Message Passing Neural Network (MPNN)

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A Message Passing Neural Network (MPNN) is a Spatial Graph Convolutional Network that treats graph convolutions as a message passing process in which information is passed from one node to another along edges.



References

2020a

$\mathbf{h}_{v}^{(k)}=U_{k}\left(\mathbf{h}_{v}^{(k-1)}, \sum_{u \in N(v)} M_{k}\left(\mathbf{h}_{v}^{(k-1)}, \mathbf{h}_{u}^{(k-1)}, \mathbf{x}_{v u}^{\tau}\right)\right)$

(21)
where $\mathbf{h}_{v}^{(0)} = \mathbf{x}_v, U_k(\cdot)$ and $M_k(\cdot)$ are functions with learnable parameters.

2020b

2019

$\mathbf{H}^{p+1}(u,:)=\displaystyle\sum_{v\in\mathcal{N}(u)}M^p\left(\mathbf{X}_p(u,:),\mathbf{X}^p(v,:),\mathbf{e}_{u,v}\right)$

(22)

$\mathbf{X}^{p+1}(u,:)=U^p\left(\mathbf{X}^p(u,:),\mathbf{H}^{p+1}(u,:)\right)$

(23)
where $M^p$,$U^p$ are the message function and the update function at the pth step, respectively, and $\mathbf{e}_{u,v}$ denotes the attributes of edge $(u, v)$. Then, the readout phase computes the feature vector for the whole graph by:

$\mathbf{\hat{y}}=R\left(\{\mathbf{X}^P(u,:)|u\in \mathcal{V}\}\right)$

(24)
where $R$ denotes the readout function.

2017