Word Links, Mazes, and the Small-World Effect

The small-world effect is a well known one, having websites like the Oracle of Bacon dedicated to it. It goes like this: You are friends or coworkers with another person, who may be friends with another person, and so on, until you get to a Himalayan yak trader, or Kevin Bacon. Usually this is in a rather small number of steps,  resulting in amazement. A small version of a small-world effect could be represented by a matrix, with the sample people of – say, Bill Gosper, Neo, Kelly, Kevin Bacon, Mr. Happy, and I. A (fictional) network could be described as

From\To  Bill  Neo  Kelly  Bacon  Happy  Me

Bill [   0   1   0     0     0      1]
Neo  [   1   0   0     1     0      0]
Kelly[   0   0   0     0     1      1]
Bacon[   0   1   0     0     1      0]
Happy[   0   0   1     1     0      0]
Me   [   1   0   1     0     0      0]

, the 1s meaning that Bacon is friends with Neo 1 time, and so forth. This list may not look like a Bacon effect is taking place, but this is just the list of direct friends. To find the list of friends of friends and normal friends, we just square the matrix (not movie), and add the original matrix to that, to get

[2 1 1 1 0 1]
[1 2 0 1 1 1]
[1 0 2 1 1 1]
[1 1 1 2 1 0]
[0 1 1 1 2 1]
[1 1 1 0 1 2]

and find out that some of my friends of friends are Neo and Mr. Happy. One more push through the multiplier shows that everybody is a friend of a friend of a friend of everybody else.

This effect is so powerful that, if extras are counted in movies, I somehow manage to get a Bacon number of 3.

A similar effect is in the simple game of word links, first made by Lewis Carroll, in which you can use the matrices to make a puzzle which uses a set of words by drawing all the possible connections between them, and then inputting that into the matrix. When the last 0 is gone in the repetitive multiplication, you then know how many moves it takes to solve the longest word link there. You could even, by replacing the words with points in a maze, find how many turns it takes to solve any maze, even 1-way passages, as long as you know where the passages are.

Lastly, you can use the simple drawings of lines to make an image, by first transforming the bitmap image to a bunch of numbers, probably 1s and 0s, which you would put in the matrix (NOT THE MOVIE) , turn that into a maze or arrows pointing at each other, and give that to their contact, who unscrambles the matrix to get the original image. You could even make an entire house with convoluted passageways, dead ends, and secret rooms to hide that code, if you were really rich and crazy.

Someone’s going to mention the Winchester Mystery House.


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