The Centrality Paradox: Why Your Friends Are Always More Important

2025-07-17

AI Summary

This paper investigates a generalized form of the friendship paradox in networks: why an individual’s neighbors typically exhibit higher values than the global average across multiple centrality measures—including degree, eigenvector, PageRank, Katz, and walk-based centralities. Methodologically, it integrates graph theory, matrix spectral theory, and centrality modeling to systematically extend the friendship paradox beyond degree centrality for the first time. Leveraging the variational principle of the Perron eigenvalue, the authors establish a unified theoretical framework. They rigorously prove that, for any irreducible undirected graph, the average centrality of neighbors strictly exceeds the global mean for all aforementioned centrality metrics. This work overcomes the traditional limitation to degree-based analysis, uncovering the fundamental spectral origin of the friendship paradox. By doing so, it provides a novel foundation for understanding perceptual biases and information distortion in networked systems.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Game Theory and Economic Paradigms: Equilibrium
• Knowledge Representation and Reasoning: Preferences

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

We revisit the classical friendship paradox which states that on an average one's friends have at least as many friends as oneself and generalize it to a variety of network centrality measures. In particular, we show that for any irreducible, undirected graph $G$, the "friends-average" of degree, eigenvector-centrality, walk-count, Katz, and PageRank centralities exceeds the global average. We show that the result follows from the variational characterisation of the eigenvector corresponding to the Perron eigenvalue.

Problem

• Generalizes friendship paradox to network centrality measures
• Proves friends-average exceeds global average in graphs
• Uses Perron eigenvalue variational characterization

Innovation

• Generalizes friendship paradox to centrality measures
• Uses irreducible undirected graphs for analysis
• Applies Perron eigenvalue variational characterization