Here's a shot:
My own sense is that they might have different stories. Sociology at Arizona has long been "under ranked" and may just face a ranking ceiling due to the (lack of) university prestige. In their case, rather than reflecting a department "on the rise" it might just reflect a great training department that chronically does better than it's ranking?
In the case of Brown, it seems more likely that it is a reflection of an increase in faculty quality and grad training in recent years -- and so may actually foreshadow an increase in department rankings. Being an elite Ivy League school, Brown shouldn't face the same counter current when trying to move up in the rankings.
So, my prediction: Brown will increase more slots in the next US News and World Report rankings than will Arizona. (They're both at 24 now).
Okay, poke all the holes you want in my arm chair theory.
Not bad. But Arizona has been as high as 9 in the US news rankings, their decline to 24 has been the result of a narrative that they can't keep faculty (even if they've been fine at this recently). So their students carry the halo of a previously very highly ranked dept. They also have a reputation as a dynamic training dept for quantitative methods, so I'm sure that helps students in this market.
Brown seems right.
One thing both programs have in common is a small size in terms of grad student body.
My sense is that NYU, Duke, and Brown are upwardly mobile based on many fronts, including strong placement. Arizona seems to be doing well on the placement front still, but they lost way too many faculty to move up anytime soon.