# Creating Message Passing Networks¶

Generalizing the convolution operator to irregular domains is typically expressed as a neighborhood aggregation or message passing scheme. With $$\mathbf{x}^{(k-1)}_i \in \mathbb{R}^F$$ denoting node features of node $$i$$ in layer $$(k-1)$$ and $$\mathbf{e}_{i,j} \in \mathbb{R}^D$$ denoting (optional) edge features from node $$i$$ to node $$j$$, message passing graph neural networks can be described as

$\mathbf{x}_i^{(k)} = \gamma^{(k)} \left( \mathbf{x}_i^{(k-1)}, \square_{j \in \mathcal{N}(i)} \, \phi^{(k)}\left(\mathbf{x}_i^{(k-1)}, \mathbf{x}_j^{(k-1)},\mathbf{e}_{i,j}\right) \right),$

where $$\square$$ denotes a differentiable, permutation invariant function, e.g., sum, mean or max, and $$\gamma$$ and $$\phi$$ denote differentiable functions such as MLPs (Multi Layer Perceptrons).

## The “MessagePassing” Base Class¶

PyTorch Geometric provides the torch_geometric.nn.MessagePassing base class, which helps in creating such kinds of message passing graph neural networks by automatically taking care of message propagation. The user only has to define the functions $$\phi$$ , i.e. message(), and $$\gamma$$ , .i.e. update(), as well as the aggregation scheme to use, .i.e. aggr='add', aggr='mean' or aggr='max'.

This is done with the help of the following methods:

• torch_geometric.nn.MessagePassing(aggr="add", flow="source_to_target"): Defines the aggregation scheme to use ("add", "mean" or "max") and the flow direction of message passing (either "source_to_target" or "target_to_source").
• torch_geometric.nn.MessagePassing.propagate(edge_index, size=None, dim=0, **kwargs): The initial call to start propagating messages. Takes in the edge indices and all additional data which is needed to construct messages and to update node embeddings. Note that propagate() is not limited to exchange messages in symmetric adjacency matrices of shape [N, N] only, but can also exchange messages in general sparse assignment matrices, .e.g., bipartite graphs, of shape [N, M] by passing size=(N, M) as an additional argument. If set to None, the assignment matrix is assumed to be symmetric. For bipartite graphs with two independent sets of nodes and indices, and each set holding its own information, this split can be marked by passing the information as a tuple, e.g. x=(x_N, x_M). Furthermore, the dim attribute indicates along which axis to propagate.
• torch_geometric.nn.MessagePassing.message(): Constructs messages to node $$i$$ in analogy to $$\phi$$ for each edge in $$(j,i) \in \mathcal{E}$$ if flow="source_to_target" and $$(i,j) \in \mathcal{E}$$ if flow="target_to_source". Can take any argument which was initially passed to propagate(). In addition, 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.
• torch_geometric.nn.MessagePassing.update(): Updates node embeddings in analogy to $$\gamma$$ 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().

Let us verify this by re-implementing two popular GNN variants, the GCN layer from Kipf and Welling and the EdgeConv layer from Wang et al..

## Implementing the GCN Layer¶

The GCN layer is mathematically defined as

$\mathbf{x}_i^{(k)} = \sum_{j \in \mathcal{N}(i) \cup \{ i \}} \frac{1}{\sqrt{\deg(i)} \cdot \sqrt{deg(j)}} \cdot \left( \mathbf{\Theta} \cdot \mathbf{x}_j^{(k-1)} \right),$

where neighboring node features are first transformed by a weight matrix $$\mathbf{\Theta}$$, normalized by their degree, and finally summed up. This formula can be divided into the following steps:

1. Add self-loops to the adjacency matrix.
2. Linearly transform node feature matrix.
3. Normalize node features in $$\phi$$.
4. Sum up neighboring node features ("add" aggregation).
5. Return new node embeddings in $$\gamma$$.

Steps 1-2 are typically computed before message passing takes place. Steps 3-5 can be easily processed using the torch_geometric.nn.MessagePassing base class. The full layer implementation is shown below:

import torch
from torch_geometric.nn import MessagePassing
from torch_geometric.utils import add_self_loops, degree

class GCNConv(MessagePassing):
def __init__(self, in_channels, out_channels):
self.lin = torch.nn.Linear(in_channels, out_channels)

def forward(self, x, edge_index):
# x has shape [N, in_channels]
# edge_index has shape [2, E]

# Step 1: Add self-loops to the adjacency matrix.
edge_index, _ = add_self_loops(edge_index, num_nodes=x.size(0))

# Step 2: Linearly transform node feature matrix.
x = self.lin(x)

# Step 3-5: Start propagating messages.
return self.propagate(edge_index, size=(x.size(0), x.size(0)), x=x)

def message(self, x_j, edge_index, size):
# x_j has shape [E, out_channels]

# Step 3: Normalize node features.
row, col = edge_index
deg = degree(row, size[0], dtype=x_j.dtype)
deg_inv_sqrt = deg.pow(-0.5)
norm = deg_inv_sqrt[row] * deg_inv_sqrt[col]

return norm.view(-1, 1) * x_j

def update(self, aggr_out):
# aggr_out has shape [N, out_channels]

# Step 5: Return new node embeddings.
return aggr_out


GCNConv inherits from torch_geometric.nn.MessagePassing with "add" propagation. All the logic of the layer takes place in forward(). Here, we first add self-loops to our edge indices using the torch_geometric.utils.add_self_loops() function (step 1), as well as linearly transform node features by calling the torch.nn.Linear instance (step 2).

We then proceed to call propagate(), which internally calls the message() and update() functions. As additional arguments for message propagation, we pass the node embeddings x.

In the message() function, we need to normalize the neighboring node features x_j. Here, x_j denotes a mapped tensor, which contains the neighboring node features of each edge. Node features can be automatically mapped by appending _i or _j to the variable name. In fact, any tensor can be mapped this way, as long as they have $$N$$ entries in its first dimension.

The neighboring node features are normalized by computing node degrees $$\deg(i)$$ for each node $$i$$ and saving $$1/(\sqrt{\deg(i)} \cdot \sqrt{\deg(j)})$$ in norm for each edge $$(i,j) \in \mathcal{E}$$.

In the update() function, we simply return the output of the aggregation.

That is all that it takes to create a simple message passing layer. You can use this layer as a building block for deep architectures. Initializing and calling it is straightforward:

conv = GCNConv(16, 32)
x = conv(x, edge_index)


## Implementing the Edge Convolution¶

The edge convolutional layer processes graphs or point clouds and is mathematically defined as

$\mathbf{x}_i^{(k)} = \max_{j \in \mathcal{N}(i)} h_{\mathbf{\Theta}} \left( \mathbf{x}_i^{(k-1)}, \mathbf{x}_j^{(k-1)} - \mathbf{x}_i^{(k-1)} \right),$

where $$h_{\mathbf{\Theta}}$$ denotes a MLP. Analogous to the GCN layer, we can use the torch_geometric.nn.MessagePassing class to implement this layer, this time using the "max" aggregation:

import torch
from torch.nn import Sequential as Seq, Linear, ReLU
from torch_geometric.nn import MessagePassing

class EdgeConv(MessagePassing):
def __init__(self, in_channels, out_channels):
super(EdgeConv, self).__init__(aggr='max') #  "Max" aggregation.
self.mlp = Seq(Linear(2 * in_channels, out_channels),
ReLU(),
Linear(out_channels, out_channels))

def forward(self, x, edge_index):
# x has shape [N, in_channels]
# edge_index has shape [2, E]

return self.propagate(edge_index, size=(x.size(0), x.size(0)), x=x)

def message(self, x_i, x_j):
# x_i has shape [E, in_channels]
# x_j has shape [E, in_channels]

tmp = torch.cat([x_i, x_j - x_i], dim=1)  # tmp has shape [E, 2 * in_channels]
return self.mlp(tmp)

def update(self, aggr_out):
# aggr_out has shape [N, out_channels]

return aggr_out


Inside the message() function, we use self.mlp to transform both the target node features x_i and the relative source node features x_j - x_i for each edge $$(j,i) \in \mathcal{E}$$.

The edge convolution is actual a dynamic convolution, which recomputes the graph for each layer using nearest neighbors in the feature space. Luckily, PyTorch Geometric comes with a GPU accelerated batch-wise k-NN graph generation method named torch_geometric.nn.knn_graph():

from torch_geometric.nn import knn_graph

class DynamicEdgeConv(EdgeConv):
def __init__(self, in_channels, out_channels, k=6):
super(DynamicEdgeConv, self).__init__(in_channels, out_channels)
self.k = k

def forward(self, x, batch=None):
edge_index = knn_graph(x, self.k, batch, loop=False, flow=self.flow)
return super(DynamicEdgeConv, self).forward(x, edge_index)


Here, knn_graph() computes a nearest neighbor graph, which is further used to call the forward() method of EdgeConv.

This leaves us with a clean interface for initializing and calling this layer:

conv = DynamicEdgeConv(3, 128, k=6)
x = conv(pos, batch)