The Friendship Paradox
In 1991 the sociologist Scott L. Feld published Why Your Friends Have More Friends Than You Do, observing a network phenomena he termed the “friendship paradox.” The paradox reads like this: the mean number of friends for any individual will (statistically) always be lower than the mean number of friends for an individual’s friends. In other words, on average, people have fewer friends than their friends, or to switch it around, our friends are likely to be more popular than we are.
But how can this be?
The reason is selection bias. Suppose you poll a classroom of students asking each for their friends’ names (as the sociologist James Coleman did in a classic 1961 study cited by Feld); the number of people who consider a student their friend will determine the number of times that student is nominated.
Setting aside the fact that some students may share a name, each student’s name appears in the true distribution of students (the class roster, say) exactly once. However, anyone with one or more friends will, by definition, be named to the distribution of friends at least once. The most popular students, whom many consider their friend, will be nominated repeatedly, while a student without any friends will not be named at all. The list, or distribution, of friends will therefore not match the actual distribution of students.
Clearly, if we select a name from the distribution of friends, our selection will not be random (in the sense of selecting any student’s name with equal odds). Instead, the odds are stacked in favor of selecting a more popular student.
The above network, which Feld assembled from Coleman’s study, depicts this selection bias at work. The leftmost number beside each name represents that student’s quantity of friends and the number to the right (in parenthesis) is that student’s mean number of friends’ friends.
At the network’s periphery is Tina, who can only name one friend, Carol. Carol, in turn, can name two friends: Tina and Pam; and Pam can name three: Carol, Sue, and Dale. Tina will appear in the distribution of friends once, Carol will appear twice, and Pam three times.
Carol’s friends have a mean of two friends each (Tina’s one plus Pam’s three, divided by two), and Pam’s friends have a mean of 3.3 friends. In all, these eight students each have a mean number of 2.5 friends but the students each have a mean number of friends’ friends of 2.98125 each.
Already, we see the friendship paradox playing out. Tina and Carol both can name fewer friends than their friends are able to (as can Betty, Jane, and Dale, for a total of five out of eight students with fewer friends than their friends’ friends). Someone in Carol’s position, being Tina’s only connection to the rest of the group, can always name at least two friends, twice that of someone in Tina’s spot (in the case of Betty and Sue, the ratio becomes four to one).
This phenomenon holds true for any network. Peripheral nodes will have a degree as little as one (degree meaning the node’s number of connections), but the nodes connecting the periphery to the rest of the network will always be of degree two (or higher ).
In effect, any random node’s connections are, on average, more central than that node (which is the same as saying a random person’s friends are more popular than that random person). For instance, rather than mapping out Curb Your Enthusiasm’s entire cast of characters (see above), we can quickly reach Larry from any random point just by examining that character’s connections (and possibly their connections’ connections).
The friendship paradox has many corollaries. Because our friends are more popular than we are, they also enjoy larger shares of popularity’s many perks. For instance, our friends are more likely to be perceived as physically attractive and to earn more money than we do.
A decade ago, the network theorist Nicholas Christakis proposed another such corollary, one with timely implications, when he used the friendship paradox to predict the spread of H1N1 (swine) flu on campus at Harvard (where he was a faculty member). Whereas contact tracing occurs retroactively, after an outbreak begins, Christakis was able to pinpoint who would have the highest risk of transmission in advance by polling students for their friends’ names.
Christakis advises employing the same methodology when prioritizing whom first to vaccinate: select random individuals in a group of people but, rather than vaccinating the people you pick, ask who their friends are and then vaccinate the people they name.
With last week’s announcement that COVID vaccines will initially be in limited supply, it is quite possible that the friendship paradox will save lives, maybe even the life of one of your friends.
