A graph isomorphism network (GIN) layer is defined as

We can write this layer in terms of concatenate function $\phi$:

And aggregate function $\varphi$

The motivation for this architecture comes from the fact that injective GNNs are as powerful as the WL test, which holds in this case as long as $\varphi ,\phi$ as defined above are injective.

${\varphi }_{\ell }\circ \sum {\phi }_{\ell -1}$ is a perceptron or MLP over the neighborhood multiset

• Hornik 1989 says an MLP can model any injective function - this is due to the universal approximation theorem
• Injective ${\varphi }_{\ell }\circ \sum _X{\varphi }_{\ell -1}$ exist for multiset $X$ (proven in paper where GINs are introduced)