Thursday, January 26, 2012

Lecture 6

Today we focused on the remaining two universal properties of networks -- small diameter and high clustering.  Remember that, in isolation, neither small diameter or high clustering coefficient is particularly "surprising" or difficult to explain.  It is only the combination that is "surprising"... and this is what is termed "small world".

We saw that one possible explanation for small world properties is a combination of local correlation with long-range random links.  However, this combination only explains the existence of short paths, it doesn't explain why we can find them so easily.

To explain this second observation (and to me the more interesting observation) we saw that the long range links have to be proportional to the distance in a very precise manner.  So, it is in some sense much more rare that we can find short paths with myopic algorithms than it is for the short paths to exist in the first place.

Remember though that the models we have discussed for small world graphs DO NOT have heavy-tailed degree distributions!  So, we really haven't given a "natural" mechanism that explains all four of our "universal" properties, we've just given mechanisms for each individually.   Can you think of how to combine our heavy-tailed degree distribution ideas with the small world ideas?

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