Six degrees of separation?

In light of the estimated world population passing 6,666,666,666 a couple of days ago, and partially inspired by the inter-character linkages in the movie Love Actually, which I just happened to watch again last night, I got to thinking about the mathematical possibility of the six degrees of separation concept.

So, let's pretend for a moment, that every person on the planet has the same number of unique distinct people that they are "in contact" with. Let's define α to be that number. So if we let n be the number of degrees of separation, then the upper-bound of the total number of people within n degrees can be given by:

αn = α × α × ... × α = αn

Now, I currently have 83 "friends" listed on my FaceBook profile - a lot of people have many more than that, and some people have less. But let's imagine for a second, that I was the "norm", and so everyone had 83 "friends" ;)

Then: α6 = α6 = 836 ≈ 3.3 × 1011 ≈ 330 billion

So, yes, it would seem that the six degrees of separation concept is at least feasible (ignoring the obvious flaws in the above assumptions).

The next question in my mind is: how many "friends" does everyone need to have? Well, that should be very easy to work out... the lower-bound would be given by:

α = 6√(6.7 billion) ≈ 43

So, it would seem, that if eveyone on the planet knows at least 43 other people, that no-one else knows (kind of silly, I know), then with only six degrees of separation, we would reach the earth's current (estimated) population of 6.7 billion people.

Hardly scientific, nor exhaustive in accuracy, but still... interesting thought.

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