The Generalized Friendship Paradox for Spectral Centralities

TL;DR

This paper generalizes the friendship paradox to various spectral centralities in networks, showing that a node's neighbors typically have higher centrality than the average node.

Contribution

It extends the classical friendship paradox to spectral centralities like eigenvector, Katz, and PageRank, providing a unified theoretical framework.

Findings

Neighbors' average spectral centrality exceeds global average

The result applies to multiple spectral centrality measures

Valid for connected undirected graphs

Abstract

We revisit the classical friendship paradox which states that on an average ones friends have at least as many friends as oneself and generalize it to a variety of network centrality indices. For a broad class of spectral centralities on connected undirected graphs degree, eigenvector centrality, walk counts, Katz centrality and PageRank, we show that the average centrality of a nodes neighbours always exceeds the global average centrality.