Social loops


It has happened again!

I’m friends on Facebook with people called TK and VBV. They have never met, don’t know each other, and aren’t Facebook Friends™.

There exists a person with initials PC, whose Facebook profile I noticed because he’d organized a tournament of Through The Ages. Now PC is friends with both TK and VBV, but I have never met PC nor knew about his existence until about five minutes ago.

Interestingly, TK and VBV are in a similar situation: each of them knows both me and PC, while not knowing each other.

Such situations, in which each of the four people knows exactly two others but not the third, apparently arise pretty often. I’m not a very advanced Facebook stalker, yet I’ve seen this happen at least five times. Wild, isn’t it?

Some back-o-the-napkin calculations:

Say there’s 10 million Czechs.1 Say an average Czech has 200 other Czechs friended on Facebook.2 Say the distribution is random. To count the number of friends of friends, 200×200=40k, but I guess there’s vast overlap among those, so say 10k friends of friends per person? So for each friend-of-a-friend, there is a one in 10m/10k=1k chance they’re also a friend-of-another-friend. With 10k friends of friends, about 10 should be connected through two different friends.3

However, the distribution is not random. The people one hangs out with and the other people one hangs out with probably have something in common. With TK and VBV and PC, we all come from towns with 10k+ inhabitants, and each of us has lived in one of the two largest cities in the Czech Republic.4 This clustering further increases the number of shared friends-of-friends. So perhaps this is not so surprising after all?

  1. For simplicity, let’s limit this to the Czechs. I’ve seen this happen with a person where one of our common connections was a Czech non-go-player and the other was a foreign go player, which is even wilder. 

  2. In 2012, an average Facebook user had 245 friends according to WaPo. By 2019 supposedly 338 friends on average. Not all of them will have been Czechs. Furthermore, I don’t care about accuracy. Ballpark is fine. 

  3. For 10k unique friends of friends, it’s 10 connected through two different friends. For 20k it’s 40, for 30k it’s 90, and for 40k it’s 160. The number of shared friends of friends grows exponentially with how many friends people have. 

  4. In fact I’m the only one of us four who has never lived in Prague! And it just began to dawn on me – maybe I’m the weird one! They all lived in Prague, no surprise some of them met. But, how did it happen I met both VBV and TK? 

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