"So for example, the transition 4->1->5->6 is one of the most popular ones" The graph does not show this or I'm misunderstanding something. The most popular looks like 1,4,1,5 Also, what does the double, triple numbers mean? 664, 66 in the fist column. I know some music theory, but the article's explanation is rather confusing.
the first number is the chord number (which should have been notated as a roman numeral, and should have differentiated between major and minor with capital and lowercase). The rest of the arabic numberals indicate the inversion using figured bass http://en.wikipedia.org/wiki/Figured_bass
'There are several limitations to this assessment since the Hooktheory API wasn’t really intended for this type of analysis. For example, it doesn’t mention whether “6” is “vi” (minor) or “VI” (major), which is kind of a big deal.'