Six degrees, more or less, sort of

In reference to a previous post on degrees of separation, I went looking through Wikipedia and found what I was pretty sure I'd seen before about graph theory and the "small world" phenomenon. A few points:

• Actual social networks, and a wide variety of similar networks from all sorts of different fields, don't act like classic random graphs, where each object in the network has about as many connections as any other. Rather, there tend to be a few objects with lots of connections and a lot with relatively few connections. But ...
• ... not a lot is known for certain, especially when it comes to networks of real live people. How connected people are depends on what kind of connections you count. There have been various efforts to measure connections in networks like FaceBook and from looking at instant messaging traffic (anonymized, I would hope!), but it's not clear what you can or can't tell about people in general from that. However ...
• ... networks in general seem to behave roughly similarly, keeping in mind that nice, regular networks are the exception both in theory and real life. In particular, as you add to the network, the diameter (the largest number of hops required to connect two objects in the network) tends to increase logarithmically. To double the diameter, you generally have to square the number of objects. One intriguing result is that if you start with a nice, regular network with a high diameter and add just a few connections here and there, the diameter drops sharply. So ...
• ... the "small world" phenomenon looks to be a general property of networks and not necessarily a product of our modern age. My hunch about human bandwidth is that there are only so many connections a person can keep up and that limit was hit quite some time ago. The question is more to what extent scattered groups of people might be more connected than in times past. And finally ...
• ... the famous "six degrees of separation" is not some sort of deep magic but rather a shorthand description of some early and not particularly rigorous experiments that suggested that most people are probably more or less six hops from each other. There's nothing particularly special about the number six, there are still small pockets of people who are not meaningfully connected to the outside world at all, and in any case the actual upper limit might just as well be twelve or five, depending on how you count (leaving isolates aside).