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

Manjish Pal

TL;DR

The notion of Subpopulation Boolean Isomorphism (SBI) is introduced as an invariant that strictly subsumes existing expressivity measures such as Weisfeiler-Lehman (WL), biconnectivity-based, and homomorphism-based frameworks.

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.

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

TL;DR

The notion of Subpopulation Boolean Isomorphism (SBI) is introduced as an invariant that strictly subsumes existing expressivity measures such as Weisfeiler-Lehman (WL), biconnectivity-based, and homomorphism-based frameworks.

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, NC), 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.
Paper Structure (14 sections, 11 theorems, 24 equations, 2 figures, 1 table, 1 algorithm)

This paper contains 14 sections, 11 theorems, 24 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work and Background
  3. Expressive Power of GNNs.
  4. Approximation Ability of GNNs
  5. Fairness Aware Graph Learning.
  6. Fourier Analysis of Boolean Function.
  7. Preliminaries
  8. Boolean Fairness Functions Expressivity of GNNs
  9. Formal Expressivity Framework
  10. Expressivity in terms of Efficiently Computable Boolean Functions
  11. Expressivity using Fourier Analysis of Boolean Functions
  12. Application to Fair Graph Learning
  13. Experiments
  14. Conclusion

Key Result

Theorem 1

The graph isomorphism problem ($\mathsf{GI}$) reduces in polynomial time to the subpopulation boolean function isomorphism problem ($\mathsf{SubIso}$).

Figures (2)

  • Figure 1: The left part of the figure shows an example of an input Subpopulation Boolean Function along with the corresponding circuit involving several attributes. The right part shows the FairSBF framework which takes as input the (training) graph as well as the SBF circuit to a GNN algorithm, and creates a new regularizer based training algorithm to produce the final trained model.
  • Figure 2: Performance comparison of Fair GNNs on the Pokec-z dataset with complex subpopulations. Our method FairSBF achieves the best trade-off, showing the highest accuracy and the lowest fairness violations (DDP, DEO). Unlike other methods which degrade on high-complexity Boolean functions, FairSBF remains robust due to its circuit-theoretic fairness layer.

Theorems & Definitions (26)

  • Definition 1: 1-WL Expressivity
  • Definition 2: Function-level Expressivity
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Definition 3: Expressivity Invariant
  • Theorem 4
  • ...and 16 more