#### Tom Rivlin investigates the science of how most people in the world are connected through a chain of six

We’ve all experienced it – you meet someone at a party, stalk them on Facebook, and see that phrase: *1 mutual friend*. Turns out they went to summer camp with that guy you knew in school. “Wow,” you think, “it’s a small world!” ‘Six degrees of separation’ is the well-known term that describes this phenomenon. It claims that every person in the world is connected to each other through about six links. But exactly how true is this?

As the world has became more and more connected over the course of the past century, this topic is increasingly subject to much research and the results have proved surprising. The phrase itself comes from the 1990 play *Six Degrees of Separation* by John Guare, famously adapted into a 1993 film starring Will Smith. It’s likely that Guare guessed at the number of degrees of separation being six, and for some reason it stuck. So is the key number really six? To find out, we investigated some interesting theoretical models.

#### Graph Theory

The study of networks, called Graph Theory, was first established in 1736 by maths superstar Leonhard Euler. Graph theory takes the idea of ‘nodes’ representing for example, people, and ‘edges’, representing the connections between them. It can be applied to countless systems such as the internet, power grids, crystal lattice structures, and unsurprisingly, social networks.

*A random graph: ‘nodes’ (the round dots) connected to each via ‘edges’ chaotically*

#### So exactly how ‘small’ is our world?

If you drew a graph to represent which of your Facebook friends know each other, what would it look like? It would be highly *clustered*; you would have friendship groups and within these groups most people know each other, so the nodes connect and form clusters. Then, by belonging to different friendship groups, such as your family, your classmates, your old school friends, etc. *you* create a connection between these groups.

*A ‘Small World’ graph – note how the outer ‘nodes’ are clustered, and some inner ones are connected to distant clusters. Your Facebook friendship graph will look something like this.*

In 1998, two researchers named Watts & Strogatz created a model they called the ‘small world’ network, and it was designed to mimic exactly these kinds of social networks (in 1998!). It is a surprisingly good simulation of real social networks, and so by understanding the properties of the model, we can subsequently apply it to the real world.

#### Maths to the rescue

Now we’re at a stage where we can answer the six degrees question – how many links actually connect any two people on the planet? Mathematically, the question we’re asking is what’s the shortest path length between two people going to be, on average? In the small world network model, the answer turns out to be, well, not a very long path. By employing a logarithm (a function that is the opposite of an exponentiation) we take a really big number (like seven billion) and it gives you a really small number. So the general idea of the *six degrees* phrase is right – no matter how many people there are, the world will still be ‘small’. But we were looking for a specific number, and sadly, it’s not six. For the global social network of humanity (7 billion people), the model, and the best research, suggests that it’s *about *10-20ish, which is still remarkably low.

When we claim that everyone is connected, we know for a fact that it’s true. The mathematical predictions of graph theory really do prove that people on the other side of the world should not be seen just as faceless strangers. It’s a good argument for being considerate towards everyone, whether they’re here or thousands of miles away – you don’t know them, but you are definitely closely connected. Research on this topic is still on going in sociology, mathematics, and even physics departments around the world. Are there outliers and exceptions? Definitely, (we’re talking about the *average* shortest path, remember.) But the general result is true: the world is a big place, but because of our friendships, it’s actually a lot smaller than it seems.

*Featured Image Credit: Wikipedia*

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