Do we have any data on distribution of shortest path lengths between nodes in any real-world networks?

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Facebook, Twitter, genomic network of yeast, etc.

social networks

graphs

society

understand

tynamite

The Small World Experiment is designed to test the hypothesis that anyone in the world can get a message to anyone else in just "six degrees of separation" by passing it from friend to friend. Sociologists have tried to prove (or disprove) this claim for decades, but it is still unresolved.

Now, using Facebook we finally have the technology to put the hypothesis to a proper scientific test. By participating in this experiment, you'll not only get to see how you're connected to people you might never otherwise encounter, you will also be helping to advance the science of social networks.
http://smallworld.sandbox.yahoo.com/index.php?lang=en

The following video is a sample demonstration of replies and retweets over time. “The spiral represents the time axis. Each blue dot is a tweet, with the size of the blue dot representing the number of retweets and replies to that tweet. Each green line shows a retweet to the original tweet and each orange line shows @reply to the original tweet,” according to Sysomos.

http://mashable.com/2010/09/29/twitter-replies-retweets/

Update
You can view the written essay about Yahoo's results here.
http://research.yahoo.com/pub/2765