Some Might Say All You Need Is Sum

The expressivity of Graph Neural Networks (GNNs) is dependent on the aggregation functions they employ. Theoretical works have pointed towards Sum aggregation GNNs subsuming every other GNNs, while certain practical works have observed a clear advantage to using Mean and Max. An examination of the theoretical guarantee identifies two caveats. First, it is size-restricted, that is, the power of every specific GNN is limited to graphs of a certain maximal size. Successfully processing larger graphs may require an other GNN, and so on. Second, it concerns the power to distinguish non-isomorphic graphs, not the power to approximate general functions on graphs, and the former does not necessarily imply the latter. It is important that a GNN's usability will not be limited to graphs of any certain maximal size. Therefore, we explore the realm of unrestricted-size expressivity. We prove that simple functions, which can be computed exactly by Mean or Max GNNs, are inapproximable by any Sum GNN. We prove that under certain restrictions, every Mean or Max GNNs can be approximated by a Sum GNN, but even there, a combination of (Sum, [Mean/Max]) is more expressive than Sum alone. Lastly, we prove further expressivity limitations of Sum-GNNs.