Eight friends are enough.

Paper: http://www.cl.cam.ac.uk/~jcb82/8_friends_paper.pdf

The paper argues that the Facebook tradition of showing eight friends in a profile page reveals far too much. There is also an interesting view about applying graph theory to social networks.

There was one part I particularly found interestesing regarding the degree in relation to Statistics..
Specifically talking about statistics, the paper analyzes, "Information about the degree d of nodes in the net- work is exposed in the sampled graph, particularly for low-degree nodes. This is not surprising, because nodes with d < k must be shown with fewer than k links in their public listing, leaking their precise degree. For most social networks, however, nding the set of users with few friends is less interesting than nding the most popular users in the network, who are useful for marketing purposes. Degree information is leaked for users with d < k because they will likely be displayed in some of their friends' listings, meaning that there will be (greater than/equal to) k edges
for most nodes in the sampled graph."

Who would've thought!

Here is the Conclusions section of the paper:
We have examined the difficulty of computing graph
statistics given a random sample of k edges from each
node, and found that many interesting properties can be
accurately approximated. This has disturbing implica-
tions for online privacy, since leaking graph information
enables transitive privacy loss: insecure friends' pro les
can be correlated to a user with a private pro le. Social
network operators should be aware of the importance of
protecting not just user pro le data, but the structure
of the social graph. In particular, they shouldn't assist
data aggregators by giving away public listings.