1: oneurope - only friend of the temporary nick_at_esc account.
2: sjbranford - immensely famous UK Harry Potter fan.
3: hanacandi - famous UK Harry Potter fan.
4: ivyblossom - very famous Harry Potter slash writer.
5: drbear - game show fan.
6: aegeus - famous Harry Potter fan.
7: julietk - Oxbridge posse person.
8: perfectlyvague - friend of mrstrellis and daweaver.
9: legomymalfoy - famous Harry Potter fan.
10: jlh - famous Harry Potter fan.
11: skykid - game show fan.
12: chrestomancy - Oxford RPGsoc type.
13: calnhobbes - syndicated comic.
14: dilbert_feed - syndicated comic.
15: mon_starling - very famous Harry Potter (etc.) fanartist.
16: helenbr - Oxford Invariant Society type.
17: msscribe - famous Harry Potter fan.
18: pegkerr - famous author associated with the Harry Potter fandom.
19: praetorianguard - famous Harry Potter fan.
20: iscaris - very famous Harry Potter slash writer.
...so a lot of different origins for people to be on the list. Now compare that to the top twenty entries on my popwithfriends chart (which might be a paid-members-only tool):
...all of whom are Harry Potter fans. (I think all fifty on my popwithfriends are associated with Harry Potter somehow.) The difference is interesting. It's also easy to understand; the Harry Potter fandom is very big and cohesive compared to my other spheres of interest on LiveJournal - there are lots of people who all link to each other. (I would further note that popwithfriends is very silly for not recognising that I have chosen not to befriend myself, and that the reason why I haven't beFriended owlman, lovely bloke that he is, is that his journal is just about Friends-only and he deFriended me some time ago. No, no angst between us.)
Reading through the explanation, I have a wild-assed but plausible guess as to how the trust metric works in practice, in simple(?) terms. Some of the details are likely wrong (for instance, it may treat communities and syndicated feeds differently; it may also do clever things about not counting you and your Friends where they ought not to be counted) but I think I get the general principle.
Everyone gets one vote, which they split evenly among all their Friends. If you only have one Friend - for instance, nick_at_esc - then that one vote is counted towards your single Friend's total; in this case, that's a full vote for oneurope. A full vote is a lot in this.
If you only have four Friends - for instance, mrstrellis - then that one vote is split among your four Friends' totals; in this case, that's a quarter vote for perfectlyvague. (She also picks up 1/88 of a vote from daweaver.) In a similar fashion, tkb has only five friends, so transmits one fifth of a vote to julietk (who picks up sundry fractions elsewhere too).
Conversely, if you have mad numbers of Friends - for instance, queerasjohn - then you transmit 1/328 of a vote to each of your 328 friends in this scheme. Truly it is One Member, One Vote. Accordingly, owlman is picking up 1/328th of a vote here, 1/189th of a vote there, 1/76th of a vote from someone else and so on from 37 different people, with these 37 people's votes adding up to enough to place him highly in the list.
Anyway, so each person sends a single vote flying around, whether as a one-vote block or 328 0.003-vote splinters. When you put your username into the TrustFlow metric box, it looks at all your friends, works out where they are casting their votes, adds up the votes that each friendsfriend has received and ranks the friendsfriends in descending order.
Some might argue that the TrustFlow metric gives too much importance to the opinions of those who only have a very small number of friends; some might argue that popwithfriends gives too much importance to the opinions of those with a very large number of friends. It's possible that a Third Way metric, where votes are cast in proportion to (eg) the number of Friends to the power of -0.5, might give "more accurate" (in quotes because it's always going to be objective) results still. It's possible that people might judge the results of a system which only casts votes when the two people have beFriended each other to be more useful still.
It would be interesting to take the system a step further and provide the rankings of the people who were your friends already as well, though of course I can't think of a way to properly credit a very close friendship between two people who happen to move in completely different LJ circles. Perhaps a metric involving credit for common interests, for geographical closeness, for reciprocal mentions (hmm - hard to distinguish positive ones from negative...) and for pointing to the same URL or user might bring more joy still? I'm also sure it's got to be possible to adapt this further and come up with another LiveJournal Top 40 still - or a Top 40 of those who have declared an interest in Harry Potter, for general values of Harry Potter.
There's one interesting little point raised at the end of this thread - LJ's search operation, er, operates in what some might consider a curious way. Some time back we were discussing the LiveJournal Browser and commenting that it didn't seem to pick up all the friendships that were in operation. I suspect that this may explain the anomaly; it seems to work well at getting friendships for people where there is mutual Friendship and less predictably where the Friendship is one-way only.
As ever, much fanpersoning to all those who spend time, effort and thought on these LJ toys for our fun and benefit. :-)