Fairness-Sensitive PageRank Approximation

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.

💡 Research Summary

The paper addresses the scalability problem of Fairness‑Sensitive PageRank (FSPR), a variant of PageRank that modifies the teleportation vector to satisfy group‑level fairness constraints. While FSPR can equalize the total PageRank mass assigned to protected and unprotected groups, its computation requires solving a dense matrix inversion followed by a quadratic convex optimization over all nodes, leading to O(N³) time and O(N²) memory complexity—untenable for large graphs.

To overcome this, the authors propose a mean‑field approximation that eliminates the need for matrix inversion and global optimization. The key steps are:

1. Degree‑Class Aggregation – Nodes are grouped into classes defined by (in‑degree, out‑degree, group label). Protected nodes form a set K_P, unprotected nodes a set K_U. All nodes within a class share identical structural attributes.

2. Jump‑Vector Approximation – The teleportation probability for a node is approximated using its in‑degree and the total in‑degree of its group:

   • For protected nodes: v_i ≈ ϕ·k_in(i)/D_P
   • For unprotected nodes: v_i ≈ (1‑ϕ)·k_in(i)/D_U
     where ϕ is the fraction of protected nodes, D_P and D_U are the sums of in‑degrees in each group.

3. Mean‑Field Transition Modeling – The expected number of edges from class k′ to class k is expressed as E_{k′→k}=k_in·P_in(k′|k)·N. Assuming an uncorrelated network, the conditional transition probability simplifies to P_in(k′|k)=k_out′·P(k′)/⟨k_in⟩.

4. Closed‑Form Approximation – Substituting the above approximations into the PageRank fixed‑point equation and taking the limit n→∞ yields a simple node‑level formula:
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