Friendship network image

Look to this page http://www.facebook.com/note.php?note_id=469716398919

It is a very interesting entry of a person in Facebook who has made a diagram on the friendship network in the world (using facebook data). I have it right now as my screen saver. It is great.

I am not sure but looking to the diagram I think you can appreciate it is small-world and scale free.

  • Small world: there are clear big connections but between the big connected points there are also clear small connections to many other points. So, the typical patterns of small world is maintained, a big number of links to your neighborhood plus always a very small number to other remote locations.
  • Scale free: It is difficult to say it without looking to the distribution of the number of links coming out of any single point. If this distribution follows a power law  P(k) \sim k^{-c}, then, we can say that the network is scale free. For the Barabassi & Adler scale free networks the constant c is between 2 and 3. Well, this is difficult to say looking to the diagram, but another curious property of these networks is that the number of links per node looks like a fractal, meaning that if you erase the nodes with highest number of links the remaining nodes maintain the link connectivity architecture (the overall aspect of the network). A consequence of that: the distribution of links coming out of a node (what is called the degree of the node) is a heavy tail distribution, that is, it is quite possible to have nodes with very, very small number of links and with very big, big numbers as well. In contrast, with the light tail distributions where it is impossible to have so small and big values (for example, the human heights distribution is light tail as it is impossible to find people  10 m tall or 10 cm tall, but, curiously 🙂 the richness distribution is heavy tail, I don’t need to give examples….)
Well, too much by now…….I hope it was interesting
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