It's a Small World
thanks to therealchips

Okay, here's the idea.
You go to a big party and meet somebuddy new.
You start to say, "I know this guy, Jim Martin of Hogsville, and ..."
The other person says, "Jim Martin ... of Hogsville? I know Jim!"
You then say (and this is compulsory), "What a small world!"

>That's it?
No, of course not. The point is, it's not a curious ritual. This small world thing has a mathematical underpinning.
>Does Jim Martin know that?
Pay attention.

  • Suppose you know several hundred people and each of these acquaintances knows 50 people that you don't know.
  • Each of these 50 know another 50 not known to any of the others.
  • Each of these 50 know another 50 ...

>Not known to any of the others? What ...?
Yes. I don't want to count people twice. If I know 50, including you, then I'm interested in your acquaintances that I don't know.
They're called non-redundant acquaintances.
>So those coloured 50s, 50 and 50 and ...
They're non-redundant. They don't know each other. Each time we move to another 50-person clique, it's a new collection of 50 people.
So, after moving from me to you to your acquaintances etc. we keep adding to the total number of people, each time multiplying by 50.
>You know 50 ... uh, non-redundant. Each of them know 50. So far we've got 50x50 = 2500 different people.
And, in six steps (or degrees of separation as it's called) we arrive at the population of the world.
>You're kidding, right?
Try it!
Number of (non-redundant) people known by the average person in the country (world?)=
Total number of people in the country (world?) =
Degrees of separation =
In the last few years this Small World Theory has become quite a hot topic.
It apparently started with Duncan Watts (born in Ontario!), a graduate student at Cornell University (in 1996).
He was working on the chirps of crickets and asked:
"How come the crickets fall into step so quickly? Was each listening to all his fellow crickets, or just to his closest neighbours?"

>That link says that I'm just a few handshakes away from knowing the president of the United States.
As well as every Australian Aborigine, eh?

>Do Aborigines have 50 non-redundants?
Go back to sleep.


In 1967 a Harvard professor, Stanley Milgram, sent a letter to several people in the mid-West with the goal of eventually getting the letters to a Boston stockbroker (A) and a Harvard divinity student's wife (B). He asked the recipients to forward the letter to whomever they thought might know A or B. Milgram found that the letters usually got to the intended recipients ... and that it took about five intermediaries (that's six degrees of separation).

Then Milgram sent the letters to white people, where the intended recipient was black, thinking that different or racially isolated communities might significantly increase the number of intermediaries. It didn't! It took only a couple of intermediaries to bridge the gap between white and black ... and that's what Duncan Watts found.