A Boolean Function-Theoretic Framework for Expressivity in GNNs with Applications to Fair Graph Mining
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866918295183032320 |
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| author | Pal, Manjish |
| author_facet | Pal, Manjish |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12751 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Boolean Function-Theoretic Framework for Expressivity in GNNs with Applications to Fair Graph Mining Pal, Manjish Machine Learning 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. |
| title | A Boolean Function-Theoretic Framework for Expressivity in GNNs with Applications to Fair Graph Mining |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2601.12751 |