# Chapter 2: Message Passing¶

Let $$x_v\in\mathbb{R}^{d_1}$$ be the feature for node $$v$$, and $$w_{e}\in\mathbb{R}^{d_2}$$ be the feature for edge $$({u}, {v})$$. The message passing paradigm defines the following node-wise and edge-wise computation at step $$t+1$$:

$\text{Edge-wise: } m_{e}^{(t+1)} = \phi \left( x_v^{(t)}, x_u^{(t)}, w_{e}^{(t)} \right) , ({u}, {v},{e}) \in \mathcal{E}.$
$\text{Node-wise: } x_v^{(t+1)} = \psi \left(x_v^{(t)}, \rho\left(\left\lbrace m_{e}^{(t+1)} : ({u}, {v},{e}) \in \mathcal{E} \right\rbrace \right) \right).$

In the above equations, $$\phi$$ is a message function defined on each edge to generate a message by combining the edge feature with the features of its incident nodes; $$\psi$$ is an update function defined on each node to update the node feature by aggregating its incoming messages using the reduce function $$\rho$$.

## Built-in Functions and Message Passing APIs¶

In DGL, message function takes a single argument edges, which has three members src, dst and data, to access features of source node, destination node, and edge, respectively.

reduce function takes a single argument nodes. A node can access its mailbox to collect the messages its neighbors send to it through edges. Some of the most common reduce operations include sum, max, min, etc.

update function takes a single argument nodes. This function operates on the aggregation result from reduce function, typically combined with a node’s feature at the the last step, and save the output as a node feature.

DGL has implemented commonly used message functions and reduce functions as built-in in the namespace dgl.function. In general, we suggest using built-in functions whenever possible since they are heavily optimized and automatically handle dimension broadcasting.

If your message passing functions cannot be implemented with built-ins, you can implement user-defined message/reduce function (aka. UDF).

Built-in message functions can be unary or binary. We support copy for unary for now. For binary funcs, we now support add, sub, mul, div, dot. The naming convention for message built-in funcs is u represents src nodes, v represents dst nodes, e represents edges. The parameters for those functions are strings indicating the input and output field names for the corresponding nodes and edges. The list of supported built-in functions can be found in DGL Built-in Function. For example, to add the hu feature from src nodes and hv feature from dst nodes then save the result on the edge at he field, we can use built-in function dgl.function.u_add_v('hu', 'hv', 'he') this is equivalent to the Message UDF:

def message_func(edges):
return {'he': edges.src['hu'] + edges.dst['hv']}


Built-in reduce functions support operations sum, max, min, prod and mean. Reduce functions usually have two parameters, one for field name in mailbox, one for field name in destination, both are strings. For example, dgl.function.sum('m', 'h') is equivalent to the Reduce UDF that sums up the message m:

import torch
def reduce_func(nodes):
return {'h': torch.sum(nodes.mailbox['m'], dim=1)}


In DGL, the interface to call edge-wise computation is apply_edges(). The parameters for apply_edges are a message function and valid edge type as described in the API Doc (by default, all edges will be updated). For example:

import dgl.function as fn


the interface to call node-wise computation is update_all(). The parameters for update_all are a message function, a reduce function and a update function. update function can be called outside of update_all by leaving the third parameter as empty. This is suggested since the update function can usually be written as pure tensor operations to make the code concise. For example：

def updata_all_example(graph):
# store the result in graph.ndata['ft']
graph.update_all(fn.u_mul_e('ft', 'a', 'm'),
fn.sum('m', 'ft'))
# Call update function outside of update_all
final_ft = graph.ndata['ft'] * 2
return final_ft


This call will generate the message m by multiply src node feature ft and edge feature a, sum up the message m to update node feature ft, finally multiply ft by 2 to get the result final_ft. After the call, the intermediate message m will be cleaned. The math formula for the above function is:

${final\_ft}_i = 2 * \sum_{j\in\mathcal{N}(i)} ({ft}_j * a_{ij})$

update_all is a high-level API that merges message generation, message reduction and node update in a single call, which leaves room for optimizations, as explained below.

## Writing Efficient Message Passing Codes¶

DGL optimized memory consumption and computing speed for message passing. The optimization includes:

• Merge multiple kernels in a single one: This is achieved by using update_all to call multiple built-in functions at once. (Speed optimization)

• Parallelism on nodes and edges: DGL abstracts edge-wise computation apply_edges as a generalized sampled dense-dense matrix multiplication (gSDDMM) operation and parallelize the computing across edges. Likewise, DGL abstracts node-wise computation update_all as a generalized sparse-dense matrix multiplication (gSPMM) operation and parallelize the computing across nodes. (Speed optimization)

• Avoid unnecessary memory copy into edges: To generate a message that requires the feature from source and destination node, one option is to copy the source and destination node feature into that edge. For some graphs, the number of edges is much larger than the number of nodes. This copy can be costly. DGL built-in message functions avoid this memory copy by sampling out the node feature using entry index. (Memory and speed optimization)

• Avoid materializing feature vectors on edges: the complete message passing process includes message generation, message reduction and node update. In update_all call, message function and reduce function are merged into one kernel if those functions are built-in. There is no message materialization on edges. (Memory optimization)

According to the above, a common practise to leverage those optimizations is to construct your own message passing functionality as a combination of update_all calls with built-in functions as parameters.

For some cases like GATConv where we have to save message on the edges, we need to call apply_edges with built-in functions. Sometimes the message on the edges can be high dimensional, which is memory consuming. We suggest keeping the edata dimension as low as possible.

Here’s an example on how to achieve this by spliting operations on the edges to nodes. The option does the following: concatenate the src feature and dst feature, then apply a linear layer, i.e. $$W\times (u || v)$$. The src and dst feature dimension is high, while the linear layer output dimension is low. A straight forward implementation would be like:

linear = nn.Parameter(th.FloatTensor(size=(1, node_feat_dim*2)))
def concat_message_function(edges):
{'cat_feat': torch.cat([edges.src.ndata['feat'], edges.dst.ndata['feat']])}
g.apply_edges(concat_message_function)
g.edata['out'] = g.edata['cat_feat'] * linear


The suggested implementation will split the linear operation into two, one applies on src feature, the other applies on dst feature. Add the output of the linear operations on the edges at the final stage, i.e. perform $$W_l\times u + W_r \times v$$, since $$W \times (u||v) = W_l \times u + W_r \times v$$, where $$W_l$$ and $$W_r$$ are the left and the right half of the matrix $$W$$, respectively:

linear_src = nn.Parameter(th.FloatTensor(size=(1, node_feat_dim)))
linear_dst = nn.Parameter(th.FloatTensor(size=(1, node_feat_dim)))
out_src = g.ndata['feat'] * linear_src
out_dst = g.ndata['feat'] * linear_dst
g.srcdata.update({'out_src': out_src})
g.dstdata.update({'out_dst': out_dst})


The above two implementations are mathematically equivalent. The later one is much efficient because we do not need to save feat_src and feat_dst on edges, which is not memory-efficient. Plus, addition could be optimized with DGL’s built-in function u_add_v, which further speeds up computation and saves memory footprint.

## Apply Message Passing On Part Of The Graph¶

If we only want to update part of the nodes in the graph, the practice is to create a subgraph by providing the ids for the nodes we want to include in the update, then call update_all on the subgraph. For example:

nid = [0, 2, 3, 6, 7, 9]
sg = g.subgraph(nid)
sg.update_all(message_func, reduce_func, apply_node_func)


This is a common usage in mini-batch training. Check Chapter 6: Stochastic Training on Large Graphs user guide for more detailed usages.

## Apply Edge Weight In Message Passing¶

A commonly seen practice in GNN modeling is to apply edge weight on the message before message aggregation, for examples, in GAT and some GCN variants. In DGL, the way to handle this is:

• Save the weight as edge feature.

• Multiply the edge feature with src node feature in message function.

For example:

graph.edata['a'] = affinity
graph.update_all(fn.u_mul_e('ft', 'a', 'm'),
fn.sum('m', 'ft'))


In the above, we use affinity as the edge weight. The edge weight should usually be a scalar.

## Message Passing on Heterogeneuous Graph¶

Heterogeneous (user guide for 1.5 Heterogeneous Graphs), or heterographs for short, are graphs that contain different types of nodes and edges. The different types of nodes and edges tend to have different types of attributes that are designed to capture the characteristics of each node and edge type. Within the context of graph neural networks, depending on their complexity, certain node and edge types might need to be modeled with representations that have a different number of dimensions.

The message passing on heterographs can be split into two parts:

1. Message computation and aggregation within each relation r.

2. Reduction that merges the results on the same node type from multiple relationships.

DGL’s interface to call message passing on heterographs is multi_update_all(). multi_update_all takes a dictionary containing the parameters for update_all within each relation using relation as the key, and a string represents the cross type reducer. The reducer can be one of sum, min, max, mean, stack. Here’s an example:

for c_etype in G.canonical_etypes:
srctype, etype, dsttype = c_etype
Wh = self.weight[etype](feat_dict[srctype])
# Save it in graph for message passing
G.nodes[srctype].data['Wh_%s' % etype] = Wh
# Specify per-relation message passing functions: (message_func, reduce_func).
# Note that the results are saved to the same destination feature 'h', which
# hints the type wise reducer for aggregation.
funcs[etype] = (fn.copy_u('Wh_%s' % etype, 'm'), fn.mean('m', 'h'))
# Trigger message passing of multiple types.
G.multi_update_all(funcs, 'sum')
# return the updated node feature dictionary
return {ntype : G.nodes[ntype].data['h'] for ntype in G.ntypes}