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

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.