š š tiger tiger for ultimate November, also St. Andrew’s Day (Scotland’s national day); meanwhile, I bring you two dinosaurs trading ideas about popularity and sluttiness
A pair of Ryan North’s Dinosaur Comics strips, coming in succession on 11/10 and 11/13, in which T-Rex rambles on to his buddy Utahraptor about a fairly well-known paradoxical-sounding phenomenon in social networks, the friendship paradox. Actually, it applies more generally, and I’ll talk you through the (apparent) paradox in the general case. Yes, I’ll get to the comics, and to the way T-Rex uses the adjective slutty, but first let’s talk about your lunch partners.
The symmetric-relation paradox. Brace yourself for some mathematician-talk, but don’t despair: I’ll work up a concrete example (about you and your lunch partners) along the way.
Consider a a set N (for example, the set of people in a social network) and a symmetric relation R between members of N; R might be being friends with, say, or having gone to grade school with or — my concrete example —Ā having had lunch with. Then for any member m of N (like you, for definiteness), define m’sĀ R:N-cohortĀ to be the set of members of N that m bears R to (like, the set of all your lunch partners), and m’sĀ R:N-indexĀ to be the size of m’s R:N-cohort (like, how many lunch partners you’d had). Then it can be shown that, on average, the R:N-indices of members of m’s R:N-cohort are greater than m’s R:N-index — like, on average, the number of lunch partners your lunch partners have had is greater than the number of lunch partners you have had. Yes, it sounds paradoxical. But it’s provably so.
Now, listen up: what the symmetric-relation paradox does not say is that (all) your lunch partners have more lunch partners than you do. That would be genuinely paradoxical. All it says is that the (arithmetic) mean of their lunch-partner figures is higher than yours, which is a great deal less thrilling (though it still has a whiff of the perverse about it). So let’s look at the special case, the friendship paradox,Ā where N is a social network and R is the being friends with relation (which is where T-Rex starts in his Dinosaur Comics ramble, before he goes on to the having had sex with relation (parallel to the having had lunch with relation) and to sluttiness, having had many sexual (rather than lunch) partners.
