friendship paradox

Your Friends Probably Have More Friends Than You Do: Using Math to Explore the “Friendship Paradox”

Your friends are likely to have more friends than you do, and math may be able to prove it, according to a theory proposed by a team of researchers who decided to explore what is known as the “Friendship Paradox.”

It’s reasonable to assume you have roughly an average number of friends, although some people’s friend circles are more extensive while others are smaller.

Enter the Friendship Paradox. First proposed in 1991  by Scott L. Field, the paradox dictates that, on average, the friends of individuals in a network will have more friends than those individuals do.

Building on this concept, a 2021 paper published in the Journal of Complex Networks revealed how a team of researchers from the Santa Fe Institute and the University of Michigan, including George Cantwell, Alec Kirkley, and Mark Newman, tackled the issue by creating a new mathematical theory to help us better understand the friendship paradox.

Math to the emotional rescue

According to the study, the researchers discovered that when mathematics is applied to real-world data, it helps reveal a far more detailed picture of the dynamics behind the paradox. For instance, it turns out that well-liked individuals tend to be friends, while less popular individuals also tend to be friends with others who are less popular.

friendship paradox
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“Standard analyses are concerned with average behavior, but there’s a lot of heterogeneity among people,” says Cantwell. “Could the average results, for example, be skewed by a few outliers? To get a fuller picture, we studied the full distribution describing how people compare to their friends — not simply the average.”

In other words, typical studies focus on what only the majority of people do, while a much more comprehensive range of differences exists among individuals. So could the average findings, for instance, be influenced by just a few unusual cases?

Exploring the Full-Range

To better understand the friendship paradox, Cantwell and the team examined the entire range of how people compare to their friends rather than just the most common results.

Based on their mathematical analysis, the data reflects that, generally, popular people are the most likely to be friends with one another. By contrast, less popular individuals are similarly more likely to befriend others who are less popular.

“This has a tendency to amplify the impact,” Cantwell said in a statement. “Although there are certainly other factors at play, approximately 95 percent of the differences within social networks can be accounted for by just these two [examples].”

Cantwell also adds we should all “simply be wary of impressions we get about our success and social status from looking at the people around us because we get a distorted view.”

Cantwell cautions that in the realm of face-to-face social interactions, bias can be alleviated by our tendency to gravitate toward like-minded individuals. However, the digital landscape of social media presents a different view–and often a distorted one–where this phenomenon can become more pronounced, as there are virtually no constraints on the number of individuals who can follow someone online and no real incentive to connect solely with those who share similar interests or perspectives.

“There’s virtually no limit on the number of people who can follow someone online and no reason to only look at ‘similar’ people,” Cantwell explains.

The team’s paper, “The friendship paradox in real and model networks,” appeared in the Journal of Complex Networks and can be read online.

Chrissy Newton is a PR professional and founder of VOCAB Communications. She hosts the Rebelliously Curious podcast, which can be found on The Debrief’s YouTube Channel. Follow her on X: @ChrissyNewton Or