The Polynomial Counting Capabilities of Message Passing Neural Networks

2026-05-11

AI Summary

This work investigates the capacity of Message Passing Neural Networks (MPNNs) to express polynomial counting constraints, moving beyond the limitations of existing approaches that handle only linear counts. By introducing a mean aggregation mechanism and leveraging logical expressivity analysis under structural assumptions on graphs—such as regularity or tree-likeness—the paper systematically characterizes the conditions under which MPNNs can verify both local and global polynomial counting constraints over node-labeled graphs. The primary contribution lies in establishing, for the first time, a formal correspondence between MPNNs and graded modal logic extended with polynomial counting. The authors prove that, under mild assumptions, MPNNs can effectively capture local constraints without nesting and arbitrary global polynomial counting constraints; furthermore, in tree-structured graphs, they also accurately express nested polynomial counting constraints.

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.

Problem

• Message Passing Neural Networks
• polynomial counting
• graded modal logic
• graph neural networks
• counting constraints

Innovation

• Polynomial Counting
• Message Passing Neural Networks
• Mean Aggregation
• Graded Modal Logic
• Graph Neural Networks