Readout Loses The Molecule

A message-passing network scores molecules for toxicity. After the final layer every atom in these two molecules happens to carry the same node embedding $a$ :

molecule	atoms	node embeddings
M1	4	$a, a, a, a$
M2	8	$a, a, a, a, a, a, a, a$

The graph-level vector is produced by a readout over the node embeddings, then fed to a classifier. Which readout can tell M1 from M2, and why?

Mean: dividing by the atom count keeps the vector scale stable across graph sizes, which makes it strictly more expressive than sum.

Sum: it returns $4a$ and $8a$ , keeping the multiplicity of the node multiset, while mean returns $a$ for both and max returns $a$ for both, so under those two readouts the classifier cannot see the size difference at all.

Max: the larger molecule has more atoms contributing to the maximum, so max readout separates them while sum and mean do not.

None of them: readouts are permutation invariant, so a graph-level output needs a canonical node ordering and a sorted concatenation of the embeddings.