The Polynomial Counting Capabilities of Message Passing Neural Networks

TL;DR

This paper investigates the ability of Message Passing Neural Networks to express polynomial counting constraints in graphs, extending their known counting capabilities beyond linear arithmetic.

Contribution

It characterizes conditions under which mean MPNNs can represent polynomial counting constraints, including local, global, and nested modalities in graphs.

Findings

Global polynomial counting constraints can be checked with mean MPNN under mild assumptions.

Local constraints are checkable with mean MPNN if formulas are non-nested and certain aggregation functions are used.

Nested modalities can be captured by mean MPNN on tree-like graphs with specific assumptions.

Abstract

The counting power of Message Passing Neural Networks (MPNN) has been the subject of many recent papers, showing that they can express logic that involves counting up to a threshold or more generally satisfy a linear arithmetic constraint. In this paper, we study the counting capabilities of MPNN beyond linear arithmetic, primarily utilising local and global mean aggregations. In particular, our goal is to tease out conditions required to express extensions of graded modal logic with polynomial counting constraints. We show that global polynomial counting constraints in node-labelled graphs can be checked using mean MPNN under mild assumptions. Checking local constraints is also possible, if we consider formulas with no nested modalities and additionally either (i) permit sum/max aggregations, or (ii) only restrict to regular graphs. We also show how formulas with nested modalities can be captured by mean MPNN over graphs with tree-like structures and similar assumptions.