[Paper Review] Consistency of Constrained Spectral Clustering under Graph Induced Fair Planted Partitions

Shubham Gupta, Ambedkar Dukkipati|arXiv (Cornell University)|May 8, 2021

Complex Network Analysis Techniques4 citations

TL;DR

This paper proposes representation-aware spectral clustering algorithms—URepSC and NRepSC—that enforce individual-level fairness by ensuring each node has sufficient representatives from its sensitive attributes in clusters, using a representation graph that encodes 'is representative of' relationships. The authors establish weak consistency for these methods under a novel fair planted partition model, marking the first such theoretical guarantees for individual-level fairness in constrained spectral clustering.

ABSTRACT

Spectral clustering is popular among practitioners and theoreticians alike. While performance guarantees for spectral clustering are well understood, recent studies have focused on enforcing ``fairness'' in clusters, requiring them to be ``balanced'' with respect to a categorical sensitive node attribute (e.g. the race distribution in clusters must match the race distribution in the population). In this paper, we consider a setting where sensitive attributes indirectly manifest in an auxiliary extit{representation graph} rather than being directly observed. This graph specifies node pairs that can represent each other with respect to sensitive attributes and is observed in addition to the usual extit{similarity graph}. Our goal is to find clusters in the similarity graph while respecting a new individual-level fairness constraint encoded by the representation graph. We develop variants of unnormalized and normalized spectral clustering for this task and analyze their performance under a \emph{fair} planted partition model induced by the representation graph. This model uses both the cluster membership of the nodes and the structure of the representation graph to generate random similarity graphs. To the best of our knowledge, these are the first consistency results for constrained spectral clustering under an individual-level fairness constraint. Numerical results corroborate our theoretical findings.

Motivation & Objective

• To address fairness in spectral clustering by enforcing individual-level constraints based on sensitive attributes indirectly observed through a representation graph.
• To develop new spectral clustering variants (URepSC and NRepSC) that incorporate fairness via a representation graph without requiring direct access to sensitive attributes.
• To establish theoretical consistency for these algorithms under a new fair planted partition model that integrates both cluster structure and representation graph topology.
• To demonstrate empirically that the proposed methods maintain high clustering quality while achieving strong fairness, even when the representation graph deviates from ideal regularity.

Proposed method

• Formalize a new individual-level fairness constraint—'representation constraint'—where each node must have a sufficient number of its representatives (from the representation graph) in its assigned cluster.
• Propose URepSC (unnormalized) and NRepSC (normalized) spectral clustering variants that incorporate the representation graph as a fairness constraint in the clustering objective.
• Introduce the $\mathcal{R}$-PP (representation-aware planted partition) model, which generates random similarity graphs conditioned on both cluster membership and the structure of the representation graph $\mathcal{R}$.
• Analyze weak consistency of URepSC and NRepSC under $\mathcal{R}$-PP for $d$-regular representation graphs, proving that the number of misclassified nodes grows sublinearly with $N$.
• Use low-rank approximation in approximate variants (URepSC (approx.), NRepSC (approx.)) to reduce computational cost while preserving performance.
• Validate theoretical findings empirically on synthetic, FAO trade, and air-transportation networks, showing competitive ratio-cut and high average balance.

Experimental results

Research questions

• RQ1 Can spectral clustering be made consistent under individual-level fairness constraints that are not directly observable but inferred via a representation graph?
• RQ2 How does the structure of the representation graph influence the consistency and performance of constrained spectral clustering?
• RQ3 Can a new planted partition model that integrates both cluster structure and representation graph topology yield theoretical guarantees for fairness-aware clustering?
• RQ4 How do the proposed representation-aware spectral clustering algorithms compare to existing fair clustering methods in terms of clustering quality and fairness metrics?
• RQ5 To what extent do approximate variants of the algorithms maintain performance and fairness when the $d$-regularity assumption on the representation graph is violated?

Key findings

• The proposed URepSC and NRepSC algorithms achieve weak consistency under the $\mathcal{R}$-PP model for $d$-regular representation graphs, meaning the number of misclassified nodes is $o(N)$ with high probability.
• The theoretical framework generalizes prior work on population-level fairness (e.g., Kleindessner et al., 2019), with their results recovering the latter as a special case.
• Empirical results on real-world networks (FAO trade and air-transportation) show that the algorithms maintain competitive ratio-cut while achieving high average balance.
• The approximate variants (URepSC (approx.), NRepSC (approx.)) preserve strong performance, with fairness and clustering quality trade-offs controllable via the rank $R$ of low-rank approximation.
• Even when the $d$-regularity assumption on the representation graph is violated, the algorithms remain effective, demonstrating robustness in practical settings.
• Separate plots of ratio-cut and average balance confirm that individual fairness comes at a modest cost in clustering quality but yields substantial fairness gains.