# Why is the Graph Isomorphism Network powerful?

I am reading a paper known as GIN, How powerful are graph neural networks?, Xu et al. 2019

The paper, Lemma 5 and Corollary 6, introduces Graph Isomorphism Network (GIN).

In Lemma 5,

Moreover, any multiset function $$g$$ can be decomposed as $$g (X) = \phi(\sum_{x\in X} f(x))$$ for some function $$\phi$$

Similarly, in Corollary 6,

Moreover, any function $$g$$ over such pairs can be decomposed as $$g (c, X) = \varphi((1+\epsilon) f(c)+ \sum_{x\in X} f(x))$$ for some function $$\varphi$$.

Finally, it makes MLP by compositing $$f^{(k+1)}$$ and $$\varphi^{(k)}$$. i.e

$$f^{(k+1)} \varphi^{(k)}$$

I know that $$h(X)$$ or $$h(c,X)$$ is injective, because they are unique to $$c$$ and $$X$$(in Lemma 5 and Corollary 6, respectively).

What I don't understand is: in the statements in Lemma 5 and Cor 6, $$\phi$$ and $$\varphi$$, I don't know whether they are injective or not.

My question is then: why is GIN powerful? i.e, Why does GIN preserve injectivity?

This paper explains that injectivity should be preserved to get powerful GNN. How can I answer this question?