One of my all-time favorites among all the scientific papers that I have ever read in my life is “Why your friends have more friends than you do,” published in the American Journal of Sociology in 1991 by my old sociology friend Scott L. Feld, who is now Professor of Sociology at Purdue University.
The title of Feld’s paper says it all, and here’s a little demonstration you can do to confirm his conclusion. List all of your friends. Then ask each of your friends how many friends they have. No matter who you are, whether you are a man or a woman, where you live, how many (or few) friends you have, and who your friends are, you will very likely discover that your friends on average have more friends than you do.
But how can this be? Friendships are bilateral (unless you are a stalker): If X is friends with Y, then Y is friends with X. How can Y and other friends of X have more friends than X does?
Feld demonstrates (and explains) the seeming paradox with a simple example in his paper. In this example, there are eight girls, and their mutual (bilateral) friendships are denoted by solid lines in the sociogram. So, for example, Betty has only one friend (Sue), but Sue has four friends (Betty, Alice, Pam, and Dale). The table summarizes the pattern of friendships among these friends. It shows that, on average, these eight girls have 2.5 friends. But the friends of these eight girls (who are the same eight girls themselves) on average have 3 friends.
If you think about it for a moment, you’ll figure out the source of this seeming paradox (although this simple insight did not occur to anyone before Feld published his paper in 1991). You are more likely to be friends with someone who has more friends than with someone who has fewer friends. There are 12 people who have a friend who has 12 friends, but there is only one person who has a friend who has only one friend. And, of course, there is no one who has a friend who doesn’t have any friend. Yet there is actually only one person who has 12 friends. So “12” gets counted only once when you compute the average number of friends that people have, but it gets counted 12 times when you compute the average number of friends that their friends have. Hence the seeming paradox that your friends have more friends than you do.
This, incidentally, is the reason why a man often gets depressed after he has sex with a woman for the first time and then she tells him how many lovers she has had because she has had more lovers than he has. A mating version of Feld’s discovery may be termed “Why your lover has had more lovers than you have.” And the reason is the same. There are 12 men who have had a lover who has had (or will have had) 12 lovers, but there is only one man who has had a lover who has had only one lover. But you should be grateful. The reason you got to be her lover in the first place is because she has had (and will likely have) many lovers. You are 12 times as likely to have sex with a woman who has had 12 lovers as you are to have sex with a woman who has had only one lover. Quite paradoxically, if your lover only had one lover, you are probably not him. And if your lover has never had a lover, you are definitely not him.
There is also an intergenerational version of Feld’s dictum (although it is expressed less elegantly): “Why our mothers had more children than women in her generation did.” There are 12 children whose mother had 12 children, but there is only one child whose mother had one child. And, of course, there are no children whose mother had no children. Yet there is only one woman who had 12 children. So if we ask around how many children everyone’s mother had (or how many siblings we have), we get the erroneous impression that our mothers were much more fertile than they actually were. Feld’s original and highly insightful observation can explain these and many other seeming numerical paradoxes.