The friendship paradox in scale-free networks

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.