# Some Might Say All You Need Is Sum

## TL;DR

This paper investigates the expressivity of different GNN aggregation functions, showing that Sum GNNs are limited compared to Mean and Max, especially for larger graphs and certain functions.

## Contribution

It provides theoretical proofs that Sum GNNs cannot approximate basic functions computed by Mean or Max GNNs, highlighting the limitations of Sum aggregation.

## Key findings

- Sum GNNs cannot approximate functions computed by Mean or Max GNNs.
- Mean and Max GNNs are more expressive than Sum GNNs.
- Combination of Sum with Mean or Max increases expressivity.

## Abstract

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 specific 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 desired that a GNN's usability will not be limited to graphs of any specific size. Therefore, we explore the realm of unrestricted-size expressivity. We prove that basic 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 GNN 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 for GNNs with a broad class of aggregations.