On per-site metas, users are given a rank based on meta participation, and this ranking is seen on the default tab of the users page on each per-site meta. How is this participation numerically calculated to give a ranking? Is it by votes? Something else? Are community wiki posts included in this?
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This is pretty much answered by the tooltip you get when you hover over the "participation" sort link:
-- that's all it is. Take the sum of those four values and sort the users based on those sums. The only minor details that are not in the tooltip are things like "don't count deleted stuff", and the fact that "votes" includes answer acceptance and that "edits" excludes edits to your own posts.
Contrary to ire_and_curses's observations, whatever the secret sauce is seems to be working. I reviewed the participation tab for a site I moderate and the rankings make perfect sense. All the top users are clearly the most active folks on our meta, and I don't see anybody listed who is a non-participant.
The exact workings of the mechanism remain a mystery. I think we need an official voice to chime in with the backend details. Even though meta reputation is typically just a reflection of the main site, your posts on meta do have votes and the reputation number calculated from these used to be discoverable. I don't know where it is now, but it could well be a factor. Also the raw number of posts, comments, edits, votes etc may be a factor.
A good hint as to what is likely considered can be found on Election nominations. Some detailed metrics are shown under each entry. You can check the latest SO election for a sample:
I think it is probable that the magic sauce you want to know about is a formula applied to that data with some kind of weighting applied.
I'm not so sure those users are being ranked. On the board and card games meta users tab, user Daenyth is placed 4th, despite a low main site rep, having never asked or answered a meta question, and with only 18 votes.
Compare this for example, to my 'position' (8th), with more participation in every category, and the ordering hypothesis doesn't seem to make much sense.