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On the use of graph models to achieve individual and group fairness

Arturo Pérez-Peralta, Sandra Benítez-Peña, Rosa E. Lillo

TL;DR

This work addresses the challenge of achieving both individual and group fairness in machine learning by introducing Fair Sheaf Diffusion (FSD), a topological framework that uses cellular sheaves and graph-based diffusion to encode and enforce fairness constraints. By shaping the diffusion dynamics through tailored sheaves and topologies (e.g., identity, vector/bottleneck, kNN, unit ball, and subset topologies), the method projects data into a bias-free space while providing closed-form SHAP explanations for interpretability. The paper develops a unified theory of fair classification with group and individual metrics, derives diffusion-based mechanisms to minimize a fairness-driven energy, and demonstrates trade-offs between accuracy and fairness on simulations and standard benchmarks (German, Compas, Adult) via Pareto fronts and grid searches. A key strength is the ability to combine multiple topologies to balance multiple fairness criteria, offering pre-/in-/post-processing flexibility and insightful explanations, with potential extensions to more complex topologies and non-linear diffusion. Overall, FSD advances responsible AI by tying algebraic-topological constructs to practical fairness objectives and interpretability, enabling principled fairness-accuracy trade-offs in real-world deployments.

Abstract

Machine Learning algorithms are ubiquitous in key decision-making contexts such as justice, healthcare and finance, which has spawned a great demand for fairness in these procedures. However, the theoretical properties of such models in relation with fairness are still poorly understood, and the intuition behind the relationship between group and individual fairness is still lacking. In this paper, we provide a theoretical framework based on Sheaf Diffusion to leverage tools based on dynamical systems and homology to model fairness. Concretely, the proposed method projects input data into a bias-free space that encodes fairness constrains, resulting in fair solutions. Furthermore, we present a collection of network topologies handling different fairness metrics, leading to a unified method capable of dealing with both individual and group bias. The resulting models have a layer of interpretability in the form of closed-form expressions for their SHAP values, consolidating their place in the responsible Artificial Intelligence landscape. Finally, these intuitions are tested on a simulation study and standard fairness benchmarks, where the proposed methods achieve satisfactory results. More concretely, the paper showcases the performance of the proposed models in terms of accuracy and fairness, studying available trade-offs on the Pareto frontier, checking the effects of changing the different hyper-parameters, and delving into the interpretation of its outputs.

On the use of graph models to achieve individual and group fairness

TL;DR

This work addresses the challenge of achieving both individual and group fairness in machine learning by introducing Fair Sheaf Diffusion (FSD), a topological framework that uses cellular sheaves and graph-based diffusion to encode and enforce fairness constraints. By shaping the diffusion dynamics through tailored sheaves and topologies (e.g., identity, vector/bottleneck, kNN, unit ball, and subset topologies), the method projects data into a bias-free space while providing closed-form SHAP explanations for interpretability. The paper develops a unified theory of fair classification with group and individual metrics, derives diffusion-based mechanisms to minimize a fairness-driven energy, and demonstrates trade-offs between accuracy and fairness on simulations and standard benchmarks (German, Compas, Adult) via Pareto fronts and grid searches. A key strength is the ability to combine multiple topologies to balance multiple fairness criteria, offering pre-/in-/post-processing flexibility and insightful explanations, with potential extensions to more complex topologies and non-linear diffusion. Overall, FSD advances responsible AI by tying algebraic-topological constructs to practical fairness objectives and interpretability, enabling principled fairness-accuracy trade-offs in real-world deployments.

Abstract

Machine Learning algorithms are ubiquitous in key decision-making contexts such as justice, healthcare and finance, which has spawned a great demand for fairness in these procedures. However, the theoretical properties of such models in relation with fairness are still poorly understood, and the intuition behind the relationship between group and individual fairness is still lacking. In this paper, we provide a theoretical framework based on Sheaf Diffusion to leverage tools based on dynamical systems and homology to model fairness. Concretely, the proposed method projects input data into a bias-free space that encodes fairness constrains, resulting in fair solutions. Furthermore, we present a collection of network topologies handling different fairness metrics, leading to a unified method capable of dealing with both individual and group bias. The resulting models have a layer of interpretability in the form of closed-form expressions for their SHAP values, consolidating their place in the responsible Artificial Intelligence landscape. Finally, these intuitions are tested on a simulation study and standard fairness benchmarks, where the proposed methods achieve satisfactory results. More concretely, the paper showcases the performance of the proposed models in terms of accuracy and fairness, studying available trade-offs on the Pareto frontier, checking the effects of changing the different hyper-parameters, and delving into the interpretation of its outputs.
Paper Structure (47 sections, 2 theorems, 53 equations, 45 figures, 12 tables)

This paper contains 47 sections, 2 theorems, 53 equations, 45 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Contributions
  3. Structure
  4. Related work
  5. Fairness
  6. Sheaf models
  7. Network topologies
  8. Theoretical background
  9. Fair classification
  10. Group Fairness metrics
  11. Independence
  12. Separation
  13. Sufficiency
  14. Individual Fairness Metrics
  15. Lipschitz constant
  16. ...and 32 more sections

Key Result

Theorem 3.1

Suppose a signal $x^t\in \mathcal{C}^{0}(G;\mathcal{F})$ under a sheaf diffusion process with initial condition $x^0$. In the limit $t\longrightarrow \infty$ the signal $x^t$ tends to the orthogonal projection of $x^0$ onto $\ker \Delta^{\mathcal{F}}$.

Figures (45)

  • Figure 1: Graph with two disjoint communities. Sheaf Diffusion is unable to reconcile fairness metrics between individuals belonging to different communities.
  • Figure 2: Examples of the subset topologies. The top row represents two partition topologies with different number of elements, while the bottom row showcases a more exotic configuration with four pairwise independent agregators.
  • Figure 3: Boxplots with fairness metrics and accuracy for one hundred simulations. The top row displays fairness metrics while the bottom row shows accuracy.
  • Figure 4: Average SHAP variable influence of one hundred simulations. The left panel shows the average absolute SHAP contributions while the right panel influences are normalized.
  • Figure 5: Fairness-accuracy trade-offs in the Pareto frontier for the grid search on the German dataset. The left figure shows trade-offs between independence and accuracy, while the right displays consistency and accuracy.
  • ...and 40 more figures

Theorems & Definitions (11)

  • Theorem 3.1
  • Definition 4.1: Fair Sheaf Diffusion
  • Definition 4.2
  • Example 4.1
  • Example 4.2
  • Example 4.3
  • Definition 4.3
  • Definition 4.4: Unit ball graph
  • Definition 4.5
  • Lemma 4.6
  • ...and 1 more