Some of my recent work has led me to think about the concentration of measure phenomenon. The standard example for concentration of measure is that most of the surface area of a (high dimensional) planet is close to the equator. This led me to try to think of real-world examples of the phenomenon. There are plenty of realistic situations supporting the phenomenon, but I’m looking for more striking examples. More precisely, I would like an example where people have noticed the “unusually high” concentration and have sought or proposed explanations when there really isn’t much to explain.
The most obvious would be the fact that most of the world’s population lives around the equator. However, this has a completely different valid explanation (as witnessed by the concentration of population in Canada) related to weather and similar issues. Population density doesn’t appear to be nicely distributed along meridians either, mostly because humans tend to live on land rather than water. Looking for examples in terms of population density looks like a dead end.
Concentration of measure also occurs in graphs. On the vertices of the unit -cube, concentration of measure says that most vertices are nearly half-way between and its antipodal point . (Indeed, the central limit theorem predicts that most vertices have equally many ‘s and ‘s with an error proportional to .) One real world graph is the so-called social web of human connections. The small world phenomenon says that any two people are at most 6 steps apart in the social web. Could this be a manifestation of the concentration of measure phenomenon?
I’m not sure. I always thought that the numbers for the small world phenomenon didn’t add up. (Though that strongly depends on which numbers you decide to use.) Looking at the small world phenomenon as a manifestation of concentration of measure seems to give enough wiggle room for the numbers to fit together, but this is just circumstantial evidence. Is there any concrete evidence that supports this point of view?