2517756c5a9be6ac007fe9bb7fb92611-Supplemental.pdf

GCN 1 6E-8 0 0 PPGN 1 0.97 0 6E-8 GCN2 1 1 1 6E-8 Table 3: Rate of pairs of graphs in set found dissimilar in expressivenessexperiment. Experiments are run on a NVidiaGeForceRTX2080GPU. Theorem 2. Let k be a local NGN kernel between node representationsρ and ρ1, consisting of for each node neighbourhoodGpq a map kpq: ρpGpq Ñ ρ1pGqq satisfying for any local edge Weneedtoshowthatforanyfeature v P ˆρpGq,that ˆρ1pφqpKGpvqq"KG1pˆρpφqpvqqP ˆρ1pG1q, which we do by showing the node features are equal at eachq1 PVpG1q. Todefine the message network, we first need to define node featuresρ,ρ1 and edge featuresτ,τ1, completely analog to how node features are defined. When Ψ is an equivariant graph network, we have thatσΨpv,Aq " Ψpσv,σAq for any permutationσ in the appropriate permutation representation. The local NGN kernel is then defined askpqpvpq " βpqpΨpαpqpvpq,Apqqq.