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Table 2 Aggregation function special cases

From: Building attention and edge message passing neural networks for bioactivity and physical–chemical property prediction

ModelHidden statesAggregation form
MPNN\(h_{v}^{\left( t \right)}\)\(M_{t} \left( {h_{v}^{\left( t \right)} ,h_{w}^{\left( t \right)} ,e_{vw} } \right) = f_{NN}^{{\left( {e_{vw} } \right)}} \left( {h_{w}^{\left( t \right)} } \right)\)
AMPNN\(h_{v}^{\left( t \right)}\)\(A_{t} \left( {h_{v}^{\left( t \right)} ,\left\{ {\left( {h_{w}^{\left( t \right)} ,e_{vw} } \right)} \right\}} \right) = \mathop \sum \limits_{w \in N\left( v \right)} f_{NN}^{{\left( {e_{vw} } \right)}} \left( {h_{w}^{\left( t \right)} } \right) \odot \frac{{{ \exp }\left( {g_{NN}^{{\left( {e_{vw} } \right)}} \left( {h_{w}^{\left( t \right)} } \right)} \right)}}{{\mathop \sum \nolimits_{w' \in N\left( v \right)} { \exp }\left( {g_{NN}^{{\left( {e_{vw'} } \right)}} \left( {h_{w'}^{\left( t \right)} } \right)} \right)}}\)
EMNN\(h_{vw}^{\left( t \right)}\)\(A_{t} \left( {e{^{\prime}}_{vw} ,S_{vw}^{\left( t \right)} } \right) = \mathop \sum \limits_{{x \in S{^{\prime}}_{vw}^{\left( t \right)} }} f_{NN} \left( x \right) \odot \frac{{{ \exp }\left( {g_{NN} \left( x \right)} \right)}}{{\mathop \sum \nolimits_{{x{^{\prime}} \in S{^{\prime}}_{vw}^{\left( t \right)} }} { \exp }\left( {g_{NN} \left( {x{^{\prime}}} \right)} \right)}}\)
\(S{^{\prime}}_{vw}^{\left( t \right)} = S_{vw}^{\left( t \right)} \rm{\bigcup }\left\{ {e_{vw}{^{\prime}} } \right\}\)