The friendship paradox in scale-free networks

TL;DR

This paper investigates the friendship paradox in scale-free networks by deriving an expression for the difference between the mean degree of vertices and the mean degree of friends of friends, highlighting how degree distribution influences this relationship.

Contribution

It provides a mathematical expression for the friendship paradox in scale-free networks with power-law degree distributions, linking network structure to the paradox.

Findings

Derived an expression for <k_FF> - <k> in scale-free networks.

Quantified how degree distribution affects the friendship paradox.

Analyzed the impact of network parameters on the paradox.

Abstract

Our friends have more friends than we do. That is the basis of the friendship paradox. In mathematical terms, the mean number of friends of friends is higher than the mean number of friends. In the present study, we analyzed the relationship between the mean degree of vertices (individuals), <k>, and the mean number of friends of friends, <k_FF>, in scale-free networks with degrees ranging from a minimum degree (k_min) to a maximum degree (k_max). We deduced an expression for <k_FF> - <k> for scale-free networks following a power-law distribution with a given scaling parameter (alpha). Based on this expression, we can quantify how the degree distribution of a scale-free network affects the mean number of friends of friends.