# Chapter 2: Message Passing¶

(中文版)

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 $$\rho$$.