The friendship paradox in real and model networks

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.