Why are your friends more popular than you?

DO YOU ever feel like your friends are more popular than you are? That may be because it is true—for nearly everyone. This odd result, dubbed the "friendship paradox", has most recently been observed to apply on Twitter. When researchers from the University of Southern California looked at 5.8m microbloggers (and 194m links between them) they found that, on average, both the people a user follows and, worse, those who follow him, have more followers than he does. How can this be?

The friendship paradox was first identified in 1991 by Scott Feld, a sociologist working at the State University of New York at Stony Brook. Back then, of course, Dr Feld was looking at real-world social networks rather than online ones. Then, last year, scientists from Cornell University confirmed that the result holds for Facebook's active users (721m people at the time of the research, joined by 69 billion virtual bonds of friendship). In fact, it obtains for any network where some members are more popular than others. And it stems from basic arithmetic.

Consider a simple social network composed of four people: Alice, Bob, Chloe and Dave. Alice's only friend is Bob. Bob is also friends with both Chloe and Dave, who are friends with each other, but not with Alice. This means that Alice has one friend (Bob); Chloe and Dave each have two friends (one another and Bob); and Bob has three. On average, then, each person in the network has two friends (eight friends divided by four people). But now consider how many friends each person's friends have (in other words, friends of friends). Alice has one friend, Bob, who in turn has three friends. Chloe's friends are Bob, who has three friends, and Dave, who has two, which means that Chloe's friends have five friends between them (even though their lists of friends overlap). The situation is analogous for Dave. Bob's friends, Alice, Chloe and Dave, have five friends in all. So the total number of friends of friends is 18. But the total number of friends in the network is eight, as before. So the average number of friends of friends (ie, how many friends each person's friends have) is 2.25 friends each (18 divided by eight), more than the two friends, on average, of the four people in the network. The reason, of course, is that Bob, who has most friends in the first place, is also counted most often in the friends-of-friends category, raising the average. The same is true for other networks: a few well-connected individuals have more friends than most people, and they skew the average for everyone in whose network they appear (which, because of their connectedness, is a lot of people).

This number-crunching has some intriguing consequences—other than to justify not getting worked up about your relative social status. During the H1N1 flu outbreak in 2009, for instance, Nicholas Christakis of Harvard University and James Fowler of the University of California, San Diego, kept tabs on a large group of randomly picked Harvard undergraduates. They also monitored the people those participants named as friends. Remarkably, the friends became ill about two weeks before the random undergraduates, probably because they were, on average, better connected. With the world only imperfectly prepared for a pandemic, being able to spot trends in this way could be useful.

This article is extracted from The Economist Magazine (Online content).