One of the privacy options Facebook allows is “restrict to friends of friends.” I was discussing with Tom Scocca the question of how many people this actually amounts to. FB doesn’t seem to offer an easy way to get a definitive accounting, so I decided to use the new Facebook Graph Search to make a quick and dirty estimate. If you ask it to show you all the friends of your friends, it just tells you that there are more than 1000, but doesn’t supply an exact number. If you want a count, you have to ask it something more specific, like “How many friends of my friends are named Constance?”
In my case, the answer is 25.
So what does that mean? Well, according to the amazing NameVoyager, between 100 and 300 babies per million are named Constance, at least in the birthdate range that contains most of Facebook’s user base and, I expect, most of my friends-of-friends (herafter, FoFs) as well. So under the assumption that my FoFs are as likely as the average American to be named Constance, there should be between 85,000 and 250,000 FoFs.
That assumption is massively unlikely, of course; name choices have strong correlations with geography, ethnicity, and socioeconomic thingamabobs. But you can just do this redundantly to get a sense of what’s going on. 59 of my FoFs are named Marianne, a name whose frequency ranges from 150-300 parts per million; that suggests a FoF range of about 200-400K.
I did this for a few names (50 Geralds, 18 Charitys (Charities??)) and the overlaps of the ranges seemed to hump at around 250,000, so that’s my vague estimate for the number.
Bu then I remembered that there was actually a paper about this on the arXiv, “The Anatomy of the Facebook Graph,” by Ugander, Karrer, Backstrom, and Marlow, which studies exactly this question. They found something which is, to me, rather surprising; that the number of FoFs grows approximately linearly in the number of friends. The appropriate coefficients have surely changed since 2011, but they get a good fit with
#FoF = 355(#friends) – 15057.
For me, with 680 friends, that’s 226,343. Good fit!
This 2012 study from Pew (on which Marlow is also an author) studies a sample in which the respondents had a mean 245 Facebook friends, and finds that the mean number of FoFs was 156,569. Interestingly, the linear model from the earlier paper gives only 72,000, though to my eye it looks like 245 is well within the range where the fit to the line is very good.
The math question this suggests: in the various random-graph models that people like to use to study social networks, what is the mean size of the 2-neighborhood of x (i.e. the number of FoFs) conditional on x having degree k? Is it ever linear in k, or approximately linear over some large range of k?