Your Friends Have More Friends

Here is a claim that sounds like an insult and is a theorem. On average, your friends have more friends than you do.

Not your friends in particular, and not because you are unusually solitary. In almost every social network anyone has measured, most people have fewer friends than the average among their friends. Scott Feld, a sociologist, set out why in 1991, in a paper called Why Your Friends Have More Friends Than You Do, and the reason is not psychological at all.

Take a small group. One person, call her Ana, knows four others, and those four know only Ana. Ana has four friends and each of the others has one, so the average number of friends in the group is eight divided by five, or 1.6. Now ask each person about their friends. Ana's friends each have one. Each of the other four has a single friend, Ana, who has four. Four people out of five have a friend more popular than they are, and the one exception is the most popular person there.

The mechanism is in the way you reach people. When you look at the population directly, everyone counts once. When you look at it through friendships, someone with fifty friends is counted fifty times and someone with two is counted twice. Sampling by following a link over-represents the well connected, in exact proportion to how well connected they are. Statisticians call this size-biased sampling, and the friendship paradox is what it looks like on a social network.

How big is the effect? It grows with inequality. If everyone had exactly the same number of friends, the paradox would vanish: your friends would have precisely as many as you. Averaged over friendships, the gap between the two figures equals the variance of the friend counts divided by their mean, so the more unequal the network, the bigger the gap. Social networks are very unequal, with a few people knowing enormous numbers, which is why the effect is large and nearly universal.

Once you see the mechanism you find it everywhere.

A college runs one lecture of 100 students and nine seminars of 10. Its brochure can truthfully say the average class has 19 students: 190 students in 10 classes. Ask the students instead. A hundred of them are in the big lecture and ninety are in small seminars, so the average class a student sits in has about 57 people in it. Both figures are correct. They answer different questions, one about classes and one about students, and the brochure chose the question with the flattering answer.

The gym seems full when you go because you are more likely to be there at the times when many people are, and so is everyone else. The bus seems always to be late because you are more likely to arrive during a long gap between buses than a short one; if buses come at random, with an average gap of ten minutes, your average wait is not five minutes but ten. Each of these is the same slip: an average taken over the experiences of individuals, compared with an average taken over the things they experience.

The useful turn is that a bias you understand is a tool. If you want to find the well connected in a network you cannot see, you do not need to see it. Pick people at random and ask each to name a friend. The friends are, on average, better connected than the people who named them.

Nicholas Christakis and James Fowler tried this at Harvard during the 2009 flu pandemic. They took a random sample of students, asked each to name friends, and followed both groups through the autumn. The epidemic reached the named friends about two weeks before it reached the random sample. The friends were not more careless or more sickly. They were more connected, so whatever was circulating got to them earlier, and a monitoring scheme built on them gets a warning that a random sample cannot give.

The same idea applies to vaccines. If you have only a few doses and want to slow the spread through a network you cannot map, giving them to people chosen at random wastes most of them on the poorly connected. Give them instead to a randomly chosen acquaintance of each person you picked, and they land disproportionately on hubs. Reuven Cohen, Shlomo Havlin and Daniel ben-Avraham showed in 2003 that this acquaintance strategy beats random immunisation by a wide margin in networks where a few people have very many contacts.

So the paradox has two faces. Looked at one way it is a reason not to measure yourself against what your friends seem to have, since you are comparing a fair sample with a biased one. Looked at the other way it is one of the cheapest instruments in epidemiology, and it costs nothing but a question: who is your friend?