Fairness-Sensitive PageRank Approximation

TL;DR

This work tackles fairness in PageRank on real networks by addressing the computational bottlenecks of Fairness-Sensitive PageRank (FSPR), which couples a fairness constraint to the standard PageRank via a dense inverse operator $Q = \nu [I - (1 - \nu) P]^{-1}$. It introduces a mean-field approximation that partitions the network into two groups and further into degree-based classes, yielding a closed-form approximate PageRank $\hat{p}(u)$ that depends only on $u$'s in-degree and global group proportions: $\hat{p}(u) = \nu \phi_{C(u)} \frac{k_{in}(u)}{D_{C(u)}} + (1 - \nu) \frac{k_{in}(u)}{\langle k_{in} \rangle}$. The paper also derives a variance expression for intra-class fluctuations, offering insight into reliability when degrees are small, and proves that the mean-field method reduces the complexity from cubic time and quadratic space to linear time and space $O(M)$ and $O(N+M)$, enabling fairness-constrained ranking at scale. Empirical studies on real networks show a high agreement with exact FSPR (e.g., correlations $\tau \ge 0.94$) and negligible utility loss and fairness gaps, demonstrating practicality for large graphs. The approach thus delivers scalable, fairness-aware ranking with rigorous support for both accuracy and group-level fairness.

Abstract

Real-world social networks have structural inequalities, including the majority and minorities, and fairness-agnostic centrality measures often amplify these inequalities by disproportionately favoring majority nodes. Fairness-Sensitive PageRank aims to balance algorithmic influence across structurally and demographically diverse groups while preserving the link-based relevance of classical PageRank. However, existing formulations require solving constrained matrix inversions that scale poorly with network size. In this work, we develop an efficient mean-field approximation for Fairness-Sensitive PageRank (FSPR) that enforces group-level fairness through an estimated teleportation (jump) vector, thereby avoiding the costly matrix inversion and iterative optimization. We derive a closed-form approximation of FSPR using the in-degree and group label of nodes, along with the global group proportion. We further analyze intra-class fluctuations by deriving expressions for the variance of approximated FSPR scores. Empirical results on real-world networks demonstrate that the proposed approximation efficiently estimates the FSPR while reducing runtime by an order of magnitude, enabling fairness-constrained ranking at scale.

Figures (3)

• Figure 1: Mean field (MF) approximated FSPR is plotted against empirical average FSPR per degree-class.
• Figure 2: Fairness-Sensitive PageRank plotted against in-degree. The dashed blue line shows the approximation derived from Eq. (\ref{['eq:equationB15']}).
• Figure 3: Coefficient of variation of FSPR with respect to in-degree.