3. After running $K$ iterations of the GNN message passing, we can use the output of the final layer to define the embeddings for each node, i.e.,

    $z_u = h^{(K)}_u, \forall u \in V$  

Carleton College

Neural Message Passing can be broken down into three steps

Steps of Neural Message Passing

Hamilton, W.L. Graph Representation Learning. *Synthesis Lectures on Artificial Intelligence and Machine Learning*. 2020, 14(3), 1–159. https://doi.org/10.2200/S01045ED1V01Y202009AIM046

Graph Representation Learning by William Hamilton

1. At each iteration $k$ of the GNN, the $AGGREGATE$ function takes as input the set of embeddings of the nodes in $u$’s graph neighborhood $N(u)$ and generates a message $m^{(k)}_{N(u)}$ based on this aggregated neighborhood information

Step 1 of Neural Message Passing

2. The update function $UPDATE$ then combines the message $m^{(k)}_{N(u)}$ with the previous embedding hu of node u to generate the updated embedding $h^{(k-1)}_u$ of node $u$ to generate the updated embedding $h^{(k)}_u$. The initial embeddings at $k$ = 0 are set to the input features for all the nodes, i.e., $h^{(0)}_u$ = **x**$_u$, $\forall u \in V$.

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