The maths quirk that can cheer you up if you’re feeling unpopular

Your friends are likely to have more friends than you do. Don’t worry, it’s nothing personal. It’s just about how networks organise.

We can represent a friendship group as a network. Draw a node (dot) for each person and a line between two nodes if those two people are friends. By doing this for a group of people who interact in person or online, we can build a representation of friendship connections.

This network allows us to explore questions such as the number of degrees of separation. If someone is a friend of your friend, they are connected to you at degree 2. Their friends are at degree 3, and so on.

How many links must we follow to get from one person to another? Connections tend to cluster together. Think of a group of friends – people you live near, some of your work colleagues or people who attend your astrophotography club. It’s likely that a lot of these people are friends with each other, so many of your “friends of friends” in the group are also your friends directly.

But there are also far-reaching connections. Your old friend who moved to another country has their own dense cluster of friends who all go to their soap-carving club. All these people are your degree-2 connections, even if you have never met them.

This is the source of the famous six degrees of separation claim. If you follow those more distant connections, you can quickly reach beyond your own network. An old colleague who took a job in London goes wargaming with a barista working near parliament, and suddenly you are only a few degrees away from a handshake with the prime minister.

What about popular people? In a friendship network, some people will naturally have more connections than others. Imagine a group of 20 people, 15 of whom are friends with Sandy and five with Charlie. If we pick someone at random, there is a ¾ probability they are friends with Sandy and only a ¼ chance they are friends with Charlie. Your friends aren’t a random selection from your group: you are more likely to be friends with the more popular people, and so find your friends have more friends than you do.

This phenomenon, called the friendship paradox, can be useful when sampling to find influential individuals. If you choose a random selection from a group of people, you would expect them to have an average number of connections. But if you ask them each to name a friend at random, chances are they will name someone who has more friends than they do. This new group is likely to have an above-average number of connections.

So, if your friends seem to go to more parties, friends at work have more contacts and friends in your art class are in more hobby groups than you, don’t feel inadequate – it’s a quirk of network dynamics.

Peter Rowlett is a mathematics lecturer, podcaster and author based at Sheffield Hallam University in the UK. Follow him @peterrowlett

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