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[Doc] doc page for builtin functions (#642)  
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.. currentmodule:: dgl

Builtin message passing functions
=================================

In DGL, message passing is expressed by two APIs:

- ``send(edges, message_func)`` for computing the messages along the given edges.
- ``recv(nodes, reduce_func)`` for collecting the in-coming messages, perform aggregation and so on.

Although the two-stage abstraction can cover all the models that are defined in the message
passing paradigm, it is inefficient due to storing explicit messages. See our
`blogpost <https://www.dgl.ai/blog/2019/05/04/kernel.html>`_ for more
details and performance results.

Our solution, also explained in the blogpost, is to fuse the two stages into one kernel so no
explicit messages are generated and stored. To achieve this, we recommend using our builtin
message/reduce functions so that DGL can analyze and map them to fused dedicated kernels. Here
are some examples (in pytorch syntax):

.. code:: python
   
   import dgl
   import dgl.function as fn
   import torch as th
   g = ... # create a DGLGraph
   g.ndata['h'] = th.randn((g.number_of_nodes(), 10)) # each node has feature size 10
   g.edata['w'] = th.randn((g.number_of_edges(), 1))  # each edge has feature size 1
   # collect features from source nodes and aggregate them in destination nodes
   g.update_all(fn.copy_u('h', 'm'), fn.sum('m', 'h_sum'))
   # multiply source node features with edge weights and aggregate them in destination nodes
   g.update_all(fn.u_mul_e('h', 'w', 'm'), fn.max('m', 'h_max'))
   # compute edge embedding by multiplying source and destination node embeddings
   g.apply_edges(fn.u_mul_v('h', 'h', 'w_new'))

``fn.copy_u``, ``fn.u_mul_e``, ``fn.u_mul_v`` are builtin message functions, while ``fn.sum``
and ``fn.max`` are builtin reduce functions. We use ``u``, ``v`` and ``e`` to represent
source nodes, destination nodes and edges among them, respectively. Hence, ``copy_u`` copies the source
node data as the messages, ``u_mul_e`` multiplies source node features with edge features,
so on and so forth.

To define a unary message function (e.g. ``copy_u``) requires one input feature name and one output
message name. To define a binary message function (e.g. ``u_mul_e``) requires
two input feature names and one output message name. During the computation,
the message function will read the data under the given names, perform computation, and return
the output using the output name. For example, the above ``fn.u_mul_e('h', 'w', 'm')`` is
the same as the following user-defined function:

.. code:: python

   def udf_u_mul_e(edges):
      return {'m' : edges.src['h'] * edges.data['w']}

To define a reduce function, one input message name and one output node feature name
need to be specified. For example, the above ``fn.max('m', 'h_max')`` is the same as the
following user-defined function:

.. code:: python

   def udf_max(nodes):
      return {'h_max' : th.max(nodes.mailbox['m'], 1)[0]}

Broadcasting is supported for binary message function, which means the tensor arguments
can be automatically expanded to be of equal sizes. The supported broadcasting semantic
is standard as in `numpy's <https://docs.scipy.org/doc/numpy/user/basics.broadcasting.html>`_
and `pytorch's <https://pytorch.org/docs/stable/notes/broadcasting.html>`_. For unfamiliar
users, we highly suggest reading those documents as broadcasting is very useful. In the
above example, ``fn.u_mul_e`` will perform broadcasted multiplication automatically because
the node feature ``'h'`` and the edge feature ``'w'`` are of different, but broadcastable shapes.

All DGL's builtin functions support both CPU and GPU and backward computation so they
can be used in any autograd system. Also, builtin functions can be used not only in ``update_all``
or ``apply_edges`` as shown in the example, but wherever message/reduce functions are
required (e.g. ``pull``, ``push``, ``send_and_recv``, etc.).

Here is a cheatsheet of all the DGL builtins.

+-------------------------+----------------------------------------------------+-----------------------+
| Category                | Functions                                          | Memo                  |
+=========================+====================================================+=======================+
| Unary message function  | ``copy_u``                                         |                       |
|                         +----------------------------------------------------+-----------------------+
|                         | ``copy_e``                                         |                       |
|                         +----------------------------------------------------+-----------------------+
|                         | ``copy_src``                                       |  alias of ``copy_u``  |
|                         +----------------------------------------------------+-----------------------+
|                         | ``copy_edge``                                      |  alias of ``copy_e``  |
+-------------------------+----------------------------------------------------+-----------------------+
| Binary message function | ``u_add_v``, ``u_sub_v``, ``u_mul_v``, ``u_div_v`` |                       |
|                         +----------------------------------------------------+-----------------------+
|                         | ``u_add_e``, ``u_sub_e``, ``u_mul_e``, ``u_div_e`` |                       |
|                         +----------------------------------------------------+-----------------------+
|                         | ``v_add_u``, ``v_sub_u``, ``v_mul_u``, ``v_div_u`` |                       |
|                         +----------------------------------------------------+-----------------------+
|                         | ``v_add_e``, ``v_sub_e``, ``v_mul_e``, ``v_div_e`` |                       |
|                         +----------------------------------------------------+-----------------------+
|                         | ``e_add_u``, ``e_sub_u``, ``e_mul_u``, ``e_div_u`` |                       |
|                         +----------------------------------------------------+-----------------------+
|                         | ``e_add_v``, ``e_sub_v``, ``e_mul_v``, ``e_div_v`` |                       |
|                         +----------------------------------------------------+-----------------------+
|                         | ``src_mul_edge``                                   |  alias of ``u_mul_e`` |
+-------------------------+----------------------------------------------------+-----------------------+
| Reduce function         | ``max``                                            |                       |
|                         +----------------------------------------------------+-----------------------+
|                         | ``min``                                            |                       |
|                         +----------------------------------------------------+-----------------------+
|                         | ``sum``                                            |                       |
|                         +----------------------------------------------------+-----------------------+
|                         | ``prod``                                           |                       |
+-------------------------+----------------------------------------------------+-----------------------+

Next Step
---------
* Checkout our :mod:`dgl.nn` module for how builtin functions are used to implement Graph Neural
  Network layers.