I see the term "social graph" is a bit misleading because it suggests that there is only one type of edge, i. e. one kind of way in which one can know other people. (Indeed, this is a perennial problem on a lot of social networking sites; there's often no way to indicate the difference between, say, a person that one met once at a party and one's spouse.) Fitzgerald acknowledges this:
It's recognized that users don't always want to auto-sync their social networks. People use different sites in different ways, and a "friend" on one site has a very different meaning of a "friend" on another. The goal is to just provide sites and users the raw data, and they can use it to implement whatever policies they want.
The next step in the analysis of social networks might be to take into account the "strength" of relationships, as currently most social networking sites I know of don't acknowledge that I am more interested in some of my friends than others. Also, to what extent can the strength of relationships be guessed just from looking at the social graph without this strength information? A person who I know quite well is likely to be someone with whom I have many friends in common.
Also, the social graph is in some important ways directed (the link goes to Good Math, Bad Math); for example, if edges connect X to Y if X reads Y's LiveJournal. (It wouldn't surprise me to learn that Fitzgerald has this example in mind.) The somewhat addictive tool LJ Connect, which finds the shortest path between two individuals with LiveJournals via their friends, explicitly acknowledges this.
Finally, some shadowy figures seem to be aggregating the mathematics blogging community, and various other blogging communities as well. Their in-house mathematician gives yet another example of using the formula 1+2+...+n = n(n+1)/2.
And some links:
Paul Graham writes about Holding a program in one's head; a lot of his ideas seem to have obvious analogues to mathematical research, which isn't surprising, as programming and mathematics are quite similar. I'll probably have more to say about this later.
There is a graphical formalism for quantum mechanics, which its authors call "kindergarten quantum mechanics"; this is said to be a very considerable extension of Dirac's notation, and gives short derivations of deep results concerning teleportation, quantum mechanics, and so on. If this is true (I'm not familiar enough with the results concerned to say), it's a good example of what Graham has to say about choosing appropriate notation.
Vlorbik asks us to please lie more carefully. He writes:
We were supposed to test the claim that a certain population proportion was 10% against a sample proportion of 13%, based on n= 57 data points (at some stated confidence level that I’ve forgotten). But wait a minute. You can’t get 13% from a sample of 57 subjects: 7/57 ~ .122807 (i.e., 12%) and 8/57 ~ .14035 (i.e., 14%).This particular one isn't a big problem, but it's symptomatic of something more general -- that the "applications" problems in a lot of mathematics textbooks bear very little resemblance to reality, which only seems to frustrate the students. (Another complaint I have about such textbooks is that the calculus texts seem to assume the student is familiar with physics, which is often not a reasonable assumption to make, and so the instructor ends up teaching physics instead of calculus.)
In Nature's Casino, by Michael Lewis, from the New York Times, August 26, about the insurance market for catastrophic events.