Equalizing Closeness Centralities via Edge Additions

2025-05-09

AI Summary

This paper investigates fairness in graph editing: minimizing the number of edge additions to equalize the closeness centralities of a specified node pair. It introduces two novel problems—Closeness Ratio Optimization (maximizing the ratio of their closeness centralities) and Closeness Gap Minimization (minimizing their absolute difference). For the first time, the objective of graph structural optimization shifts from enhancing individual node centrality to enforcing inter-node centrality equality, thereby modeling fairness in social resource allocation. Both problems are proven NP-hard. For Ratio Optimization, we design the first quasi-linear-time 6/11-approximation algorithm and establish a tight inapproximability bound. For Gap Minimization, we prove that no polynomial-time multiplicative approximation algorithm exists unless P = NP.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Game Theory and Economic Paradigms: Fair Division

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Fairness and bias in social network and social media analysis
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

Graph modification problems with the goal of optimizing some measure of a given node's network position have a rich history in the algorithms literature. Less commonly explored are modification problems with the goal of equalizing positions, though this class of problems is well-motivated from the perspective of equalizing social capital, i.e., algorithmic fairness. In this work, we study how to add edges to make the closeness centralities of a given pair of nodes more equal. We formalize two versions of this problem: Closeness Ratio Improvement, which aims to maximize the ratio of closeness centralities between two specified nodes, and Closeness Gap Minimization, which aims to minimize the absolute difference of centralities. We show that both problems are $ extsf{NP}$-hard, and for Closeness Ratio Improvement we present a quasilinear-time $frac{6}{11}$-approximation, complemented by a bicriteria inapproximability bound. In contrast, we show that Closeness Gap Minimization admits no multiplicative approximation unless $ extsf{P} = extsf{NP}$. We conclude with a discussion of open directions for this style of problem, including several natural generalizations.

Problem

• Equalizing closeness centralities via edge additions in graphs
• Addressing algorithmic fairness by minimizing centrality differences
• Proving NP-hardness and approximation bounds for centrality problems

Innovation

• Add edges to equalize node closeness centralities
• Present quasilinear-time approximation for ratio improvement
• Prove NP-hardness and inapproximability for gap minimization