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?

“59 of my friends are named Marianne.” Either you meant FoF here, or you have a serious Leonard Cohen fetish.

Thanks, fixed!

I may be being thick, but wouldn’t you in the absence of any other information expect your friends to have the average number of friends. The mutual friend problem would be a second order (local) effect. I know nothing about modeling social networks, but I’d expect the graph that you get by choosing edges at random from a large complete graph to have the property (as long as you choose few enough edges)

No, you’re being quite perceptive! In a random graph chosen as you suggest (more or less Erdos-Renyi) you’d indeeed see linear growth, I think — but we know that these social graphs are NEVER like that, and the mean number of friends your friends have grows notably — even pretty close to linearly, to the naked eye — with the number of friends you yourself have. In other words, the local effects you postulate are really there, and they’re strong! That’s where the (incorrect) intuition comes from that the order-2 neighborhoods should be growing pretty quickly with degree.

LinkedIn actually tells you how many friends of friends (AKA 2nd order connections) you have. For example, I’ve got 345 1st order connections and 91,145 2nd order connections. The latter is a little less than the linear equation you quote predicts, but reasonably close. (It makes sense that it would overestimate in my case because an unusually high number of my connections come from one company.)

P.S. 25 of my LinkedIn FOF are named Marianne. And I do have a mild Leonard Cohen fetish, though it is not nearly as bad as my son’s. (You get one guess what his favorite Leonard song is.)

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