You're looking at different snapshots in time - when Google indexes your homepage that links to 10 internal pages, those will rank a bit better for a while (until the page is indexed again and the PageRank of the target pages are recalculated).
So it would be short lived and chances are it might not even be noticable. Depends on a bunch of other factors. High competition areas for example would mean that the little boost wouldn't be enough to jump up a position.
There's also the question whether or not the loss of the link would count as an additional negative factor, causing the pages then to drop lower in rankings. I've seen this happen on large sites (but in a different manner to what you are describing).
To be honest, the only reason I've ever displayed random links is to get deep content in larger sites indexed a bit faster.
IMO if you want the ranking boost, then link to your most popular pages permanently. That keeps the busy pages ranking well (and perhaps link from them down to poorer performing pages). The permanent approach is always going to be better than juggling the pages around.