Table 2.

Aggregation function special cases

Model	Hidden states	Aggregation form
MPNN	$h_v^{\left(t\right)}$	$M_t\left(h_v^{\left(t\right)},h_w^{\left(t\right)},e_{\mathit{vw}}\right)=f_{\mathit{NN}}^{\left(e_{\mathit{vw}}\right)}\left(h_w^{\left(t\right)}\right)$
AMPNN	$h_v^{\left(t\right)}$	$A_t\left(h_v^{\left(t\right)},\left\{\left(h_w^{\left(t\right)},e_{\mathit{vw}}\right)\right\}\right)=\sum _{w\in N\left(v\right)}{f}_{\mathit{NN}}^{\left(e_{\mathit{vw}}\right)}\left(h_w^{\left(t\right)}\right)\odot \frac{exp\left(g_{\mathit{NN}}^{\left(e_{\mathit{vw}}\right)}\left(h_w^{\left(t\right)}\right)\right)}{{\sum }_{w^{\prime }\in N\left(v\right)}exp\left(g_{\mathit{NN}}^{\left(e_{v{w}^{\prime }}\right)}\left(h_{w^{\prime }}^{\left(t\right)}\right)\right)}$
EMNN	$h_{\mathit{vw}}^{\left(t\right)}$	$A_t\left(e{{}^{\prime }}_{\mathit{vw}},S_{\mathit{vw}}^{\left(t\right)}\right)=\sum _{x\in S{{}^{\prime }}_{\mathit{vw}}^{\left(t\right)}}{f}_{\mathit{NN}}\left(x\right)\odot \frac{exp\left(g_{\mathit{NN}}\left(x\right)\right)}{{\sum }_{x{}^{\prime }\in S{{}^{\prime }}_{\mathit{vw}}^{\left(t\right)}}exp\left(g_{\mathit{NN}}\left(x{}^{\prime }\right)\right)}$ $S{{}^{\prime }}_{\mathit{vw}}^{\left(t\right)}=S_{\mathit{vw}}^{\left(t\right)}\bigcup \left\{{\mathrm{e}}_{\mathrm{vw}}{}^{\prime }\right\}$