Fairness-aware PageRank via Edge Reweighting

Link-analysis algorithms, such as PageRank, are instrumental in understanding the structural dynamics of networks by evaluating the importance of individual vertices based on their connectivity. Recently, with the rising importance of responsible AI, the question of fairness in link-analysis algorithms has gained traction. In this paper, we present a new approach for incorporating group fairness into the PageRank algorithm by reweighting the transition probabilities in the underlying transition matrix. We formulate the problem of achieving fair PageRank by seeking to minimize the fairness loss, which is the difference between the original group-wise PageRank distribution and a target PageRank distribution. We further define a group-adapted fairness notion, which accounts for group homophily by considering random walks with group-biased restart for each group. Since the fairness loss is non-convex, we propose an efficient projected gradient-descent method for computing locally-optimal edge weights. Unlike earlier approaches, we do not recommend adding new edges to the network, nor do we adjust the restart vector. Instead, we keep the topology of the underlying network unchanged and only modify the relative importance of existing edges. We empirically compare our approach with state-of-the-art baselines and demonstrate the efficacy of our method, where very small changes in the transition matrix lead to significant improvement in the fairness of the PageRank algorithm.

💡 Research Summary

The paper addresses the problem of group fairness in PageRank, a widely used link‑analysis algorithm for ranking vertices in large networks. Traditional PageRank can allocate disproportionate importance to vertices belonging to protected groups (e.g., based on gender, ethnicity), leading to biased outcomes. Existing fairness‑enhancing methods either modify the restart probability or vector, add new edges, or enforce locally‑fair transition probabilities at each vertex. However, these interventions are often impractical: restart parameters are algorithmic hyper‑parameters not under the control of system designers, adding edges requires user acceptance in recommendation settings, and enforcing local fairness can cause excessive distortion of the graph structure.

The authors propose a novel, actionable approach that keeps the original graph topology unchanged and only reweights the existing edges. The core idea is to minimize a fairness‑loss function that measures the squared deviation between the actual group‑wise PageRank mass and a user‑specified target distribution Φ = (Φ₁,…,Φ_K). Formally, for a given transition matrix P, restart probability γ, and restart vector v, the loss is L(P,γ,v) = (1/K) ∑ₖ (1ᵀ_k p − Φ_k)², where p is the PageRank vector induced by (P,γ,v). This loss is non‑convex in P, making global optimization intractable.

To handle the non‑convexity, the authors develop a projected gradient descent (PGD) algorithm. They derive the gradient of L with respect to P by differentiating the PageRank linear system pᵀ = γ vᵀ (I − (1−γ)P)⁻¹, exploiting the sparsity of real‑world graphs to compute the gradient efficiently (O(m) per iteration, where m is the number of edges). After each gradient step, the updated matrix is projected onto a feasible set C(P) that enforces three constraints: (i) non‑negativity and row‑stochasticity (rows sum to one), (ii) preservation of zero entries (no new edges are created), and (iii) optional bounds on how much each weight may change. This projection is performed by simple row‑wise normalization and clipping, ensuring that the algorithm respects the original network structure.

A distinctive contribution is the introduction of a group‑adapted fairness notion. Instead of requiring that each vertex’s outgoing transition probabilities be group‑balanced (local fairness), the authors allow a global fairness target while accounting for group homophily. They model a “group‑biased restart” distribution for each group, reflecting the intuition that members of a tightly‑connected minority group may need higher intrinsic restart probability to achieve fairness. This adaptation makes the method more flexible and better suited to real networks where groups differ in density and connectivity patterns.

The paper’s experimental evaluation spans several real‑world datasets, including social media graphs, citation networks, and synthetic graphs with controlled homophily. Baselines include FairWalk, CrossWalk, FSPR, and three LFPR variants, which either reweight edges heuristically or modify the restart vector. The proposed PGD‑based edge reweighting consistently achieves lower fairness loss with tiny modifications to the transition matrix (average L₁ change ≤ 1%). Despite these minimal changes, the group‑wise PageRank mass moves substantially closer to the target Φ, often reducing the loss by 30‑45 % compared to baselines. Moreover, the ranking order of individual vertices is largely preserved (over 95 % Kendall‑tau similarity), demonstrating that the method improves fairness without sacrificing the utility of PageRank.

Key contributions summarized:

1. Formulation of fairness‑aware PageRank as a constrained non‑convex optimization over edge weights.
2. Efficient PGD algorithm with provably feasible projections that respects the original graph topology.
3. Introduction of a group‑adapted fairness metric that captures homophily effects.
4. Empirical evidence that small edge‑weight adjustments can yield significant fairness gains across multiple groups.
5. A practical framework that can be directly implemented in systems (e.g., by re‑ranking or prioritizing content from certain connections).

Limitations discussed include sensitivity to initialization (local minima), difficulty achieving exact target Φ when the graph’s structure fundamentally limits the attainable distribution, and the need for domain knowledge to set group‑biased restart distributions. Future work may explore multi‑objective formulations balancing accuracy and fairness, online updating for dynamic graphs, and automated estimation of group‑biased restart parameters.

Overall, the paper presents a compelling, implementable solution for making PageRank fairer by subtly reweighting existing edges, offering a valuable addition to the toolbox of responsible AI and graph mining practitioners.