Do you ever feel like your friends are more popular, richer, and happier than you?

Well it turns out they probably are. But don't go home and cry yourself to sleep, because paradoxically this is somehow true for almost everyone.

Two social network scientists, Young-Ho Eom at the University of Toulouse and Hang-Hyun Jo at Aalto University in Finland, have figured out a simple mathematical explanation for why.

The friendship paradox

The fact that most of our friends have more friends than we do is old news. Scott Feld figured out in the 90s that most people have fewer friends than their friends. He called it the friendship paradox.

Compare the number of friends one person has to the average number of friends their friends have, and the second number is always bigger. Most of us only have a few friends, but some people that have tons of friends. It's those super-popular people that creates the paradoxical effect.

It makes sense when you think about it: People who have a ton of friends are more likely to be your friend in the first place. They have a greater tendency to make friends. People with a lot of friends drive the average number of friends up in tons of other people's social networks because they are connected to so many other social networks.

In the image below you can see the paradox illustrated. Sue has the most friends and is driving the average up among the group. The number above each name is how many friends each girl has, the number in parentheses is the average number of friends her friends have.

Everyone but Sue and Alice, who are the super-popular girls, has fewer friends than their friends do.

You can apply the same idea to online social networks like Facebook and Twitter. The people a person follows on Twitter almost always have more followers than they do, because we are more likely to follow users who are popular.

The same is true for intimacy. Your sexual partners have probably had more partners than you have. The more partners someone has the more likely they are to eventually become your partner.

But how do network scientists know that our friends are not just more popular than us, they're richer and happier too? It turns out the friendship paradox not only applies to the sheer numbers of a given social network, it applies to other characteristics of the group too.

Other characteristics of a social network follow the same paradox

Eom and Jo have worked out the mathematical conditions under which a characteristics like wealth and happiness of a social group will follow the same trend as the friendship paradox. Their paper was published on the open access pre-publication database arXiv.

The scientists examined the networks of physicists and "network scientists" like themselves, counting how many co-authors each scientist had on papers they had written. True to the friendship paradox each scientist's co-authors had a higher average number of co-authors than they did.

But here's where it gets interesting. Your co-authors not only have more co-authors than you, they also have more citations and more publications. Eom and Jo are calling this the generalized friendship paradox (GFP). The scientists suggest that other features of social networks, like wealth and happiness, are likely to behave the same way as citations and publications.

They call these "high characteristics" and they have a positive correlation with the number of friends of friends, i.e. your friends are wealthier and happier than you. These kinds of characteristics of a social network follow the same kind of pattern that we see in the original friendship paradox, the scientists say, and it happens because we're looking at a biased sample of our friends.

This paradox can make us feel bad about ourselves

"When we compare our characteristics like popularity, income, reputation, or happiness to those of our friends, our perception of ourselves might be distorted as expected by the GFP," Eom and Jo say in the paper.

It's similar to going to the gym and feeling like the most out of shape person there. The reason that everyone around you is so in shape is because they're at the gym all the time — that's why you're seeing them. Everyone else is at home relaxing and not getting in shape. You're looking at a very biased sample of people.

In our social networks we are seeing a group biased toward those that are more socially-active. Wealth and happiness likely follow the same pattern — we see a biased sampling of those characteristics too. So when we look at our friends we see a group of people that are much more popular, wealthy, and happy and it has a big impact on how we perceive ourselves.

Online social networks make it even easier to see how much better off our well-off friends are.

There are several studies showing that heavy online social network users are actually less happy than lighter users, and the researchers think constantly seeing updates from this biased sample of friends that are more popular, wealthy, and happy than them could be the reason.

So next time you're worrying about the number of twitter followers you have or how much happier your friends seem than you, remember that most people are in the same boat.