An simplification of message passing omit the explicit update, but simply add a self-loop: $\mathbf{h}_u^{(k)}=\text{AGGREGATE}(\{\mathbf{h}_v^{(k-1)},\forall v\in \mathcal{N}(u)\cup\{u\}\})$ Where $\mathbf{h}_u^{(k)}$ is the $k^{th}$ hidden layer of node $u$.

In the case of basic GNN, it can be wriiten as the following graph level update: $\mathbf{H}^{(t)}=\sigma((\mathbf{A}+\mathbf{I})\mathbf{H}^{(t-1)}\mathbf{W}^{(t)})$ Where $\mathbf{H}^{(t)}$ is the graph level hidden layer representation.