A Boolean Function-Theoretic Framework for Expressivity in GNNs with Applications to Fair Graph Mining

TL;DR

This paper introduces a Boolean function-theoretic framework for analyzing GNN expressivity, enabling better understanding and handling of complex subpopulation structures for fair graph mining.

Contribution

It proposes the Subpopulation Boolean Isomorphism as a new expressivity measure and develops a circuit-based fairness algorithm that outperforms existing methods.

Findings

The framework subsumes existing expressivity measures.

The circuit-based algorithm handles high-complexity subpopulations.

Experiments show improved fairness across intersectional groups.

Abstract

We propose a novel expressivity framework for Graph Neural Networks (GNNs) grounded in Boolean function theory, enabling a fine-grained analysis of their ability to capture complex subpopulation structures. We introduce the notion of \textit{Subpopulation Boolean Isomorphism} (SBI) as an invariant that strictly subsumes existing expressivity measures such as Weisfeiler-Lehman (WL), biconnectivity-based, and homomorphism-based frameworks. Our theoretical results identify Fourier degree, circuit class (AC$^0$, NC$^1$), and influence as key barriers to expressivity in fairness-aware GNNs. We design a circuit-traversal-based fairness algorithm capable of handling subpopulations defined by high-complexity Boolean functions, such as parity, which break existing baselines. Experiments on real-world graphs show that our method achieves low fairness gaps across intersectional groups where state-of-the-art methods fail, providing the first principled treatment of GNN expressivity tailored to fairness.