Named in a paper
The sociologist Scott Feld described the effect in "Why Your Friends Have More Friends Than You Do", published in the American Journal of Sociology in 1991. He showed the cause is the maths of averages, not envy or low self-esteem.
Count your friends, then the average number of friends they have. In almost any network your friends come out more popular. It's not about you: it's how the average works. A person with 50 friends is counted 50 times, a person with one friend just once. Check it on a 36-person network, on networks of thousands, and through five explanations.
The friendship paradox was described by the sociologist Scott Feld in 1991. It isn't a contradiction. It's called a paradox because the conclusion sounds like an insult, yet it follows from simple arithmetic.
There is a group of people: a class, a company, a social network. Some pairs of people are friends. Friendship is mutual: if Anna is friends with Boris, Boris is friends with Anna.
You count how many friends you have. Say 4. Mathematicians call this number the degree of a node.
You ask each of your 4 friends how many friends they have and take the average. For example, 3, 5, 9 and 15 average 8.
For how many people in the network do their friends have more friends, on average, than they do? And does a random friend have more friends than a random person?
"Friends are ordinary people too. On average it should come out even"
In fact, in any network where some people have many friends and others few, a random friend is always more popular than a random person. They are equal only when everyone has the same number of friends.
For follows, as on Twitter/X, where links go one way, a similar effect applies. More in round 5.
Take a typical social network, with a few very popular people and many people with a handful of friends. What share of people have fewer friends than their friends do on average? Lock in your answer.
Your answer will be compared here with a simulated network of 3,000 people.
Each circle is a person, each line a friendship, and the number is how many friends they have. Magenta marks people whose friends have more friends on average. Switch to the "fair" network, where everyone has exactly 4 friends, and the paradox disappears.
The computer builds networks with the same average number of friends, 6, but different inequality. In the "fair" network everyone has 6 friends. In an Erdős–Rényi random network pairs become friends at random. In a Barabási–Albert network newcomers prefer people who already have many friends, so "stars" appear. The line is the formula; the dots are simulated networks.
Horizontal: inequality, the coefficient of variation CV, which is the standard deviation of friend counts divided by the mean. Vertical: how many times more friends a random friend has than a random person.
| Network | Mean | Friend's | CV | Below friends | Max |
|---|
The key is how we pick the people we compare with. When we pick a friend, popular people come up more often, because they are friends with many. Here are five ways to see it.
When everyone lists their friends, a person with 50 friends appears on 50 lists, and a person with one friend on just one. So in the list of "friends", popular people show up as many times as they have friends. The numbers come from the 3,000-person Barabási–Albert network in round 1.
Five of the six people have fewer friends than their friend. The network's average is 10/6 ≈ 1.67 friends. But pick one of the 10 entries in everyone's friend lists at random, and that friend has 30/10 = 3 friends on average.
If everyone has exactly k friends, every friend has exactly k friends too, and there is nothing to compare. As soon as some people have more friends than others, the average over friends shifts up. The bigger the spread, the stronger the effect: no difference in the fair network, about one extra friend in the random one, and 2.5 to 3 times as many in a network with "stars".
Almost every real network is unequal: friends, followers, co-authors, flights between cities.
The first line is the ordinary average. In the second, each person is weighted by their number of friends, so d becomes d². The gap between them equals the variance divided by the mean. It is zero only when the variance is zero, which means everyone has the same number of friends. The last line shows the same for the other way of counting: take each person's friends' average, then average over people, and the result is still at least E[d].
The same biased sample shows up wherever we look "from the inside": a student sees their class, a passenger their bus, and a friend belongs to many circles. It can also be put to use: friends of randomly chosen people get sick earlier.
The university averages over classes: each class counts once. Students average over themselves: a class of 210 is surveyed 210 times, a class of 10 only 10 times. It's the same weight d as with friends, and the same formula E[s²]/E[s].
That's why buses, restaurants and gyms feel crowded more often than they are: you tend to be there exactly when many people are, because all those people are there too.
A Barabási–Albert network of 3,000 people. We pick 150 people at random, and each names one friend. Then an epidemic runs: each day a sick person infects each healthy friend with probability 4% and recovers with probability 12%. The chart shows the share of each group that has been sick so far.
| Where | What we compare | Why the average is higher |
|---|---|---|
| Twitter/X | your followers and the accounts you follow | many people follow popular accounts, so they turn up more often in your lists |
| Science | co-authors and their citations | prolific scientists write with many people, so they show up as co-authors more often |
| University | class size | a big class is surveyed as many times as it has students |
| Transport | how full buses and planes are | a full trip has more passengers to notice it |
| Epidemics | when a person gets infected | a person with many contacts has more chances to catch it early |
The sociologist Scott Feld described the effect in "Why Your Friends Have More Friends Than You Do", published in the American Journal of Sociology in 1991. He showed the cause is the maths of averages, not envy or low self-esteem.
In 2011 Johan Ugander, Brian Karrer, Lars Backstrom and Cameron Marlow analysed the Facebook friendship network. More than 90% of users had fewer friends than their friends had on average.
During the 2009 flu epidemic, Nicholas Christakis and James Fowler followed Harvard students. Friends of randomly chosen students got the flu earlier, on average, than the chosen students themselves. That makes friends an early warning signal for epidemics.
Nine seminars of 10 students and one lecture of 210 give an average class of 30. But 70% of students sit in the lecture, and the average class a student sees is 150. Both numbers are correct; they just average different things.
If 9 trips carry 5 people each and one carries 95, the average bus has 14 passengers. But the average passenger rides with about 66 people. It's the same reason a café seems busy exactly when you go there.
Studies of Twitter found that most users have fewer followers than the accounts they follow. This "generalised friendship paradox" works for other traits too, such as the citations of your co-authors.
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