Linking my way to fame and fortune

As I may have mentioned, I'm not exactly a social networking party animal (more a party vegetable, really), but I happened to log back into LinkedIn after I'm-not-sure-how-long and lo and behold, I'm a 3rd-degree connection to the President of the United States himself.

I doubt this puts me in particularly distinguished company. LinkedIn itself appears to be on a first-name basis with the prez, and tells me that "Barack's connections" number "500+".
Alas, "Barack Obama is not currently open to receiving Introductions or InMail™" so I doubt I'll be joining that select 500+ anytime soon.

A few statistics about my own network may make the three degrees of separation somewhat less surprising. Keep in mind that I haven't tried particularly hard to make connections. They just seem to accumulate:

• I have 83 direct connections.
• About a dozen of those have 500+ connections.
• 43 of the 83, so just over half, are two degrees from Obama
• I have 14,000+ 2nd-degree connections and about 1.4 million 3rd-degree connections.

The president's statistics are probably more like

• So many direct connections you need a staffer to keep them straight
• Given the nature of politics, dozens if not hundreds of those have 500+ connections. Granted, there's going to be a lot of overlap.

Now, how many 2nd-degree connections, and how many of those direct connections are two degrees from me?

There are approximately 45 million LinkedIn users overall, making it a proper megacommunity by my earlier definition. If we assume that being two degrees from one person is independent of being two degrees from another -- a dicey assumption for a number of reasons -- then the POTUS's 2nd-degree connections would number about half that, or 22 million+, based on about half of my connections being his 2nd-degree connections.

That's not completely implausible, but it's also possible that through the luck of the draw I'm connected to a cluster of people with somewhat closer than usual ties to the president. Nonetheless, if you're an active social networker with 500+ connections yourself, your odds of being 2nd-degree or closer look pretty good. The odds of any random LinkedIn member being in my position also look quite good.

Likewise, the purely random model would suggest that about 1/3,000th of Barack's connections would be two degrees from me, so likely a handful of people, maybe just one, depending on just how far over 500 the number really is. Again plausible, but again, we're not dealing with strictly random samples.

Paul Erdős, Alfréd Rényi, Béla Bollobás and others proved some very interesting results about random graphs starting in 1959, but social network graphs don't appear to fit the usual model. I recall running across more relevant work while trolling through Wikipedia a while ago. I might have to go back for another look. [There's more on social network graphs here and here --D.H. Dec 2015]

So, does this demonstrate the awesome power of social networking, that a random none-too-social geek can find himself three steps away from one of the world's most influential offices? Well, just what use am I meant to make of this connectivity beyond getting a longer-than-expected blog post out of it? Whatever use I might want to put it to, pretty much the rest of LinkedIn has the same shot if not better. I and the rest of the teeming masses can't see Obama's connections, or send him InMail, or do anything else particularly impressive. In other words, we're in about the same situation as any other private citizen of the US, which is where we were without LinkedIn.

There's a general principle at work here: You can't deliver privileged access to everyone.