The friendship paradox∗

Why do your friends have more friends than you do? The question may sound offensive. We don’t even know you. How can we assume than you have fewer friends than your friends have on average? Because most people do. This so-called friendship paradox has first been described and studied in [1]. It does seem counterintuitive: If we are talking about the average number of friends of average friends of an average person, shouldn’t this average out to the average number of friends of an average person? Enough loose talk about averages that makes the average person’s head spin. Let’s steady our thoughts with some solid mathematical definitions. Consider a graph G that represents friendships between persons numbered 1, . . . , N . The degree ki of node i represents the number of i’s friends. The “average” number of friends of a randomly chosen person can be most naturally interpreted as the mean degree 〈k〉 that is given by