bias=True)

The ARMA graph convolutional operator from the “Graph Neural Networks with Convolutional ARMA Filters” paper

\[\mathbf{X}^{\prime} = \frac{1}{K} \sum_{k=1}^K \mathbf{X}_k^{(T)},\]

with \(\mathbf{X}_k^{(T)}\) being recursively defined by

\[\mathbf{X}_k^{(t+1)} = \sigma \left( \mathbf{\hat{L}} \mathbf{X}_k^{(t)} \mathbf{W} + \mathbf{X}^{(0)} \mathbf{V} \right),\]

where \(\mathbf{\hat{L}} = \mathbf{I} - \mathbf{L} = \mathbf{D}^{-1/2} \mathbf{A} \mathbf{D}^{-1/2}\) denotes the modified Laplacian \(\mathbf{L} = \mathbf{I} - \mathbf{D}^{-1/2} \mathbf{A} \mathbf{D}^{-1/2}\).

Parameters

• in_channels (int) – Size of each input sample \(\mathbf{x}^{(t)}\).

• out_channels (int) – Size of each output sample \(\mathbf{x}^{(t+1)}\).

• num_stacks (int, optional) – Number of parallel stacks \(K\). (default: 1).

• num_layers (int, optional) – Number of layers \(T\). (default: 1)

• act (callable, optional) – Activation function \(\sigma\). (default: torch.nn.functional.ReLU())

• shared_weights (int, optional) – If set to True the layers in each stack will share the same parameters. (default: False)

• dropout (float, optional) – Dropout probability of the skip connection. (default: 0)

• bias (bool, optional) – If set to False, the layer will not learn an additive bias. (default: True)

forward(x, edge_index, edge_weight=None)

reset_parameters()

class SGConv(in_channels, out_channels, K=1, cached=False, bias=True, **kwargs)

The simple graph convolutional operator from the “Simplifying Graph Convolutional Networks” paper

\[\mathbf{X}^{\prime} = {\left(\mathbf{\hat{D}}^{-1/2} \mathbf{\hat{A}} \mathbf{\hat{D}}^{-1/2} \right)}^K \mathbf{X} \mathbf{\Theta},\]

where \(\mathbf{\hat{A}} = \mathbf{A} + \mathbf{I}\) denotes the adjacency matrix with inserted self-loops and \(\hat{D}_{ii} = \sum_{j=0} \hat{A}_{ij}\) its diagonal degree matrix.

Parameters

• in_channels (int) – Size of each input sample.

• out_channels (int) – Size of each output sample.

• K (int, optional) – Number of hops \(K\). (default: 1)

• cached (bool, optional) – If set to True, the layer will cache the computation of \({\left(\mathbf{\hat{D}}^{-1/2} \mathbf{\hat{A}} \mathbf{\hat{D}}^{-1/2} \right)}^K\) on first execution, and will use the cached version for further executions. This parameter should only be set to True in transductive learning scenarios. (default: False)

• bias (bool, optional) – If set to False, the layer will not learn an additive bias. (default: True)

• **kwargs (optional) – Additional arguments of torch_geometric.nn.conv.MessagePassing.

forward(x, edge_index, edge_weight=None)

reset_parameters()

class APPNP(K, alpha, bias=True, **kwargs)

The approximate personalized propagation of neural predictions layer from the “Predict then Propagate: Graph Neural Networks meet Personalized PageRank” paper

\[ \begin{align}\begin{aligned}\mathbf{X}^{(0)} &= \mathbf{X}\\\mathbf{X}^{(k)} &= (1 - \alpha) \mathbf{\hat{D}}^{-1/2} \mathbf{\hat{A}} \mathbf{\hat{D}}^{-1/2} \mathbf{X}^{(k-1)} + \alpha \mathbf{X}^{(0)}\\\mathbf{X}^{\prime} &= \mathbf{X}^{(K)},\end{aligned}\end{align} \]

where \(\mathbf{\hat{A}} = \mathbf{A} + \mathbf{I}\) denotes the adjacency matrix with inserted self-loops and \(\hat{D}_{ii} = \sum_{j=0} \hat{A}_{ij}\) its diagonal degree matrix.

Parameters

• K (int) – Number of iterations \(K\).

• alpha (float) – Teleport probability \(\alpha\).

• bias (bool, optional) – If set to False, the layer will not learn an additive bias. (default: True)

• **kwargs (optional) – Additional arguments of torch_geometric.nn.conv.MessagePassing.

forward(x, edge_index, edge_weight=None)

class RGCNConv(in_channels, out_channels, num_relations, num_bases, root_weight=True, bias=True, **kwargs)

The relational graph convolutional operator from the “Modeling Relational Data with Graph Convolutional Networks” paper

\[\mathbf{x}^{\prime}_i = \mathbf{\Theta}_{\textrm{root}} \cdot \mathbf{x}_i + \sum_{r \in \mathcal{R}} \sum_{j \in \mathcal{N}_r(i)} \frac{1}{|\mathcal{N}_r(i)|} \mathbf{\Theta}_r \cdot \mathbf{x}_j,\]

where \(\mathcal{R}\) denotes the set of relations, i.e. edge types. Edge type needs to be a one-dimensional torch.long tensor which stores a relation identifier \(\in \{ 0, \ldots, |\mathcal{R}| - 1\}\) for each edge.

Parameters

• in_channels (int) – Size of each input sample.

• out_channels (int) – Size of each output sample.

• num_relations (int) – Number of relations.

• num_bases (int) – Number of bases used for basis-decomposition.

• root_weight (bool, optional) – If set to False, the layer will not add transformed root node features to the output. (default: True)

• bias (bool, optional) – If set to False, the layer will not learn an additive bias. (default: True)

• **kwargs (optional) – Additional arguments of torch_geometric.nn.conv.MessagePassing.

forward(x, edge_index, edge_type, edge_norm=None, size=None)

reset_parameters()

class SignedConv(in_channels, out_channels, first_aggr, bias=True, **kwargs)

The signed graph convolutional operator from the “Signed Graph Convolutional Network” paper

\[ \begin{align}\begin{aligned}\mathbf{x}_v^{(\textrm{pos})} &= \mathbf{\Theta}^{(\textrm{pos})} \left[ \frac{1}{|\mathcal{N}^{+}(v)|} \sum_{w \in \mathcal{N}^{+}(v)} \mathbf{x}_w , \mathbf{x}_v \right]\\\mathbf{x}_v^{(\textrm{neg})} &= \mathbf{\Theta}^{(\textrm{neg})} \left[ \frac{1}{|\mathcal{N}^{-}(v)|} \sum_{w \in \mathcal{N}^{-}(v)} \mathbf{x}_w , \mathbf{x}_v \right]\end{aligned}\end{align} \]

if first_aggr is set to True, and

\[ \begin{align}\begin{aligned}\mathbf{x}_v^{(\textrm{pos})} &= \mathbf{\Theta}^{(\textrm{pos})} \left[ \frac{1}{|\mathcal{N}^{+}(v)|} \sum_{w \in \mathcal{N}^{+}(v)} \mathbf{x}_w^{(\textrm{pos})}, \frac{1}{|\mathcal{N}^{-}(v)|} \sum_{w \in \mathcal{N}^{-}(v)} \mathbf{x}_w^{(\textrm{neg})} , \mathbf{x}_v^{(\textrm{pos})} \right]\\\mathbf{x}_v^{(\textrm{neg})} &= \mathbf{\Theta}^{(\textrm{pos})} \left[ \frac{1}{|\mathcal{N}^{+}(v)|} \sum_{w \in \mathcal{N}^{+}(v)} \mathbf{x}_w^{(\textrm{neg})}, \frac{1}{|\mathcal{N}^{-}(v)|} \sum_{w \in \mathcal{N}^{-}(v)} \mathbf{x}_w^{(\textrm{pos})} , \mathbf{x}_v^{(\textrm{neg})} \right]\end{aligned}\end{align} \]

otherwise. In case first_aggr is False, the layer expects x to be a tensor where x[:, :in_channels] denotes the positive node features \(\mathbf{X}^{(\textrm{pos})}\) and x[:, in_channels:] denotes the negative node features \(\mathbf{X}^{(\textrm{neg})}\).

Parameters

• in_channels (int) – Size of each input sample.

• out_channels (int) – Size of each output sample.

• first_aggr (bool) – Denotes which aggregation formula to use.

• bias (bool, optional) – If set to False, the layer will not learn an additive bias. (default: True)

• **kwargs (optional) – Additional arguments of torch_geometric.nn.conv.MessagePassing.

forward(x, pos_edge_index, neg_edge_index)

reset_parameters()

class DNAConv(channels, heads=1, groups=1, dropout=0, cached=False, bias=True, **kwargs)

The dynamic neighborhood aggregation operator from the “Just Jump: Towards Dynamic Neighborhood Aggregation in Graph Neural Networks” paper

\[\mathbf{x}_v^{(t)} = h_{\mathbf{\Theta}}^{(t)} \left( \mathbf{x}_{v \leftarrow v}^{(t)}, \left\{ \mathbf{x}_{v \leftarrow w}^{(t)} : w \in \mathcal{N}(v) \right\} \right)\]

based on (multi-head) dot-product attention

\[\mathbf{x}_{v \leftarrow w}^{(t)} = \textrm{Attention} \left( \mathbf{x}^{(t-1)}_v \, \mathbf{\Theta}_Q^{(t)}, [\mathbf{x}_w^{(1)}, \ldots, \mathbf{x}_w^{(t-1)}] \, \mathbf{\Theta}_K^{(t)}, \, [\mathbf{x}_w^{(1)}, \ldots, \mathbf{x}_w^{(t-1)}] \, \mathbf{\Theta}_V^{(t)} \right)\]

with \(\mathbf{\Theta}_Q^{(t)}, \mathbf{\Theta}_K^{(t)}, \mathbf{\Theta}_V^{(t)}\) denoting (grouped) projection matrices for query, key and value information, respectively. \(h^{(t)}_{\mathbf{\Theta}}\) is implemented as a non-trainable version of torch_geometric.nn.conv.GCNConv.

Note

In contrast to other layers, this operator expects node features as shape [num_nodes, num_layers, channels].

Parameters

• channels (int) – Size of each input/output sample.

• heads (int, optional) – Number of multi-head-attentions. (default: 1)

• groups (int, optional) – Number of groups to use for all linear projections. (default: 1)

• dropout (float, optional) – Dropout probability of attention coefficients. (default: 0)

• cached (bool, optional) – If set to True, the layer will cache the computation of \(\mathbf{\hat{D}}^{-1/2} \mathbf{\hat{A}} \mathbf{\hat{D}}^{-1/2}\) on first execution, and will use the cached version for further executions. This parameter should only be set to True in transductive learning scenarios. (default: False)

• bias (bool, optional) – If set to False, the layer will not learn an additive bias. (default: True)

• **kwargs (optional) – Additional arguments of torch_geometric.nn.conv.MessagePassing.

forward(x, edge_index, edge_weight=None)

reset_parameters()

class PointConv(local_nn=None, global_nn=None, **kwargs)

The PointNet set layer from the “PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation” and “PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space” papers

\[\mathbf{x}^{\prime}_i = \gamma_{\mathbf{\Theta}} \left( \max_{j \in \mathcal{N}(i) \cup \{ i \}} h_{\mathbf{\Theta}} ( \mathbf{x}_j, \mathbf{p}_j - \mathbf{p}_i) \right),\]

where \(\gamma_{\mathbf{\Theta}}\) and \(h_{\mathbf{\Theta}}\) denote neural networks, .i.e. MLPs, and \(\mathbf{P} \in \mathbb{R}^{N \times D}\) defines the position of each point.

Parameters

• local_nn (torch.nn.Module, optional) – A neural network \(h_{\mathbf{\Theta}}\) that maps node features x and relative spatial coordinates pos_j - pos_i of shape [-1, in_channels + num_dimensions] to shape [-1, out_channels], e.g., defined by torch.nn.Sequential. (default: None)

• global_nn (torch.nn.Module, optional) – A neural network \(\gamma_{\mathbf{\Theta}}\) that maps aggregated node features of shape [-1, out_channels] to shape [-1, final_out_channels], e.g., defined by torch.nn.Sequential. (default: None)

• **kwargs (optional) – Additional arguments of torch_geometric.nn.conv.MessagePassing.

forward(x, pos, edge_index)

Parameters

• x (Tensor) – The node feature matrix. Allowed to be None.

• pos (Tensor or tuple) – The node position matrix. Either given as tensor for use in general message passing or as tuple for use in message passing in bipartite graphs.

• edge_index (LongTensor) – The edge indices.

reset_parameters()

class GMMConv(in_channels, out_channels, dim, kernel_size, separate_gaussians=False, aggr='mean', root_weight=True, bias=True, **kwargs)

The gaussian mixture model convolutional operator from the “Geometric Deep Learning on Graphs and Manifolds using Mixture Model CNNs” paper

\[\mathbf{x}^{\prime}_i = \frac{1}{|\mathcal{N}(i)|} \sum_{j \in \mathcal{N}(i)} \frac{1}{K} \sum_{k=1}^K \mathbf{w}_k(\mathbf{e}_{i,j}) \odot \mathbf{\Theta}_k \mathbf{x}_j,\]

where

\[\mathbf{w}_k(\mathbf{e}) = \exp \left( -\frac{1}{2} {\left( \mathbf{e} - \mathbf{\mu}_k \right)}^{\top} \Sigma_k^{-1} \left( \mathbf{e} - \mathbf{\mu}_k \right) \right)\]

denotes a weighting function based on trainable mean vector \(\mathbf{\mu}_k\) and diagonal covariance matrix \(\mathbf{\Sigma}_k\).

Parameters

• in_channels (int) – Size of each input sample.

• out_channels (int) – Size of each output sample.

• dim (int) – Pseudo-coordinate dimensionality.

• kernel_size (int) – Number of kernels \(K\).

• separate_gaussians (bool, optional) – If set to True, will learn separate GMMs for every pair of input and output channel, inspired by traditional CNNs. (default: False)

• aggr (string, optional) – The aggregation operator to use ("add", "mean", "max"). (default: "mean")

• root_weight (bool, optional) – If set to False, the layer will not add transformed root node features to the output. (default: True)

• bias (bool, optional) – If set to False, the layer will not learn an additive bias. (default: True)

• **kwargs (optional) – Additional arguments of torch_geometric.nn.conv.MessagePassing.

forward(x, edge_index, pseudo)

reset_parameters()

class SplineConv(in_channels, out_channels, dim, kernel_size, is_open_spline=True, degree=1, aggr='mean', root_weight=True, bias=True, **kwargs)

The spline-based convolutional operator from the “SplineCNN: Fast Geometric Deep Learning with Continuous B-Spline Kernels” paper

\[\mathbf{x}^{\prime}_i = \frac{1}{|\mathcal{N}(i)|} \sum_{j \in \mathcal{N}(i)} \mathbf{x}_j \cdot h_{\mathbf{\Theta}}(\mathbf{e}_{i,j}),\]

where \(h_{\mathbf{\Theta}}\) denotes a kernel function defined over the weighted B-Spline tensor product basis.

Note

Pseudo-coordinates must lay in the fixed interval \([0, 1]\) for this method to work as intended.

Parameters

• in_channels (int) – Size of each input sample.

• out_channels (int) – Size of each output sample.

• dim (int) – Pseudo-coordinate dimensionality.

• kernel_size (int or [int]) – Size of the convolving kernel.

• is_open_spline (bool or [bool], optional) – If set to False, the operator will use a closed B-spline basis in this dimension. (default True)

• degree (int, optional) – B-spline basis degrees. (default: 1)

• aggr (string, optional) – The aggregation operator to use ("add", "mean", "max"). (default: "mean")

• root_weight (bool, optional) – If set to False, the layer will not add transformed root node features to the output. (default: True)

• bias (bool, optional) – If set to False, the layer will not learn an additive bias. (default: True)

• **kwargs (optional) – Additional arguments of torch_geometric.nn.conv.MessagePassing.

forward(x, edge_index, pseudo)

reset_parameters()

class NNConv(in_channels, out_channels, nn, aggr='add', root_weight=True, bias=True, **kwargs)

The continuous kernel-based convolutional operator from the “Neural Message Passing for Quantum Chemistry” paper. This convolution is also known as the edge-conditioned convolution from the “Dynamic Edge-Conditioned Filters in Convolutional Neural Networks on Graphs” paper (see torch_geometric.nn.conv.ECConv for an alias):

\[\mathbf{x}^{\prime}_i = \mathbf{\Theta} \mathbf{x}_i + \sum_{j \in \mathcal{N}(i)} \mathbf{x}_j \cdot h_{\mathbf{\Theta}}(\mathbf{e}_{i,j}),\]

where \(h_{\mathbf{\Theta}}\) denotes a neural network, .i.e. a MLP.

Parameters

• in_channels (int) – Size of each input sample.

• out_channels (int) – Size of each output sample.

• nn (torch.nn.Module) – A neural network \(h_{\mathbf{\Theta}}\) that maps edge features edge_attr of shape [-1, num_edge_features] to shape [-1, in_channels * out_channels], e.g., defined by torch.nn.Sequential.

• aggr (string, optional) – The aggregation scheme to use ("add", "mean", "max"). (default: "add")

• root_weight (bool, optional) – If set to False, the layer will not add the transformed root node features to the output. (default: True)

• bias (bool, optional) – If set to False, the layer will not learn an additive bias. (default: True)

• **kwargs (optional) – Additional arguments of torch_geometric.nn.conv.MessagePassing.

forward(x, edge_index, edge_attr)

reset_parameters()

ECConv

alias of torch_geometric.nn.conv.nn_conv.NNConv

class CGConv(channels, dim, aggr='add', bias=True, **kwargs)

The crystal graph convolutional operator from the “Crystal Graph Convolutional Neural Networks for an Accurate and Interpretable Prediction of Material Properties” paper

\[\mathbf{x}^{\prime}_i = \mathbf{x}_i + \sum_{j \in \mathcal{N}(i)} \sigma \left( \mathbf{z}_{i,j} \mathbf{W}_f + \mathbf{b}_f \right) \odot g \left( \mathbf{z}_{i,j} \mathbf{W}_s + \mathbf{b}_s \right)\]

where \(\mathbf{z}_{i,j} = [ \mathbf{x}_i, \mathbf{x}_j, \mathbf{e}_{i,j} ]\) denotes the concatenation of central node features, neighboring node features and edge features. In addition, \(\sigma\) and \(g\) denote the sigmoid and softplus functions, respectively.

Parameters

• channels (int) – Size of each input sample.

• dim (int) – Edge feature dimensionality.

• aggr (string, optional) – The aggregation operator to use ("add", "mean", "max"). (default: "add")

• bias (bool, optional) – If set to False, the layer will not learn an additive bias. (default: True)

• **kwargs (optional) – Additional arguments of torch_geometric.nn.conv.MessagePassing.

forward(x, edge_index, edge_attr)

reset_parameters()

class EdgeConv(nn, aggr='max', **kwargs)

The edge convolutional operator from the “Dynamic Graph CNN for Learning on Point Clouds” paper

\[\mathbf{x}^{\prime}_i = \sum_{j \in \mathcal{N}(i)} h_{\mathbf{\Theta}}(\mathbf{x}_i \, \Vert \, \mathbf{x}_j - \mathbf{x}_i),\]

where \(h_{\mathbf{\Theta}}\) denotes a neural network, .i.e. a MLP.

Parameters

• nn (torch.nn.Module) – A neural network \(h_{\mathbf{\Theta}}\) that maps pair-wise concatenated node features x of shape [-1, 2 * in_channels] to shape [-1, out_channels], e.g., defined by torch.nn.Sequential.

• aggr (string, optional) – The aggregation scheme to use ("add", "mean", "max"). (default: "max")

• **kwargs (optional) – Additional arguments of torch_geometric.nn.conv.MessagePassing.

forward(x, edge_index)

reset_parameters()

class DynamicEdgeConv(nn, k, aggr='max', **kwargs)

The dynamic edge convolutional operator from the “Dynamic Graph CNN for Learning on Point Clouds” paper (see torch_geometric.nn.conv.EdgeConv), where the graph is dynamically constructed using nearest neighbors in the feature space.

Parameters

• nn (torch.nn.Module) – A neural network \(h_{\mathbf{\Theta}}\) that maps pair-wise concatenated node features x of shape :obj:`[-1, 2 * in_channels] to shape [-1, out_channels], e.g. defined by torch.nn.Sequential.

• k (int) – Number of nearest neighbors.

• aggr (string) – The aggregation operator to use ("add", "mean", "max"). (default: "max")

• **kwargs (optional) – Additional arguments of torch_geometric.nn.conv.MessagePassing.

forward(x, batch=None)

class XConv(in_channels, out_channels, dim, kernel_size, hidden_channels=None, dilation=1, bias=True, **kwargs)

The convolutional operator on \(\mathcal{X}\)-transformed points from the “PointCNN: Convolution On X-Transformed Points” paper

\[\mathbf{x}^{\prime}_i = \mathrm{Conv}\left(\mathbf{K}, \gamma_{\mathbf{\Theta}}(\mathbf{P}_i - \mathbf{p}_i) \times \left( h_\mathbf{\Theta}(\mathbf{P}_i - \mathbf{p}_i) \, \Vert \, \mathbf{x}_i \right) \right),\]

where \(\mathbf{K}\) and \(\mathbf{P}_i\) denote the trainable filter and neighboring point positions of \(\mathbf{x}_i\), respectively. \(\gamma_{\mathbf{\Theta}}\) and \(h_{\mathbf{\Theta}}\) describe neural networks, i.e. MLPs, where \(h_{\mathbf{\Theta}}\) individually lifts each point into a higher-dimensional space, and \(\gamma_{\mathbf{\Theta}}\) computes the \(\mathcal{X}\)- transformation matrix based on all points in a neighborhood.

Parameters

• in_channels (int) – Size of each input sample.

• out_channels (int) – Size of each output sample.

• dim (int) – Point cloud dimensionality.

• kernel_size (int) – Size of the convolving kernel, i.e. number of neighbors including self-loops.

• hidden_channels (int, optional) – Output size of \(h_{\mathbf{\Theta}}\), i.e. dimensionality of lifted points. If set to None, will be automatically set to in_channels / 4. (default: None)

• dilation (int, optional) – The factor by which the neighborhood is extended, from which kernel_size neighbors are then uniformly sampled. Can be interpreted as the dilation rate of classical convolutional operators. (default: 1)

• bias (bool, optional) – If set to False, the layer will not learn an additive bias. (default: True)

• **kwargs (optional) – Additional arguments of torch_cluster.knn_graph.

forward(x, pos, batch=None)

reset_parameters()

class PPFConv(local_nn=None, global_nn=None, **kwargs)

The PPFNet operator from the “PPFNet: Global Context Aware Local Features for Robust 3D Point Matching” paper

\[\mathbf{x}^{\prime}_i = \gamma_{\mathbf{\Theta}} \left( \max_{j \in \mathcal{N}(i) \cup \{ i \}} h_{\mathbf{\Theta}} ( \mathbf{x}_j, \| \mathbf{d_{j,i}} \|, \angle(\mathbf{n}_i, \mathbf{d_{j,i}}), \angle(\mathbf{n}_j, \mathbf{d_{j,i}}), \angle(\mathbf{n}_i, \mathbf{n}_j) \right)\]

where \(\gamma_{\mathbf{\Theta}}\) and \(h_{\mathbf{\Theta}}\) denote neural networks, .i.e. MLPs, which takes in node features and torch_geometric.transforms.PointPairFeatures.

Parameters

• local_nn (torch.nn.Module, optional) – A neural network \(h_{\mathbf{\Theta}}\) that maps node features x and relative spatial coordinates pos_j - pos_i of shape [-1, in_channels + num_dimensions] to shape [-1, out_channels], e.g., defined by torch.nn.Sequential. (default: None)

• global_nn (torch.nn.Module, optional) – A neural network \(\gamma_{\mathbf{\Theta}}\) that maps aggregated node features of shape [-1, out_channels] to shape [-1, final_out_channels], e.g., defined by torch.nn.Sequential. (default: None)

• **kwargs (optional) – Additional arguments of torch_geometric.nn.conv.MessagePassing.

forward(x, pos, norm, edge_index)

Parameters

• x (Tensor) – The node feature matrix. Allowed to be None.

• pos (Tensor or tuple) – The node position matrix. Either given as tensor for use in general message passing or as tuple for use in message passing in bipartite graphs.

• norm (Tensor or tuple) – The normal vectors of each node. Either given as tensor for use in general message passing or as tuple for use in message passing in bipartite graphs.

• edge_index (LongTensor) – The edge indices.

reset_parameters()

class FeaStConv(in_channels, out_channels, heads=1, bias=True, **kwargs)

The (translation-invariant) feature-steered convolutional operator from the “FeaStNet: Feature-Steered Graph Convolutions for 3D Shape Analysis” paper

\[\mathbf{x}^{\prime}_i = \frac{1}{|\mathcal{N}(i)|} \sum_{j \in \mathcal{N}(i)} \sum_{h=1}^H q_h(\mathbf{x}_i, \mathbf{x}_j) \mathbf{W}_h \mathbf{x}_j\]

with \(q_h(\mathbf{x}_i, \mathbf{x}_j) = \mathrm{softmax}_j (\mathbf{u}_h^{\top} (\mathbf{x}_j - \mathbf{x}_i) + c_h)\), where \(H\) denotes the number of attention heads, and \(\mathbf{W}_h\), \(\mathbf{u}_h\) and \(c_h\) are trainable parameters.

Parameters

• in_channels (int) – Size of each input sample.

• out_channels (int) – Size of each output sample.

• heads (int, optional) – Number of attention heads \(H\). (default: 1)

• bias (bool, optional) – If set to False, the layer will not learn an additive bias. (default: True)

• **kwargs (optional) – Additional arguments of torch_geometric.nn.conv.MessagePassing.

forward(x, edge_index)

reset_parameters()

class HypergraphConv(in_channels, out_channels, use_attention=False, heads=1, concat=True, negative_slope=0.2, dropout=0, bias=True)

The hypergraph convolutional operator from the “Hypergraph Convolution and Hypergraph Attention” paper

\[\mathbf{X}^{\prime} = \mathbf{D}^{-1} \mathbf{H} \mathbf{W} \mathbf{B}^{-1} \mathbf{H}^{\top} \mathbf{X} \mathbf{\Theta}\]

where \(\mathbf{H} \in {\{ 0, 1 \}}^{N \times M}\) is the incidence matrix, \(\mathbf{W}\) is the diagonal hyperedge weight matrix, and \(\mathbf{D}\) and \(\mathbf{B}\) are the corresponding degree matrices.

Parameters

• in_channels (int) – Size of each input sample.

• out_channels (int) – Size of each output sample.

• use_attention (bool, optional) – If set to True, attention will be added to this layer. (default: False)

• heads (int, optional) – Number of multi-head-attentions. (default: 1)

• concat (bool, optional) – If set to False, the multi-head attentions are averaged instead of concatenated. (default: True)

• negative_slope (float, optional) – LeakyReLU angle of the negative slope. (default: 0.2)

• dropout (float, optional) – Dropout probability of the normalized attention coefficients which exposes each node to a stochastically sampled neighborhood during training. (default: 0)

• bias (bool, optional) – If set to False, the layer will not learn an additive bias. (default: True)

forward(x, hyperedge_index, hyperedge_weight=None)

Parameters

• x (Tensor) – Node feature matrix \(\mathbf{X}\)

• hyper_edge_index (LongTensor) – Hyperedge indices from \(\mathbf{H}\).

• hyperedge_weight (Tensor, optional) – Sparse hyperedge weights from \(\mathbf{W}\). (default: None)

reset_parameters()

class MetaLayer(edge_model=None, node_model=None, global_model=None)

A meta layer for building any kind of graph network, inspired by the “Relational Inductive Biases, Deep Learning, and Graph Networks” paper.

A graph network takes a graph as input and returns an updated graph as output (with same connectivity). The input graph has node features x, edge features edge_attr as well as global-level features u. The output graph has the same structure, but updated features.

Edge features, node features as well as global features are updated by calling the modules edge_model, node_model and global_model, respectively.

To allow for batch-wise graph processing, all callable functions take an additional argument batch, which determines the assignment of edges or nodes to their specific graphs.

Parameters

• edge_model (Module, optional) – A callable which updates a graph’s edge features based on its source and target node features, its current edge features and its global features. (default: None)

• node_model (Module, optional) – A callable which updates a graph’s node features based on its current node features, its graph connectivity, its edge features and its global features. (default: None)

• global_model (Module, optional) – A callable which updates a graph’s global features based on its node features, its graph connectivity, its edge features and its current global features.

from torch.nn import Sequential as Seq, Linear as Lin, ReLU
from torch_scatter import scatter_mean
from torch_geometric.nn import MetaLayer

class EdgeModel(torch.nn.Module):
    def __init__(self):
        super(EdgeModel, self).__init__()
        self.edge_mlp = Seq(Lin(..., ...), ReLU(), Lin(..., ...))

    def forward(self, src, dest, edge_attr, u, batch):
        # source, target: [E, F_x], where E is the number of edges.
        # edge_attr: [E, F_e]
        # u: [B, F_u], where B is the number of graphs.
        # batch: [E] with max entry B - 1.
        out = torch.cat([src, dest, edge_attr, u[batch]], 1)
        return self.edge_mlp(out)

class NodeModel(torch.nn.Module):
    def __init__(self):
        super(NodeModel, self).__init__()
        self.node_mlp_1 = Seq(Lin(..., ...), ReLU(), Lin(..., ...))
        self.node_mlp_2 = Seq(Lin(..., ...), ReLU(), Lin(..., ...))

    def forward(self, x, edge_index, edge_attr, u, batch):
        # x: [N, F_x], where N is the number of nodes.
        # edge_index: [2, E] with max entry N - 1.
        # edge_attr: [E, F_e]
        # u: [B, F_u]
        # batch: [N] with max entry B - 1.
        row, col = edge_index
        out = torch.cat([x[row], edge_attr], dim=1)
        out = self.node_mlp_1(out)
        out = scatter_mean(out, col, dim=0, dim_size=x.size(0))
        out = torch.cat([x, out, u[batch]], dim=1)
        return self.node_mlp_2(out)

class GlobalModel(torch.nn.Module):
    def __init__(self):
        super(GlobalModel, self).__init__()
        self.global_mlp = Seq(Lin(..., ...), ReLU(), Lin(..., ...))

    def forward(self, x, edge_index, edge_attr, u, batch):
        # x: [N, F_x], where N is the number of nodes.
        # edge_index: [2, E] with max entry N - 1.
        # edge_attr: [E, F_e]
        # u: [B, F_u]
        # batch: [N] with max entry B - 1.
        out = torch.cat([u, scatter_mean(x, batch, dim=0)], dim=1)
        return self.global_mlp(out)

op = MetaLayer(EdgeModel(), NodeModel(), GlobalModel())
x, edge_attr, u = op(x, edge_index, edge_attr, u, batch)

forward(x, edge_index, edge_attr=None, u=None, batch=None)

reset_parameters()

Dense Convolutional Layers

class DenseGCNConv(in_channels, out_channels, improved=False, bias=True)

See torch_geometric.nn.conv.GCNConv.

forward(x, adj, mask=None, add_loop=True)

Parameters

• x (Tensor) – Node feature tensor \(\mathbf{X} \in \mathbb{R}^{B \times N \times F}\), with batch-size \(B\), (maximum) number of nodes \(N\) for each graph, and feature dimension \(F\).

• adj (Tensor) – Adjacency tensor \(\mathbf{A} \in \mathbb{R}^{B \times N \times N}\). The adjacency tensor is broadcastable in the batch dimension, resulting in a shared adjacency matrix for the complete batch.

• mask (BoolTensor, optional) – Mask matrix \(\mathbf{M} \in {\{ 0, 1 \}}^{B \times N}\) indicating the valid nodes for each graph. (default: None)

• add_loop (bool, optional) – If set to False, the layer will not automatically add self-loops to the adjacency matrices. (default: True)

reset_parameters()

class DenseSAGEConv(in_channels, out_channels, normalize=False, bias=True)

See torch_geometric.nn.conv.SAGEConv.

forward(x, adj, mask=None, add_loop=True)

Parameters

• x (Tensor) – Node feature tensor \(\mathbf{X} \in \mathbb{R}^{B \times N \times F}\), with batch-size \(B\), (maximum) number of nodes \(N\) for each graph, and feature dimension \(F\).

• adj (Tensor) – Adjacency tensor \(\mathbf{A} \in \mathbb{R}^{B \times N \times N}\). The adjacency tensor is broadcastable in the batch dimension, resulting in a shared adjacency matrix for the complete batch.

• mask (BoolTensor, optional) – Mask matrix \(\mathbf{M} \in {\{ 0, 1 \}}^{B \times N}\) indicating the valid nodes for each graph. (default: None)

• add_loop (bool, optional) – If set to False, the layer will not automatically add self-loops to the adjacency matrices. (default: True)

reset_parameters()

class DenseGraphConv(in_channels, out_channels, aggr='add', bias=True)

See torch_geometric.nn.conv.GraphConv.

forward(x, adj, mask=None)

Parameters

• x (Tensor) – Node feature tensor \(\mathbf{X} \in \mathbb{R}^{B \times N \times F}\), with batch-size \(B\), (maximum) number of nodes \(N\) for each graph, and feature dimension \(F\).

• adj (Tensor) – Adjacency tensor \(\mathbf{A} \in \mathbb{R}^{B \times N \times N}\). The adjacency tensor is broadcastable in the batch dimension, resulting in a shared adjacency matrix for the complete batch.

• mask (BoolTensor, optional) – Mask matrix \(\mathbf{M} \in {\{ 0, 1 \}}^{B \times N}\) indicating the valid nodes for each graph. (default: None)

reset_parameters()

class DenseGINConv(nn, eps=0, train_eps=False)

See torch_geometric.nn.conv.GINConv.

Return type

Tensor

forward(x, adj, mask=None, add_loop=True)

Parameters

• x (Tensor) – Node feature tensor \(\mathbf{X} \in \mathbb{R}^{B \times N \times F}\), with batch-size \(B\), (maximum) number of nodes \(N\) for each graph, and feature dimension \(F\).

• adj (Tensor) – Adjacency tensor \(\mathbf{A} \in \mathbb{R}^{B \times N \times N}\). The adjacency tensor is broadcastable in the batch dimension, resulting in a shared adjacency matrix for the complete batch.

• mask (BoolTensor, optional) – Mask matrix \(\mathbf{M} \in {\{ 0, 1 \}}^{B \times N}\) indicating the valid nodes for each graph. (default: None)

• add_loop (bool, optional) – If set to False, the layer will not automatically add self-loops to the adjacency matrices. (default: True)

reset_parameters()

Normalization Layers

class BatchNorm(in_channels, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True)

Applies batch normalization over a batch of node features as described in the “Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift” paper

\[\mathbf{x}^{\prime}_i = \frac{\mathbf{x} - \textrm{E}[\mathbf{x}]}{\sqrt{\textrm{Var}[\mathbf{x}] + \epsilon}} \odot \gamma + \beta\]

Parameters

• in_channels (int) – Size of each input sample.

• eps (float, optional) – A value added to the denominator for numerical stability. (default: 1e-5)

• momentum (float, optional) – The value used for the running mean and running variance computation. (default: 0.1)

• affine (bool, optional) – If set to True, this module has learnable affine parameters \(\gamma\) and \(\beta\). (default: True)

• track_running_stats (bool, optional) – If set to True, this module tracks the running mean and variance, and when set to False, this module does not track such statistics and always uses batch statistics in both training and eval modes. (default: True)

forward(x)

class InstanceNorm(in_channels, eps=1e-05, momentum=0.1, affine=False, track_running_stats=False)

Applies instance normalization over each individual example in a batch of node features as described in the “Instance Normalization: The Missing Ingredient for Fast Stylization” paper

\[\mathbf{x}^{\prime}_i = \frac{\mathbf{x} - \textrm{E}[\mathbf{x}]}{\sqrt{\textrm{Var}[\mathbf{x}] + \epsilon}} \odot \gamma + \beta\]

Parameters

• in_channels (int) – Size of each input sample.

• eps (float, optional) – A value added to the denominator for numerical stability. (default: 1e-5)

• momentum (float, optional) – The value used for the running mean and running variance computation. (default: 0.1)

• affine (bool, optional) – If set to True, this module has learnable affine parameters \(\gamma\) and \(\beta\). (default: False)

• track_running_stats (bool, optional) – If set to True, this module tracks the running mean and variance, and when set to False, this module does not track such statistics and always uses instance statistics in both training and eval modes. (default: False)

forward(x, batch=None)

class GraphSizeNorm

Applies Graph Size Normalization over each individual graph in a batch of node features as described in the “Benchmarking Graph Neural Networks” paper

\[\mathbf{x}^{\prime}_i = \frac{\mathbf{x}_i}{\sqrt{|\mathcal{V}|}}\]

forward(x, batch=None)

Global Pooling Layers

global_add_pool(x, batch, size=None)

Returns batch-wise graph-level-outputs by adding node features across the node dimension, so that for a single graph \(\mathcal{G}_i\) its output is computed by

\[\mathbf{r}_i = \sum_{n=1}^{N_i} \mathbf{x}_n\]

Parameters

• x (Tensor) – Node feature matrix \(\mathbf{X} \in \mathbb{R}^{(N_1 + \ldots + N_B) \times F}\).

• batch (LongTensor) – Batch vector \(\mathbf{b} \in {\{ 0, \ldots, B-1\}}^N\), which assigns each node to a specific example.

• size (int, optional) – Batch-size \(B\). Automatically calculated if not given. (default: None)

Return type

Tensor

global_mean_pool(x, batch, size=None)

Returns batch-wise graph-level-outputs by averaging node features across the node dimension, so that for a single graph \(\mathcal{G}_i\) its output is computed by

\[\mathbf{r}_i = \frac{1}{N_i} \sum_{n=1}^{N_i} \mathbf{x}_n\]

Parameters

• x (Tensor) – Node feature matrix \(\mathbf{X} \in \mathbb{R}^{(N_1 + \ldots + N_B) \times F}\).

• batch (LongTensor) – Batch vector \(\mathbf{b} \in {\{ 0, \ldots, B-1\}}^N\), which assigns each node to a specific example.

• size (int, optional) – Batch-size \(B\). Automatically calculated if not given. (default: None)

Return type

Tensor

global_max_pool(x, batch, size=None)

Returns batch-wise graph-level-outputs by taking the channel-wise maximum across the node dimension, so that for a single graph \(\mathcal{G}_i\) its output is computed by

\[\mathbf{r}_i = \mathrm{max}_{n=1}^{N_i} \, \mathbf{x}_n\]

Parameters

• x (Tensor) – Node feature matrix \(\mathbf{X} \in \mathbb{R}^{(N_1 + \ldots + N_B) \times F}\).

• batch (LongTensor) – Batch vector \(\mathbf{b} \in {\{ 0, \ldots, B-1\}}^N\), which assigns each node to a specific example.

• size (int, optional) – Batch-size \(B\). Automatically calculated if not given. (default: None)

Return type

Tensor

global_sort_pool(x, batch, k)

The global pooling operator from the “An End-to-End Deep Learning Architecture for Graph Classification” paper, where node features are first sorted individually and then sorted in descending order based on their last features. The first \(k\) nodes form the output of the layer.

Parameters

• x (Tensor) – Node feature matrix \(\mathbf{X} \in \mathbb{R}^{N \times F}\).

• batch (LongTensor) – Batch vector \(\mathbf{b} \in {\{ 0, \ldots, B-1\}}^N\), which assigns each node to a specific example.

• k (int) – The number of nodes to hold for each graph.

Return type

Tensor

class GlobalAttention(gate_nn, nn=None)

Global soft attention layer from the “Gated Graph Sequence Neural Networks” paper

\[\mathbf{r}_i = \sum_{n=1}^{N_i} \mathrm{softmax} \left( h_{\mathrm{gate}} ( \mathbf{x}_n ) \right) \odot h_{\mathbf{\Theta}} ( \mathbf{x}_n ),\]

where \(h_{\mathrm{gate}} \colon \mathbb{R}^F \to \mathbb{R}\) and \(h_{\mathbf{\Theta}}\) denote neural networks, i.e. MLPS.

Parameters

• gate_nn (torch.nn.Module) – A neural network \(h_{\mathrm{gate}}\) that computes attention scores by mapping node features x of shape [-1, in_channels] to shape [-1, 1], e.g., defined by torch.nn.Sequential.

• nn (torch.nn.Module, optional) – A neural network \(h_{\mathbf{\Theta}}\) that maps node features x of shape [-1, in_channels] to shape [-1, out_channels] before combining them with the attention scores, e.g., defined by torch.nn.Sequential. (default: None)

forward(x, batch, size=None)

reset_parameters()

class Set2Set(in_channels, processing_steps, num_layers=1)

The global pooling operator based on iterative content-based attention from the “Order Matters: Sequence to sequence for sets” paper

\[ \begin{align}\begin{aligned}\mathbf{q}_t &= \mathrm{LSTM}(\mathbf{q}^{*}_{t-1})\\\alpha_{i,t} &= \mathrm{softmax}(\mathbf{x}_i \cdot \mathbf{q}_t)\\\mathbf{r}_t &= \sum_{i=1}^N \alpha_{i,t} \mathbf{x}_i\\\mathbf{q}^{*}_t &= \mathbf{q}_t \, \Vert \, \mathbf{r}_t,\end{aligned}\end{align} \]

where \(\mathbf{q}^{*}_T\) defines the output of the layer with twice the dimensionality as the input.

Parameters

• in_channels (int) – Size of each input sample.

• processing_steps (int) – Number of iterations \(T\).

• num_layers (int, optional) – Number of recurrent layers, .e.g, setting num_layers=2 would mean stacking two LSTMs together to form a stacked LSTM, with the second LSTM taking in outputs of the first LSTM and computing the final results. (default: 1)

forward(x, batch)

reset_parameters()

Pooling Layers

class TopKPooling(in_channels, ratio=0.5, min_score=None, multiplier=1, nonlinearity=<built-in method tanh of type object>)

\(\mathrm{top}_k\) pooling operator from the “Graph U-Nets”, “Towards Sparse Hierarchical Graph Classifiers” and “Understanding Attention and Generalization in Graph Neural Networks” papers

if min_score \(\tilde{\alpha}\) is None:

\[ \begin{align}\begin{aligned}\mathbf{y} &= \frac{\mathbf{X}\mathbf{p}}{\| \mathbf{p} \|}\\\mathbf{i} &= \mathrm{top}_k(\mathbf{y})\\\mathbf{X}^{\prime} &= (\mathbf{X} \odot \mathrm{tanh}(\mathbf{y}))_{\mathbf{i}}\\\mathbf{A}^{\prime} &= \mathbf{A}_{\mathbf{i},\mathbf{i}}\end{aligned}\end{align} \]

if min_score \(\tilde{\alpha}\) is a value in [0, 1]:

\[ \begin{align}\begin{aligned}\mathbf{y} &= \mathrm{softmax}(\mathbf{X}\mathbf{p})\\\mathbf{i} &= \mathbf{y}_i > \tilde{\alpha}\\\mathbf{X}^{\prime} &= (\mathbf{X} \odot \mathbf{y})_{\mathbf{i}}\\\mathbf{A}^{\prime} &= \mathbf{A}_{\mathbf{i},\mathbf{i}},\end{aligned}\end{align} \]

where nodes are dropped based on a learnable projection score \(\mathbf{p}\).

Parameters

• in_channels (int) – Size of each input sample.

• ratio (float) – Graph pooling ratio, which is used to compute \(k = \lceil \mathrm{ratio} \cdot N \rceil\). This value is ignored if min_score is not None. (default: 0.5)

• min_score (float, optional) – Minimal node score \(\tilde{\alpha}\) which is used to compute indices of pooled nodes \(\mathbf{i} = \mathbf{y}_i > \tilde{\alpha}\). When this value is not None, the ratio argument is ignored. (default: None)

• multiplier (float, optional) – Coefficient by which features gets multiplied after pooling. This can be useful for large graphs and when min_score is used. (default: 1)

• nonlinearity (torch.nn.functional, optional) – The nonlinearity to use. (default: torch.tanh)

forward(x, edge_index, edge_attr=None, batch=None, attn=None)

reset_parameters()

class SAGPooling(in_channels, ratio=0.5, GNN=<class 'torch_geometric.nn.conv.graph_conv.GraphConv'>, min_score=None, multiplier=1, nonlinearity=<built-in method tanh of type object>, **kwargs)

The self-attention pooling operator from the “Self-Attention Graph Pooling” and “Understanding Attention and Generalization in Graph Neural Networks” papers

if min_score \(\tilde{\alpha}\) is None:

\[ \begin{align}\begin{aligned}\mathbf{y} &= \textrm{GNN}(\mathbf{X}, \mathbf{A})\\\mathbf{i} &= \mathrm{top}_k(\mathbf{y})\\\mathbf{X}^{\prime} &= (\mathbf{X} \odot \mathrm{tanh}(\mathbf{y}))_{\mathbf{i}}\\\mathbf{A}^{\prime} &= \mathbf{A}_{\mathbf{i},\mathbf{i}}\end{aligned}\end{align} \]

if min_score \(\tilde{\alpha}\) is a value in [0, 1]:

\[ \begin{align}\begin{aligned}\mathbf{y} &= \mathrm{softmax}(\textrm{GNN}(\mathbf{X},\mathbf{A}))\\\mathbf{i} &= \mathbf{y}_i > \tilde{\alpha}\\\mathbf{X}^{\prime} &= (\mathbf{X} \odot \mathbf{y})_{\mathbf{i}}\\\mathbf{A}^{\prime} &= \mathbf{A}_{\mathbf{i},\mathbf{i}},\end{aligned}\end{align} \]

where nodes are dropped based on a learnable projection score \(\mathbf{p}\). Projections scores are learned based on a graph neural network layer.

Parameters

• in_channels (int) – Size of each input sample.

• ratio (float) – Graph pooling ratio, which is used to compute \(k = \lceil \mathrm{ratio} \cdot N \rceil\). This value is ignored if min_score is not None. (default: 0.5)

• GNN (torch.nn.Module, optional) – A graph neural network layer for calculating projection scores (one of torch_geometric.nn.conv.GraphConv, torch_geometric.nn.conv.GCNConv, torch_geometric.nn.conv.GATConv or torch_geometric.nn.conv.SAGEConv). (default: torch_geometric.nn.conv.GraphConv)

• min_score (float, optional) – Minimal node score \(\tilde{\alpha}\) which is used to compute indices of pooled nodes \(\mathbf{i} = \mathbf{y}_i > \tilde{\alpha}\). When this value is not None, the ratio argument is ignored. (default: None)

• multiplier (float, optional) – Coefficient by which features gets multiplied after pooling. This can be useful for large graphs and when min_score is used. (default: 1)

• nonlinearity (torch.nn.functional, optional) – The nonlinearity to use. (default: torch.tanh)

• **kwargs (optional) – Additional parameters for initializing the graph neural network layer.

forward(x, edge_index, edge_attr=None, batch=None, attn=None)

reset_parameters()

class EdgePooling(in_channels, edge_score_method=None, dropout=0, add_to_edge_score=0.5)

The edge pooling operator from the “Towards Graph Pooling by Edge Contraction” and “Edge Contraction Pooling for Graph Neural Networks” papers.

In short, a score is computed for each edge. Edges are contracted iteratively according to that score unless one of their nodes has already been part of a contracted edge.

To duplicate the configuration from the “Towards Graph Pooling by Edge Contraction” paper, use either EdgePooling.compute_edge_score_softmax() or EdgePooling.compute_edge_score_tanh(), and set add_to_edge_score to 0.

To duplicate the configuration from the “Edge Contraction Pooling for Graph Neural Networks” paper, set dropout to 0.2.

Parameters

• in_channels (int) – Size of each input sample.

• edge_score_method (function, optional) – The function to apply to compute the edge score from raw edge scores. By default, this is the softmax over all incoming edges for each node. This function takes in a raw_edge_score tensor of shape [num_nodes], an edge_index tensor and the number of nodes num_nodes, and produces a new tensor of the same size as raw_edge_score describing normalized edge scores. Included functions are EdgePooling.compute_edge_score_softmax(), EdgePooling.compute_edge_score_tanh(), and EdgePooling.compute_edge_score_sigmoid(). (default: EdgePooling.compute_edge_score_softmax())

• dropout (float, optional) – The probability with which to drop edge scores during training. (default: 0)

• add_to_edge_score (float, optional) – This is added to each computed edge score. Adding this greatly helps with unpool stability. (default: 0.5)

static compute_edge_score_sigmoid(raw_edge_score, edge_index, num_nodes)

static compute_edge_score_softmax(raw_edge_score, edge_index, num_nodes)

static compute_edge_score_tanh(raw_edge_score, edge_index, num_nodes)

forward(x, edge_index, batch)

Forward computation which computes the raw edge score, normalizes it, and merges the edges.

Parameters

• x (Tensor) – The node features.

• edge_index (LongTensor) – The edge indices.

• batch (LongTensor) – Batch vector \(\mathbf{b} \in {\{ 0, \ldots, B-1\}}^N\), which assigns each node to a specific example.

Return types:

• x (Tensor) - The pooled node features.

• edge_index (LongTensor) - The coarsened edge indices.

• batch (LongTensor) - The coarsened batch vector.

• unpool_info (unpool_description) - Information that is consumed by EdgePooling.unpool() for unpooling.

reset_parameters()

unpool(x, unpool_info)

Unpools a previous edge pooling step.

For unpooling, x should be of same shape as those produced by this layer’s forward() function. Then, it will produce an unpooled x in addition to edge_index and batch.

Parameters

• x (Tensor) – The node features.

• unpool_info (unpool_description) – Information that has been produced by EdgePooling.forward().

Return types:

• x (Tensor) - The unpooled node features.

• edge_index (LongTensor) - The new edge indices.

• batch (LongTensor) - The new batch vector.

unpool_description

alias of UnpoolDescription

max_pool(cluster, data, transform=None)

Pools and coarsens a graph given by the torch_geometric.data.Data object according to the clustering defined in cluster. All nodes within the same cluster will be represented as one node. Final node features are defined by the maximum features of all nodes within the same cluster, node positions are averaged and edge indices are defined to be the union of the edge indices of all nodes within the same cluster.

Parameters

• cluster (LongTensor) – Cluster vector \(\mathbf{c} \in \{ 0, \ldots, N - 1 \}^N\), which assigns each node to a specific cluster.

• data (Data) – Graph data object.

• transform (callable, optional) – A function/transform that takes in the coarsened and pooled torch_geometric.data.Data object and returns a transformed version. (default: None)

Return type

torch_geometric.data.Data

avg_pool(cluster, data, transform=None)

Pools and coarsens a graph given by the torch_geometric.data.Data object according to the clustering defined in cluster. Final node features are defined by the average features of all nodes within the same cluster. See torch_geometric.nn.pool.max_pool() for more details.

Parameters

• cluster (LongTensor) – Cluster vector \(\mathbf{c} \in \{ 0, \ldots, N - 1 \}^N\), which assigns each node to a specific cluster.

• data (Data) – Graph data object.

• transform (callable, optional) – A function/transform that takes in the coarsened and pooled torch_geometric.data.Data object and returns a transformed version. (default: None)

Return type

torch_geometric.data.Data

max_pool_x(cluster, x, batch, size=None)

Max-Pools node features according to the clustering defined in cluster.

Parameters

• cluster (LongTensor) – Cluster vector \(\mathbf{c} \in \{ 0, \ldots, N - 1 \}^N\), which assigns each node to a specific cluster.

• x (Tensor) – Node feature matrix \(\mathbf{X} \in \mathbb{R}^{(N_1 + \ldots + N_B) \times F}\).

• batch (LongTensor) – Batch vector \(\mathbf{b} \in {\{ 0, \ldots, B-1\}}^N\), which assigns each node to a specific example.

• size (int, optional) – The maximum number of clusters in a single example. This property is useful to obtain a batch-wise dense representation, e.g. for applying FC layers, but should only be used if the size of the maximum number of clusters per example is known in advance. (default: None)

Return type

(Tensor, LongTensor) if size is None, else Tensor

avg_pool_x(cluster, x, batch, size=None)

Average pools node features according to the clustering defined in cluster. See torch_geometric.nn.pool.max_pool_x() for more details.

Parameters

• cluster (LongTensor) – Cluster vector \(\mathbf{c} \in \{ 0, \ldots, N - 1 \}^N\), which assigns each node to a specific cluster.

• x (Tensor) – Node feature matrix \(\mathbf{X} \in \mathbb{R}^{(N_1 + \ldots + N_B) \times F}\).

• batch (LongTensor) – Batch vector \(\mathbf{b} \in {\{ 0, \ldots, B-1\}}^N\), which assigns each node to a specific example.

• size (int, optional) – The maximum number of clusters in a single example. (default: None)

Return type

(Tensor, LongTensor) if size is None, else Tensor

graclus(edge_index, weight=None, num_nodes=None)

A greedy clustering algorithm from the “Weighted Graph Cuts without Eigenvectors: A Multilevel Approach” paper of picking an unmarked vertex and matching it with one of its unmarked neighbors (that maximizes its edge weight). The GPU algoithm is adapted from the “A GPU Algorithm for Greedy Graph Matching” paper.

Parameters

• edge_index (LongTensor) – The edge indices.

• weight (Tensor, optional) – One-dimensional edge weights. (default: None)

• num_nodes (int, optional) – The number of nodes, i.e. max_val + 1 of edge_index. (default: None)

Return type

LongTensor

voxel_grid(pos, batch, size, start=None, end=None)

Voxel grid pooling from the, e.g., Dynamic Edge-Conditioned Filters in Convolutional Networks on Graphs paper, which overlays a regular grid of user-defined size over a point cloud and clusters all points within the same voxel.

Parameters

• pos (Tensor) – Node position matrix \(\mathbf{X} \in \mathbb{R}^{(N_1 + \ldots + N_B) \times D}\).

• batch (LongTensor) – Batch vector \(\mathbf{b} \in {\{ 0, \ldots, B-1\}}^N\), which assigns each node to a specific example.

• size (float or [float] or Tensor) – Size of a voxel (in each dimension).

• start (float or [float] or Tensor, optional) – Start coordinates of the grid (in each dimension). If set to None, will be set to the minimum coordinates found in pos. (default: None)

• end (float or [float] or Tensor, optional) – End coordinates of the grid (in each dimension). If set to None, will be set to the maximum coordinates found in pos. (default: None)

Return type

LongTensor

fps(x, batch=None, ratio=0.5, random_start=True)

“A sampling algorithm from the “PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space” paper, which iteratively samples the most distant point with regard to the rest points.

Parameters

• x (Tensor) – Node feature matrix \(\mathbf{X} \in \mathbb{R}^{N \times F}\).

• batch (LongTensor, optional) – Batch vector \(\mathbf{b} \in {\{ 0, \ldots, B-1\}}^N\), which assigns each node to a specific example. (default: None)

• ratio (float, optional) – Sampling ratio. (default: 0.5)

• random_start (bool, optional) – If set to False, use the first node in \(\mathbf{X}\) as starting node. (default: obj:True)

Return type

LongTensor

import torch
from torch_geometric.nn import fps

x = torch.Tensor([[-1, -1], [-1, 1], [1, -1], [1, 1]])
batch = torch.tensor([0, 0, 0, 0])
index = fps(x, batch, ratio=0.5)

knn(x, y, k, batch_x=None, batch_y=None, cosine=False)

Finds for each element in y the k nearest points in x.

Parameters

• x (Tensor) – Node feature matrix \(\mathbf{X} \in \mathbb{R}^{N \times F}\).

• y (Tensor) – Node feature matrix \(\mathbf{X} \in \mathbb{R}^{M \times F}\).

• k (int) – The number of neighbors.

• batch_x (LongTensor, optional) – Batch vector \(\mathbf{b} \in {\{ 0, \ldots, B-1\}}^N\), which assigns each node to a specific example. (default: None)

• batch_y (LongTensor, optional) – Batch vector \(\mathbf{b} \in {\{ 0, \ldots, B-1\}}^M\), which assigns each node to a specific example. (default: None)

• cosine (boolean, optional) – If True, will use the cosine distance instead of euclidean distance to find nearest neighbors. (default: False)

Return type

LongTensor

import torch
from torch_geometric.nn import knn

x = torch.Tensor([[-1, -1], [-1, 1], [1, -1], [1, 1]])
batch_x = torch.tensor([0, 0, 0, 0])
y = torch.Tensor([[-1, 0], [1, 0]])
batch_x = torch.tensor([0, 0])
assign_index = knn(x, y, 2, batch_x, batch_y)

knn_graph(x, k, batch=None, loop=False, flow='source_to_target', cosine=False)

Computes graph edges to the nearest k points.

Parameters

• x (Tensor) – Node feature matrix \(\mathbf{X} \in \mathbb{R}^{N \times F}\).

• k (int) – The number of neighbors.

• batch (LongTensor, optional) – Batch vector \(\mathbf{b} \in {\{ 0, \ldots, B-1\}}^N\), which assigns each node to a specific example. (default: None)

• loop (bool, optional) – If True, the graph will contain self-loops. (default: False)

• flow (string, optional) – The flow direction when using in combination with message passing ("source_to_target" or "target_to_source"). (default: "source_to_target")

• cosine (boolean, optional) – If True, will use the cosine distance instead of euclidean distance to find nearest neighbors. (default: False)

Return type

LongTensor

import torch
from torch_geometric.nn import knn_graph

x = torch.Tensor([[-1, -1], [-1, 1], [1, -1], [1, 1]])
batch = torch.tensor([0, 0, 0, 0])
edge_index = knn_graph(x, k=2, batch=batch, loop=False)

radius(x, y, r, batch_x=None, batch_y=None, max_num_neighbors=32)

Finds for each element in y all points in x within distance r.

Parameters

• x (Tensor) – Node feature matrix \(\mathbf{X} \in \mathbb{R}^{N \times F}\).

• y (Tensor) – Node feature matrix \(\mathbf{Y} \in \mathbb{R}^{M \times F}\).

• r (float) – The radius.

• batch_x (LongTensor, optional) – Batch vector \(\mathbf{b} \in {\{ 0, \ldots, B-1\}}^N\), which assigns each node to a specific example. (default: None)

• batch_y (LongTensor, optional) – Batch vector \(\mathbf{b} \in {\{ 0, \ldots, B-1\}}^M\), which assigns each node to a specific example. (default: None)

• max_num_neighbors (int, optional) – The maximum number of neighbors to return for each element in y. (default: 32)

Return type

LongTensor

import torch
from torch_geometric.nn import radius

x = torch.Tensor([[-1, -1], [-1, 1], [1, -1], [1, 1]])
batch_x = torch.tensor([0, 0, 0, 0])
y = torch.Tensor([[-1, 0], [1, 0]])
batch_y = torch.tensor([0, 0])
assign_index = radius(x, y, 1.5, batch_x, batch_y)

radius_graph(x, r, batch=None, loop=False, max_num_neighbors=32, flow='source_to_target')

Computes graph edges to all points within a given distance.

Parameters

• x (Tensor) – Node feature matrix \(\mathbf{X} \in \mathbb{R}^{N \times F}\).

• r (float) – The radius.

• batch (LongTensor, optional) – Batch vector \(\mathbf{b} \in {\{ 0, \ldots, B-1\}}^N\), which assigns each node to a specific example. (default: None)

• loop (bool, optional) – If True, the graph will contain self-loops. (default: False)

• max_num_neighbors (int, optional) – The maximum number of neighbors to return for each element in y. (default: 32)

• flow (string, optional) – The flow direction when using in combination with message passing ("source_to_target" or "target_to_source"). (default: "source_to_target")

Return type

LongTensor

import torch
from torch_geometric.nn import radius_graph

x = torch.Tensor([[-1, -1], [-1, 1], [1, -1], [1, 1]])
batch = torch.tensor([0, 0, 0, 0])
edge_index = radius_graph(x, r=1.5, batch=batch, loop=False)

nearest(x, y, batch_x=None, batch_y=None)

Clusters points in x together which are nearest to a given query point in y.

Parameters

• x (Tensor) – Node feature matrix \(\mathbf{X} \in \mathbb{R}^{N \times F}\).

• y (Tensor) – Node feature matrix \(\mathbf{Y} \in \mathbb{R}^{M \times F}\).

• batch_x (LongTensor, optional) – Batch vector \(\mathbf{b} \in {\{ 0, \ldots, B-1\}}^N\), which assigns each node to a specific example. (default: None)

• batch_y (LongTensor, optional) – Batch vector \(\mathbf{b} \in {\{ 0, \ldots, B-1\}}^M\), which assigns each node to a specific example. (default: None)

Return type

LongTensor

import torch
from torch_geometric.nn import nearest

x = torch.Tensor([[-1, -1], [-1, 1], [1, -1], [1, 1]])
batch_x = torch.tensor([0, 0, 0, 0])
y = torch.Tensor([[-1, 0], [1, 0]])
batch_y = torch.tensor([0, 0])
cluster = nearest(x, y, batch_x, batch_y)

Dense Pooling Layers

dense_diff_pool(x, adj, s, mask=None)

Differentiable pooling operator from the “Hierarchical Graph Representation Learning with Differentiable Pooling” paper

\[ \begin{align}\begin{aligned}\mathbf{X}^{\prime} &= {\mathrm{softmax}(\mathbf{S})}^{\top} \cdot \mathbf{X}\\\mathbf{A}^{\prime} &= {\mathrm{softmax}(\mathbf{S})}^{\top} \cdot \mathbf{A} \cdot \mathrm{softmax}(\mathbf{S})\end{aligned}\end{align} \]

based on dense learned assignments \(\mathbf{S} \in \mathbb{R}^{B \times N \times C}\). Returns pooled node feature matrix, coarsened adjacency matrix and two auxiliary objectives: (1) The link prediction loss

\[\mathcal{L}_{LP} = {\| \mathbf{A} - \mathrm{softmax}(\mathbf{S}) {\mathrm{softmax}(\mathbf{S})}^{\top} \|}_F,\]

and the entropy regularization

\[\mathcal{L}_E = \frac{1}{N} \sum_{n=1}^N H(\mathbf{S}_n).\]

Parameters

• x (Tensor) – Node feature tensor \(\mathbf{X} \in \mathbb{R}^{B \times N \times F}\) with batch-size \(B\), (maximum) number of nodes \(N\) for each graph, and feature dimension \(F\).

• adj (Tensor) – Adjacency tensor \(\mathbf{A} \in \mathbb{R}^{B \times N \times N}\).

• s (Tensor) – Assignment tensor \(\mathbf{S} \in \mathbb{R}^{B \times N \times C}\) with number of clusters \(C\). The softmax does not have to be applied beforehand, since it is executed within this method.

• mask (BoolTensor, optional) – Mask matrix \(\mathbf{M} \in {\{ 0, 1 \}}^{B \times N}\) indicating the valid nodes for each graph. (default: None)

Return type

(Tensor, Tensor, Tensor, Tensor)

dense_mincut_pool(x, adj, s, mask=None)

MinCUt pooling operator from the “Mincut Pooling in Graph Neural Networks” paper

\[ \begin{align}\begin{aligned}\mathbf{X}^{\prime} &= {\mathrm{softmax}(\mathbf{S})}^{\top} \cdot \mathbf{X}\\\mathbf{A}^{\prime} &= {\mathrm{softmax}(\mathbf{S})}^{\top} \cdot \mathbf{A} \cdot \mathrm{softmax}(\mathbf{S})\end{aligned}\end{align} \]

based on dense learned assignments \(\mathbf{S} \in \mathbb{R}^{B \times N \times C}\). Returns pooled node feature matrix, coarsened symmetrically normalized adjacency matrix and two auxiliary objectives: (1) The minCUT loss

\[\mathcal{L}_c = - \frac{\mathrm{Tr}(\mathbf{S}^{\top} \mathbf{A} \mathbf{S})} {\mathrm{Tr}(\mathbf{S}^{\top} \mathbf{D} \mathbf{S})}\]

where \(\mathbf{D}\) is the degree matrix, and (2) the orthogonality loss

\[\mathcal{L}_o = {\left\| \frac{\mathbf{S}^{\top} \mathbf{S}} {{\|\mathbf{S}^{\top} \mathbf{S}\|}_F} -\frac{\mathbf{I}_C}{\sqrt{C}} \right\|}_F.\]

Parameters

• x (Tensor) – Node feature tensor \(\mathbf{X} \in \mathbb{R}^{B \times N \times F}\) with batch-size \(B\), (maximum) number of nodes \(N\) for each graph, and feature dimension \(F\).

• adj (Tensor) – Symmetrically normalized adjacency tensor \(\mathbf{A} \in \mathbb{R}^{B \times N \times N}\).

• s (Tensor) – Assignment tensor \(\mathbf{S} \in \mathbb{R}^{B \times N \times C}\) with number of clusters \(C\). The softmax does not have to be applied beforehand, since it is executed within this method.

• mask (BoolTensor, optional) – Mask matrix \(\mathbf{M} \in {\{ 0, 1 \}}^{B \times N}\) indicating the valid nodes for each graph. (default: None)

Return type

(Tensor, Tensor, Tensor, Tensor)

Unpooling Layers

knn_interpolate(x, pos_x, pos_y, batch_x=None, batch_y=None, k=3)

The k-NN interpolation from the “PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space” paper. For each point \(y\) with position \(\mathbf{p}(y)\), its interpolated features \(\mathbf{f}(y)\) are given by

\[\mathbf{f}(y) = \frac{\sum_{i=1}^k w(x_i) \mathbf{f}(x_i)}{\sum_{i=1}^k w(x_i)} \textrm{, where } w(x_i) = \frac{1}{d(\mathbf{p}(y), \mathbf{p}(x_i))^2}\]

and \(\{ x_1, \ldots, x_k \}\) denoting the \(k\) nearest points to \(y\).

Parameters

• x (Tensor) – Node feature matrix \(\mathbf{X} \in \mathbb{R}^{N \times F}\).

• pos_x (Tensor) – Node position matrix \(\in \mathbb{R}^{N \times d}\).

• pos_y (Tensor) – Upsampled node position matrix \(\in \mathbb{R}^{M \times d}\).

• batch_x (LongTensor, optional) – Batch vector \(\mathbf{b_x} \in {\{ 0, \ldots, B-1\}}^N\), which assigns each node from \(\mathbf{X}\) to a specific example. (default: None)

• batch_y (LongTensor, optional) – Batch vector \(\mathbf{b_y} \in {\{ 0, \ldots, B-1\}}^N\), which assigns each node from \(\mathbf{Y}\) to a specific example. (default: None)

• k (int, optional) – Number of neighbors. (default: 3)

Models

class JumpingKnowledge(mode, channels=None, num_layers=None)

The Jumping Knowledge layer aggregation module from the “Representation Learning on Graphs with Jumping Knowledge Networks” paper based on either concatenation ("cat")

\[\mathbf{x}_v^{(1)} \, \Vert \, \ldots \, \Vert \, \mathbf{x}_v^{(T)}\]

max pooling ("max")

\[\max \left( \mathbf{x}_v^{(1)}, \ldots, \mathbf{x}_v^{(T)} \right)\]

or weighted summation

\[\sum_{t=1}^T \alpha_v^{(t)} \mathbf{x}_v^{(t)}\]

with attention scores \(\alpha_v^{(t)}\) obtained from a bi-directional LSTM ("lstm").

Parameters

• mode (string) – The aggregation scheme to use ("cat", "max" or "lstm").

• channels (int, optional) – The number of channels per representation. Needs to be only set for LSTM-style aggregation. (default: None)

• num_layers (int, optional) – The number of layers to aggregate. Needs to be only set for LSTM-style aggregation. (default: None)

forward(xs)

Aggregates representations across different layers.

Parameters

xs (list or tuple) – List containing layer-wise representations.

reset_parameters()

class Node2Vec(num_nodes, embedding_dim, walk_length, context_size, walks_per_node=1, p=1, q=1, num_negative_samples=None)

The Node2Vec model from the “node2vec: Scalable Feature Learning for Networks” paper where random walks of length walk_length are sampled in a given graph, and node embeddings are learned via negative sampling optimization.

Parameters

• num_nodes (int) – The number of nodes.

• embedding_dim (int) – The size of each embedding vector.

• walk_length (int) – The walk length.

• context_size (int) – The actual context size which is considered for positive samples. This parameter increases the effective sampling rate by reusing samples across different source nodes.

• walks_per_node (int, optional) – The number of walks to sample for each node. (default: 1)

• p (float, optional) – Likelihood of immediately revisiting a node in the walk. (default: 1)

• q (float, optional) – Control parameter to interpolate between breadth-first strategy and depth-first strategy (default: 1)

• num_negative_samples (int, optional) – The number of negative samples to use for each node. If set to None, this parameter gets set to context_size - 1. (default: None)

forward(subset)

Returns the embeddings for the nodes in subset.

loss(edge_index, subset=None)

Computes the loss for the nodes in subset with negative sampling.

reset_parameters()

test(train_z, train_y, test_z, test_y, solver='lbfgs', multi_class='auto', *args, **kwargs)

Evaluates latent space quality via a logistic regression downstream task.

class DeepGraphInfomax(hidden_channels, encoder, summary, corruption)

The Deep Graph Infomax model from the “Deep Graph Infomax” paper based on user-defined encoder and summary model \(\mathcal{E}\) and \(\mathcal{R}\) respectively, and a corruption function \(\mathcal{C}\).

Parameters

• hidden_channels (int) – The latent space dimensionality.

• encoder (Module) – The encoder module \(\mathcal{E}\).

• summary (callable) – The readout function \(\mathcal{R}\).

• corruption (callable) – The corruption function \(\mathcal{C}\).

discriminate(z, summary, sigmoid=True)

Given the patch-summary pair z and summary, computes the probability scores assigned to this patch-summary pair.

Parameters

• z (Tensor) – The latent space.

• sigmoid (bool, optional) – If set to False, does not apply the logistic sigmoid function to the output. (default: True)

forward(*args, **kwargs)

Returns the latent space for the input arguments, their corruptions and their summary representation.

loss(pos_z, neg_z, summary)

Computes the mutal information maximization objective.

reset_parameters()

test(train_z, train_y, test_z, test_y, solver='lbfgs', multi_class='auto', *args, **kwargs)

Evaluates latent space quality via a logistic regression downstream task.

class InnerProductDecoder

The inner product decoder from the “Variational Graph Auto-Encoders” paper

\[\sigma(\mathbf{Z}\mathbf{Z}^{\top})\]

where \(\mathbf{Z} \in \mathbb{R}^{N \times d}\) denotes the latent space produced by the encoder.

forward(z, edge_index, sigmoid=True)

Decodes the latent variables z into edge probabilities for the given node-pairs edge_index.

Parameters

• z (Tensor) – The latent space \(\mathbf{Z}\).

• sigmoid (bool, optional) – If set to False, does not apply the logistic sigmoid function to the output. (default: True)

forward_all(z, sigmoid=True)

Decodes the latent variables z into a probabilistic dense adjacency matrix.

Parameters

• z (Tensor) – The latent space \(\mathbf{Z}\).

• sigmoid (bool, optional) – If set to False, does not apply the logistic sigmoid function to the output. (default: True)

class GAE(encoder, decoder=None)

The Graph Auto-Encoder model from the “Variational Graph Auto-Encoders” paper based on user-defined encoder and decoder models.

Parameters

• encoder (Module) – The encoder module.

• decoder (Module, optional) – The decoder module. If set to None, will default to the torch_geometric.nn.models.InnerProductDecoder. (default: None)

decode(*args, **kwargs)

Runs the decoder and computes edge probabilities.

encode(*args, **kwargs)

Runs the encoder and computes node-wise latent variables.

recon_loss(z, pos_edge_index)

Given latent variables z, computes the binary cross entropy loss for positive edges pos_edge_index and negative sampled edges.

Parameters

• z (Tensor) – The latent space \(\mathbf{Z}\).

• pos_edge_index (LongTensor) – The positive edges to train against.

reset_parameters()

test(z, pos_edge_index, neg_edge_index)

Given latent variables z, positive edges pos_edge_index and negative edges neg_edge_index, computes area under the ROC curve (AUC) and average precision (AP) scores.

Parameters

• z (Tensor) – The latent space \(\mathbf{Z}\).

• pos_edge_index (LongTensor) – The positive edges to evaluate against.

• neg_edge_index (LongTensor) – The negative edges to evaluate against.

class VGAE(encoder, decoder=None)

The Variational Graph Auto-Encoder model from the “Variational Graph Auto-Encoders” paper.

Parameters

• encoder (Module) – The encoder module to compute \(\mu\) and \(\log\sigma^2\).

• decoder (Module, optional) – The decoder module. If set to None, will default to the torch_geometric.nn.models.InnerProductDecoder. (default: None)

encode(*args, **kwargs)

kl_loss(mu=None, logvar=None)

Computes the KL loss, either for the passed arguments mu and logvar, or based on latent variables from last encoding.

Parameters

• mu (Tensor, optional) – The latent space for \(\mu\). If set to None, uses the last computation of \(mu\). (default: None)

• logvar (Tensor, optional) – The latent space for \(\log\sigma^2\). If set to None, uses the last computation of \(\log\sigma^2\).(default: None)

reparametrize(mu, logvar)

class ARGA(encoder, discriminator, decoder=None)

The Adversarially Regularized Graph Auto-Encoder model from the “Adversarially Regularized Graph Autoencoder for Graph Embedding” paper. paper.

Parameters

• encoder (Module) – The encoder module.

• discriminator (Module) – The discriminator module.

• decoder (Module, optional) – The decoder module. If set to None, will default to the torch_geometric.nn.models.InnerProductDecoder. (default: None)

discriminator_loss(z)

Computes the loss of the discriminator.

Parameters

z (Tensor) – The latent space \(\mathbf{Z}\).

reg_loss(z)

Computes the regularization loss of the encoder.

Parameters

z (Tensor) – The latent space \(\mathbf{Z}\).

reset_parameters()

class ARGVA(encoder, discriminator, decoder=None)

The Adversarially Regularized Variational Graph Auto-Encoder model from the “Adversarially Regularized Graph Autoencoder for Graph Embedding” paper. paper.

Parameters

• encoder (Module) – The encoder module to compute \(\mu\) and \(\log\sigma^2\).

• discriminator (Module) – The discriminator module.

• decoder (Module, optional) – The decoder module. If set to None, will default to the torch_geometric.nn.models.InnerProductDecoder. (default: None)

encode(*args, **kwargs)

kl_loss(mu=None, logvar=None)

reparametrize(mu, logvar)

class SignedGCN(in_channels, hidden_channels, num_layers, lamb=5, bias=True)

The signed graph convolutional network model from the “Signed Graph Convolutional Network” paper. Internally, this module uses the torch_geometric.nn.conv.SignedConv operator.

Parameters

• in_channels (int) – Size of each input sample.

• hidden_channels (int) – Size of each hidden sample.

• num_layers (int) – Number of layers.

• lamb (float, optional) – Balances the contributions of the overall objective. (default: 5)

• bias (bool, optional) – If set to False, all layers will not learn an additive bias. (default: True)

create_spectral_features(pos_edge_index, neg_edge_index, num_nodes=None)

Creates in_channels spectral node features based on positive and negative edges.

Parameters

• pos_edge_index (LongTensor) – The positive edge indices.

• neg_edge_index (LongTensor) – The negative edge indices.

• num_nodes (int, optional) – The number of nodes, i.e. max_val + 1 of pos_edge_index and neg_edge_index. (default: None)

discriminate(z, edge_index)

Given node embeddings z, classifies the link relation between node pairs edge_index to be either positive, negative or non-existent.

Parameters

• x (Tensor) – The input node features.

• edge_index (LongTensor) – The edge indices.

forward(x, pos_edge_index, neg_edge_index)

Computes node embeddings z based on positive edges pos_edge_index and negative edges neg_edge_index.

Parameters

• x (Tensor) – The input node features.

• pos_edge_index (LongTensor) – The positive edge indices.

• neg_edge_index (LongTensor) – The negative edge indices.

loss(z, pos_edge_index, neg_edge_index)

Computes the overall objective.

Parameters

• z (Tensor) – The node embeddings.

• pos_edge_index (LongTensor) – The positive edge indices.

• neg_edge_index (LongTensor) – The negative edge indices.

neg_embedding_loss(z, neg_edge_index)

Computes the triplet loss between negative node pairs and sampled non-node pairs.

Parameters

• z (Tensor) – The node embeddings.

• neg_edge_index (LongTensor) – The negative edge indices.

nll_loss(z, pos_edge_index, neg_edge_index)

Computes the discriminator loss based on node embeddings z, and positive edges pos_edge_index and negative nedges neg_edge_index.

Parameters

• z (Tensor) – The node embeddings.

• pos_edge_index (LongTensor) – The positive edge indices.

• neg_edge_index (LongTensor) – The negative edge indices.

pos_embedding_loss(z, pos_edge_index)

Computes the triplet loss between positive node pairs and sampled non-node pairs.

Parameters

• z (Tensor) – The node embeddings.

• pos_edge_index (LongTensor) – The positive edge indices.

reset_parameters()

split_edges(edge_index, test_ratio=0.2)

Splits the edges edge_index into train and test edges.

Parameters

• edge_index (LongTensor) – The edge indices.

• test_ratio (float, optional) – The ratio of test edges. (default: 0.2)

test(z, pos_edge_index, neg_edge_index)

Evaluates node embeddings z on positive and negative test edges by computing AUC and F1 scores.

Parameters

• z (Tensor) – The node embeddings.

• pos_edge_index (LongTensor) – The positive edge indices.

• neg_edge_index (LongTensor) – The negative edge indices.

class RENet(num_nodes, num_rels, hidden_channels, seq_len, num_layers=1, dropout=0.0, bias=True)

The Recurrent Event Network model from the “Recurrent Event Network for Reasoning over Temporal Knowledge Graphs” paper

\[f_{\mathbf{\Theta}}(\mathbf{e}_s, \mathbf{e}_r, \mathbf{h}^{(t-1)}(s, r))\]

based on a RNN encoder

\[\mathbf{h}^{(t)}(s, r) = \textrm{RNN}(\mathbf{e}_s, \mathbf{e}_r, g(\mathcal{O}^{(t)}_r(s)), \mathbf{h}^{(t-1)}(s, r))\]

where \(\mathbf{e}_s\) and \(\mathbf{e}_r\) denote entity and relation embeddings, and \(\mathcal{O}^{(t)}_r(s)\) represents the set of objects interacted with subject \(s\) under relation \(r\) at timestamp \(t\). This model implements \(g\) as the Mean Aggregator and \(f_{\mathbf{\Theta}}\) as a linear projection.

Parameters

• num_nodes (int) – The number of nodes in the knowledge graph.

• num_rels (int) – The number of relations in the knowledge graph.

• hidden_channels (int) – Hidden size of node and relation embeddings.

• seq_len (int) – The sequence length of past events.

• num_layers (int, optional) – The number of recurrent layers. (default: 1)

• dropout (float) – If non-zero, introduces a dropout layer before the final prediction. (default: 0.)

• bias (bool, optional) – If set to False, all layers will not learn an additive bias. (default: True)

forward(data)

Given a data batch, computes the forward pass.

Parameters

data (torch_geometric.data.Data) – The input data, holding subject sub, relation rel and object obj information with shape [batch_size]. In addition, data needs to hold history information for subjects, given by a vector of node indices h_sub and their relative timestamps h_sub_t and batch assignments h_sub_batch. The same information must be given for objects (h_obj, h_obj_t, h_obj_batch).

static pre_transform(seq_len)

Precomputes history objects

\[\{ \mathcal{O}^{(t-k-1)}_r(s), \ldots, \mathcal{O}^{(t-1)}_r(s) \}\]

of a torch_geometric.datasets.icews.EventDataset with \(k\) denoting the sequence length seq_len.

reset_parameters()

test(logits, y)

Given ground-truth y, computes Mean Reciprocal Rank (MRR) and Hits at 1/3/10.

class GraphUNet(in_channels, hidden_channels, out_channels, depth, pool_ratios=0.5, sum_res=True, act=<function relu>)

The Graph U-Net model from the “Graph U-Nets” paper which implements a U-Net like architecture with graph pooling and unpooling operations.

Parameters

• in_channels (int) – Size of each input sample.

• hidden_channels (int) – Size of each hidden sample.

• out_channels (int) – Size of each output sample.

• depth (int) – The depth of the U-Net architecture.

• pool_ratios (float or [float], optional) – Graph pooling ratio for each depth. (default: 0.5)

• sum_res (bool, optional) – If set to False, will use concatenation for integration of skip connections instead summation. (default: True)

• act (torch.nn.functional, optional) – The nonlinearity to use. (default: torch.nn.functional.relu)

augment_adj(edge_index, edge_weight, num_nodes)

forward(x, edge_index, batch=None)

reset_parameters()

DataParallel Layers

class DataParallel(module, device_ids=None, output_device=None)

Implements data parallelism at the module level.

This container parallelizes the application of the given module by splitting a list of torch_geometric.data.Data objects and copying them as torch_geometric.data.Batch objects to each device. In the forward pass, the module is replicated on each device, and each replica handles a portion of the input. During the backwards pass, gradients from each replica are summed into the original module.

The batch size should be larger than the number of GPUs used.

The parallelized module must have its parameters and buffers on device_ids[0].

Note

You need to use the torch_geometric.data.DataListLoader for this module.

Parameters

• module (Module) – Module to be parallelized.

• device_ids (list of int or torch.device) – CUDA devices. (default: all devices)

• output_device (int or torch.device) – Device location of output. (default: device_ids[0])

forward(data_list)

scatter(data_list, device_ids)