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Your friends probably have more friends than you. Don’t worry, it’s nothing personal. It’s just about how networks organize themselves.
We can represent a group of friends as a network. Draw a node (dot) for each person and a line between two nodes if the two people are friends. By doing this for a group of people interacting 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 do we have to 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 go to your astrophotography club. It’s likely that many of these people are friends with each other, so many of your “friends of friends” in the group are also direct friends.
But there are also far-reaching connections. Your old friend who moved to another country has his own tight circle of friends who all go to their soap cutting club. All of these people are your second-degree connections, even if you’ve never met them.
This is the source of the famous claim of six degrees of separation. If you follow the more distant connections, you can quickly reach beyond your own network. An old colleague who took a job in London goes to war with a barista who works near Parliament, and suddenly you’re just a few degrees away from shaking hands with the Prime Minister.
What about popular people? In a friendship network, some 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 chance that they are friends with Sandy and only a ¼ chance that they are friends with Charlie. Your friends are not a random selection from your group: you are more likely to be friends with the more popular people, and therefore find that your friends have more friends than you.
This phenomenon, called the friendship paradox, can be useful when trying to find influential individuals. If you select a random sample from a group of people, you would expect them to have an average number of connections. But if you ask them to name a friend at random, chances are they’ll 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 connections, 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
These articles are published every week at
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