阅读论文1：《How Powerful Are Graph Neural Networks？》

一、 论文中定理

1、定义1 —— Multiset

A multiset is a generalized concept of a set that allows multiple instances for its elements. More formally, a multiset is a 2-tulle $X=(S,m)$ where $S$ Is the underlying set of $X$ that is formed from its distinct elements, and $m:S\to N_{\ge 1}$ gives the multiplicity of the elements.

2、引理2

Let $G_1$ and $G_2$ be any two non-isomorphic graphs. If a graph neural network $A:G\to R^d$ maps $G_1$ and $G_2$ to different embeddings, the Weisfeiler-Lehman graph isomorphism test also decides $G_1$ and $G_2$ are not isomorphic.

3、定理3

Let $A:G\to R^d$ be a GNN. With a sufficient number of GNN layers. A maps any graphs $G_1$ And $G_2$ that the WL test of isomorphism decides as non-isomorphic, to different embeddings if the following conditions hold:
a) A aggregates and updates node features iteratively with:
$h_v^{(k)}=\varphi (h_v^{(k-1)},f(\{h_u^{(k-1)}:u\in N(v)\}))$
Where the functions $f$, which operates on multisets, and $\varphi$ are invective.
b). A’s graph-level readout, which operates on the multiset of node features $\{h_v^k\}$, is invective.

4、引理4