The friendship paradox in real and model networks

TL;DR

This paper develops a comprehensive mathematical framework for the friendship paradox in networks, analyzing both average behavior and variability, and validates the theory with real-world data, also extending it to generalized friendship paradox scenarios.

Contribution

It introduces a detailed mathematical theory of the friendship paradox, including distributional analysis and application to real networks, which was not previously available.

Findings

The theory accurately predicts degree distributions in real networks.

There is strong agreement between theoretical predictions and empirical data.

The framework extends to generalized friendship paradox scenarios.

Abstract

The friendship paradox is the observation that the degrees of the neighbors of a node in any network will, on average, be greater than the degree of the node itself. In common parlance, your friends have more friends than you do. In this paper we develop the mathematical theory of the friendship paradox, both in general as well as for specific model networks, focusing not only on average behavior but also on variation about the average and using generating function methods to calculate full distributions of quantities of interest. We compare the predictions of our theory with measurements on a large number of real-world network data sets and find remarkably good agreement. We also develop equivalent theory for the generalized friendship paradox, which compares characteristics of nodes other than degree to those of their neighbors.