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

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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." 

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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 Live Science in an email.

Their equations show that the friendship paradox tends to be stronger in social networks that are made up of people with very different popularities. If a person with only two friends is in the same social network as a person with 100 friends, for example, in general, the friendship paradox will be stronger in that network than one where the most social person in a network has 10 friends and the least "friended" has three.

The takeaway is that "our social circles are biased samples of the population." It's not exactly clear how that bias may play out in specific cases, but in most cases "it's probably not appropriate to compare ourselves to our friends," Cantwell said. 

Such mathematical equations can help to explain other aspects in society such as election polling and infectious disease spread. "There are several interesting things to explore next," Cantwell said. Some studies have shown that election polling can be improved by asking about people's "social circles," but the findings are observed and haven't mathematically been calculated, he said. 

In addition, the people who you are in close physical contact with are statistically more likely to be in such close physical contact with many other people. So the friendship paradox equations could also help shed light on the spread of an infectious disease. For instance, the friendship paradox has been used in flu surveillance to detect outbreaks on average two weeks earlier than traditional surveillance methods, according to a 2010 study in the journal PLOS One.

"How, exactly, does this affect the dynamics of disease?" he asked. 

The findings were published on May 27 in the Journal of Complex Networks.

Originally published on Live Science.

Yasemin Saplakoglu
Staff Writer

Yasemin is a staff writer at Live Science, covering health, neuroscience and biology. Her work has appeared in Scientific American, Science and the San Jose Mercury News. She has a bachelor's degree in biomedical engineering from the University of Connecticut and a graduate certificate in science communication from the University of California, Santa Cruz.