chore: import upstream snapshot with attribution
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.. _guide-message-passing:
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Chapter 2: Message Passing
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==========================
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:ref:`(中文版) <guide_cn-message-passing>`
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Message Passing Paradigm
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------------------------
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Let :math:`x_v\in\mathbb{R}^{d_1}` be the feature for node :math:`v`,
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and :math:`w_{e}\in\mathbb{R}^{d_2}` be the feature for edge
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:math:`({u}, {v})`. The **message passing paradigm** defines the
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following node-wise and edge-wise computation at step :math:`t+1`:
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.. math:: \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}.
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.. math:: \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).
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In the above equations, :math:`\phi` is a **message function**
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defined on each edge to generate a message by combining the edge feature
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with the features of its incident nodes; :math:`\psi` is an
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**update function** defined on each node to update the node feature
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by aggregating its incoming messages using the **reduce function**
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:math:`\rho`.
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Roadmap
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-------
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This chapter introduces DGL's message passing APIs, and how to efficiently use them on both nodes and edges.
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The last section of it explains how to implement message passing on heterogeneous graphs.
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* :ref:`guide-message-passing-api`
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* :ref:`guide-message-passing-efficient`
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* :ref:`guide-message-passing-part`
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* :ref:`guide-message-passing-heterograph`
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.. toctree::
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:maxdepth: 1
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:hidden:
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:glob:
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message-api
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message-efficient
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message-part
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message-heterograph
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