There are many nuances to the definition of "Rivalry", but according to the dictionary, it's "competition for the same objective or for superiority in the same field.". But it really doesn't take into account tradition.
Harvard - Yale
Duke - UNC
Red Sox - Yankees
UCLA - USC
Army-Navy
Some rivalries go back a century or more, and those are the truest IMHO. So if you look at UCONN's longest rival, it's really Tennessee. We had to knock them off in 1995 to get to the top, and there have been MANY battles ever since, fortunately with UCONN coming out on top more times than not. Plus comparing Geno vs. Pat - NC's, final 4's, total wins, winning%, etc etc. And the fan bases dislike each other as well, so to me that's the truest long-time rival. We now hold a 14-9 edge over them, 23 games in our history.
Once Tennessee started faltering and the teams stopped playing, Notre Dame became our fiercest rival. Both teams were again near the top of the rankings every year and we faced them regular season, BET, and NCAA's.
ND has since fallen off, and we haven't played them in a while. Although we are 38-12 all time. Stanford, to me, hasn't really been a rival because we don't play in their conference, and we have played them 18 times going 11-7. There have certainly been some memorable games of great importance.
So now, the 3 best programs seem to be UCONN, Stanford and SC. All 3 are recruiting gangbusters and IMHO will be final 4 fixtures over the next 3-4 years. Everyone else will be fighting for the 4th spot. But we really don't play Stanford, and only SC has been schedule the past few years. Our record vs. SC is 8-1, but we know they are very good and recruiting well, but is it a rivalry, or just a big important game?
I guess it depends how you really define "rivalry"...