First, pick one person at random from a whole social media network; then pick someone that they follow. Thought about in another way, by picking two linked people, what we are actually doing is picking a random link from all the followership relationships in the network. In graph theory, these links are called the edges of the graph.


Now, since popular people have (by definition) more edges going into them, we are more likely to find a popular person on the end of any given edge.

Thus, a randomly chosen friend of a randomly chosen person (the person at the end of the edge) is likely to have more friends than a randomly chosen person, and the friendship paradox holds.

That is the mathematical theory.

So how does it work out in practice? Kristina Lerman, Research Associate Professor at the University of Southern California, decided to find out. She and her colleagues took the network of Twitter users in 2009 (at an early stage in the life of the social network when it had only 5.8 million users) and looked at follower relationships.

They found that the people followed by a typical Twitter user had around 10 times as many followers as they themselves had. Only 2 per cent of users were more popular than their followers.

Read more about social media:

Lerman and her colleagues went on to find another result that completely defies our intuition. She found that the followers of a randomly chosen Twitter user were on average 20 times better connected than they are!

While it appears reasonable that the people we follow are popular – many of them are celebrities after all – it is much more difficult to grasp how the people that follow us have become more popular than us. If they are following you, how can they be more popular? It doesn’t seem fair.

The answer lies in our tendency to create mutual followed/follower relationships.

There is a social pressure to become ‘mutuals’ when someone follows you. It is rude not to follow them back. On average, the people who follow you on Instagram, or send you friend requests on Facebook, are also likely to have sent similar requests to others. The result is that these people make up a larger part of our social network.

It gets worse. The researchers also found that your friends post more, get more likes, more shares, and they reach more people than you do.

Once you have accepted the mathematical unavoidability of not being popular, your relationship with social media should start to improve.

You are not alone. Kristina Lerman and her colleagues’ study estimated that 99 per cent of Twitter users are in the same situation as you are.

Read more about friendships:

Indeed, popular people may well be even worse off than you are. Think about it. In their never-ending search for a better social position, the ‘cool kids’ are trying to become mutuals with people who are more successful than they are. The more they do so, the more they end up surrounded by those who are more popular than they are.

More like this

It’s a small consolation, but it is good to know that even those who appear successful probably feel the same way you do. With the possible exception of Piers Morgan and J. K. Rowling, the remaining 1 per cent of Twitter users are either PR-managed celebrity accounts or, very likely, people who have been driven half-crazy by the need to keep up online appearances.

David Sumpter will be joining us 3 December 2020 for a virtual lecture on the mathematics of life. Get your ticket for just £10.

David's new book, The Ten Equations that Rule the World, is out now (£20, Allen Lane)

Cover of The Ten Equations that Rule the World © Allen Lane