Asymptotic distribution of the friendship paradox of a random geometric graph

In social networks, there is a paradox that an individual's friends tend to have more friends than the individual. The friendship paradox is a summary statistic of networks that measures the strength of the paradox. It has many applications in network data analysis. As far as we know, theoretical properties of the friendship paradox are not well studied. In this paper, we derive the asymptotic distribution of the friendship paradox in a random geometric graph. After centering and scaling, the friendship paradox converges in law to a series of independent standard normal random variables. It is interesting that the limiting distribution is not the standard normal distribution.