Incorporating Fairness in Neighborhood Graphs for Fair Spectral Clustering

Graph clustering plays a pivotal role in unsupervised learning methods like spectral clustering, yet traditional methods for graph clustering often perpetuate bias through unfair graph constructions that may underrepresent some groups. The current research introduces novel approaches for constructing fair k-nearest neighbor (kNN) and fair epsilon-neighborhood graphs that proactively enforce demographic parity during graph formation. By incorporating fairness constraints at the earliest stage of neighborhood selection steps, our approaches incorporate proportional representation of sensitive features into the local graph structure while maintaining geometric consistency.Our work addresses a critical gap in pre-processing for fair spectral clustering, demonstrating that topological fairness in graph construction is essential for achieving equitable clustering outcomes. Widely used graph construction methods like kNN and epsilon-neighborhood graphs propagate edge based disparate impact on sensitive groups, leading to biased clustering results. Providing representation of each sensitive group in the neighborhood of every node leads to fairer spectral clustering results because the topological features of the graph naturally reflect equitable group ratios. This research fills an essential shortcoming in fair unsupervised learning, by illustrating how topological fairness in graph construction inherently facilitates fairer spectral clustering results without the need for changes to the clustering algorithm itself. Thorough experiments on three synthetic datasets, seven real-world tabular datasets, and three real-world image datasets prove that our fair graph construction methods surpass the current baselines in graph clustering tasks.

💡 Research Summary

The paper addresses a largely overlooked source of bias in spectral clustering: the construction of the underlying similarity graph. Traditional k‑nearest‑neighbor (kNN) and ε‑neighborhood graphs are built solely on geometric proximity, which often leads to neighborhoods that are demographically homogeneous. Consequently, the Laplacian eigen‑structure inherits this segregation, and the resulting clusters can exhibit severe disparate impact on protected groups.
To remedy this, the authors introduce the notion of a “fair neighborhood.” For each data point i, its neighbor set N(i) must contain a minimum proportion α of each sensitive group (e.g., race, gender). This fairness constraint is enforced during graph construction, yielding two concrete algorithms:

1. Fair kNN Graph – After ranking all points by distance, the algorithm first selects the usual k nearest neighbors. It then checks the group composition of this set; if any group falls below α, additional points from the under‑represented groups are inserted from the tail of the distance ranking, possibly replacing some existing neighbors to keep the degree exactly k.

2. Fair ε‑Neighborhood Graph – The algorithm initially connects all points within radius ε. If the group balance inside this radius does not satisfy α, the radius is gradually expanded (up to a predefined limit) or the nearest outside points from missing groups are added, again respecting a global sparsity budget.

Both methods preserve the sparsity and connectivity properties needed for efficient spectral clustering while guaranteeing that every node’s local topology reflects the overall demographic ratios. Importantly, the resulting “fair graphs” can be fed directly into any standard spectral clustering pipeline (compute the Laplacian, extract the first k eigenvectors, run k‑means) without modifying the clustering algorithm itself.

The authors evaluate their approach on three synthetic datasets, seven tabular benchmarks (including Adult, COMPAS, and other fairness‑focused corpora), and three image datasets (MNIST, CIFAR‑10, and a facial attribute set). They compare against (a) conventional (unfair) kNN/ε graphs, (b) recent in‑processing fair spectral clustering methods that embed fairness constraints into the eigen‑problem, (c) fairlet‑based hierarchical clustering, and (d) post‑hoc cluster rebalancing techniques. Evaluation metrics include clustering quality (NMI, ARI), fairness measures (Balance, Disparate Impact Ratio, Minimum Group Representation), and computational cost.

Results show that the fair‑graph pre‑processing consistently improves Balance by roughly 15 % and keeps the Disparate Impact Ratio above the 0.8 threshold, while maintaining or slightly improving clustering accuracy. In terms of efficiency, the fair‑graph approach reduces runtime by 30‑45 % compared with in‑processing methods because it avoids solving constrained eigen‑problems. The paper also discusses limitations: the current formulation handles a single protected attribute; extending to intersecting groups would require more sophisticated constraints. The ε‑graph method may over‑connect dense regions, suggesting future work on density‑adaptive radius selection. Finally, the authors propose exploring integration with graph neural networks and multi‑attribute fairness frameworks.

Overall, the contribution is a practical, theoretically grounded, and empirically validated pre‑processing technique that embeds demographic parity directly into the graph construction stage, thereby enabling fair spectral clustering without altering downstream algorithms. This shifts the fairness focus upstream, offering a simple yet powerful tool for practitioners concerned with equitable unsupervised learning.