The ‘friendship paradox’ doesn’t always explain real friendships, mathematicians say

The paradox says that on average, your friends have more friends than you do.

Your friends are on average more popular than you are, according to a phenomenon known as the “friendship paradox.” Now, a group of mathematicians has come up with a new theory that takes the friendship paradox beyond averages, and they found that their equations describe real-world popularity differences among friends.

Sociologist Scott Feld first explained the idea of the “friendship paradox” in 1991 in a journal article titled “Why Your Friends Have More Friends Than You Do.” The general idea — based on a simple calculation — is that the number of friends of a person’s friends is, on average, greater than the number of friends of that individual person.

But “averages are often highly misleading or at least can fail to describe people’s experiences,” said lead author George Cantwell, a postdoctoral fellow at the Santa Fe Institute in New Mexico. “Some people are less popular than their friends, others are more so.”

To understand why, think about a person with just two friends contrasted with a person who has hundreds of friends. Now imagine entering this social bubble: You are more likely to be friends with the social butterfly than the wallflower, simply because there are more “chances” that you are one of the hundreds of the social butterfly’s friends than one of the wallflower’s two best buds. But it’s still possible for you to become friends with the wallflower, and focusing on averages can obscure when that might happen.

Now, Cantwell and his colleagues have developed new mathematical equations to make the friendship paradox better match the range of situations found in real social networks. They based their equations on two assumptions from real-world studies: There’s a significant degree of variation in how many friends people have, depending on the social network analyzed; and popular people are more likely to have popular friends, whereas unpopular people are more likely to have unpopular friends.

The researchers also developed a new mathematical theory to explain another variation of the friendship paradox known as the “generalized friendship paradox,” which states that, on average, your friends are not only more popular than you but also richer and better looking. That’s based on the assumption that popular people are more likely to be rich and good looking than unpopular people.

Their new equations, which accounted for these assumptions, could explain 95% of the variance in real-world situations, Cantwell told.

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