Friendship-paradox paradox: Do most people's friends really have more friends than they do?

TL;DR

This paper develops a framework to distinguish between mean-based and median-based friendship paradox inequalities, revealing how local majority relations can behave independently of the classical paradox in complex networks.

Contribution

It introduces a new framework separating mean-based inequalities from local majority measures, clarifying the distinction between population averages and local perceptions in network analysis.

Findings

Neither fraction is constrained by the classical friendship paradox.

Degree distribution skewness affects local majority relations.

Empirical networks illustrate independent behaviors of the fractions.

Abstract

The classical friendship paradox asserts that, on average, an individual's neighbors have a higher degree than the individual. This statement concerns network-level means and does not describe how often a typical node is locally dominated by its neighbors. Motivated by this distinction, we develop a framework that separates mean-based friendship paradox inequalities from two majority-type quantities: a global fraction measuring how many nodes have a degree smaller than the mean degree of their neighbors, and a local fraction based on hub centrality that measures how many nodes are dominated in a median-based sense. We show that neither fraction is constrained by the classical friendship paradox and that they can behave independently of each other. A simple example and two empirical networks illustrate how quadrant patterns in the joint distribution of a node's degree and its neighbors' degree determine the signs and magnitudes of the two fractions, and how left- or right-skewed degree distributions of neighboring nodes can yield opposite conclusions for mean-based and median-based comparisons. The resulting framework offers a clearer distinction between population averages and local majority relations and provides a foundation for future analyses of local advantage, disadvantage, and perception asymmetry in complex networks.