Tuesday, April 07, 2009

The small world phenomenon

I expect that just about everyone on the internet is aware of the concept of sixth degrees of separation, which is the idea that everyone in the world is connected to everyone else by no more than six steps.

The name is based on an experiment by Stanley Milgram, which basically went along the lines: A person gets a letter for a given person. He (they were all male) is not allowed to send it directly to that person, unless he knows the person already. Instead he sends it to whatever friend he has which he considers most likely to be able to get the letter closer to the recipient. The experiment showed that ever letter that reached their destination would go through no more than six people, crossing social and racial boundaries.

This was a really groundbreaking experiment at the time, and even though later evaluations of the experiment have demonstrated many fundamental flaws (see e.g. here), it is still considered the first real experiment relating to the small world problem.

So, what is the small world problem?

Well, simply put, it’s the problem of why elements in worlds are more interconnected than they should be. Let’s take the human population. It’s intuitively easy to understand why everyone is connected to everyone else through their friends, but it’s not intuitively easy to understand how everyone can be connected in just six steps. We are more than sex billion people worldwide after all, and people are both socially and geographically divided.

In other words, how can a person like me, living in Denmark, be not only six steps, at most, removed from President Barack Obama, but also from an Afghan boy in Kabul or an Aboriginal elder in Australia?

If we presume that you are only linked to people close to you geographically and socioeconomically, then it would take quite a few steps to get to any of the people I mentioned, since each link will only get you a little closer to the people in question, especially since friends tend to hang out in clusters, where most people know each other.

Even though this will be true for most of your friends, however, there will be those people who fall outside the pattern. Even though most of your friends are clustered in a few clusters, there will be a few who don’t belong to any of these clusters, and which will connect you to other clusters. In other words, these friends will become a bridge between your clusters and other clusters, allowing people in your cluster to link with those clusters, even if they have never met anyone in any of those clusters.

Suddenly your network is much larger than it appears. And since this goes for any friend of anyone in the cluster which doesn’t belong to the cluster, there are lots of links to other clusters. In other words, people might be clustered together in interlinked groups, but there are plenty of connections between these clusters.

There is obviously a lot of research into small world phenomena, and there are several good books about the subject if you’re interested in learning about this subject.

Duncan J. Watts: Six Degrees: The New Science of Networks
Duncan J. Watts: Small Worlds: The Dynamics of Networks between Order and Randomness
Mark Buchanan: Small World: Uncovering Nature's Hidden Networks
Mark Buchanan: Nexus: Small Worlds and the Groundbreaking Theory of Networks
Steven Strogatz: Sync: The Emerging Science of Spontaneous Order

Note: I plane to write more on the issue of small worlds in the future, but I thought I'd write a small post explaining the basic concept, just to get started on the subject

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