Thursday, August 7, 2008

On connections...

There are connections all around us! It makes our small world even smaller and more interesting. I'll now present two interesting anecdotes of this concept:

1) I have a new neighbor. Her renter's son's ex-girlfriend lives in the same complex and I used to work with the renter's son's ex at my previous job. After talking with her the other day, she told me that her second cousin also worked for the company that I work for now. My cubicle was across from his for six months and I had no idea.

2) Today the EPA rejected the bid by Rick Perry (Gov. Texas) to cut the ethanol subsidy. Ethanol is a curious connection between gas and chicken prices. Allow me to elucidate: Ethanol is used in 10% of gas, causing a drop in gas. It also causes a price increase in corn (heavily used in ethanol production), meaning that chickens (using corn as a food source) cost more for upkeep, meaning a loss of profit and a drop in the value.

James Burke is the undisputed master of such connections, and he had a series of shows (eponymously named) that dealt with these connections (if you haven't seen this series you must find a copy of it or look for it on PBS, it's really fascinating). He believes that the modern world would not exist without all these interconnections. There's also another cultural icon that relates to the concept of connections: six degrees of Kevin Bacon. The essence of this game is that through a series of movies less than six, link a named actor with Kevin Bacon. According to Wikipedia the game was invented by three snowbound college students, but the mathematics world did the same thing with Paul Erdos in 1969 and called it an Erdos Number.

The connotation that a connection usually "shrinks" the size of our perceivable universe is a quaint notion. I'm not sure how to prove it mathematically, because I'm not necessarily convinced that it's true. First we must define the domain of the problem. The closest concept to the "size of our percievable universe" is a diameter of a graph. A graph (our stand in for the universe) is a mathematical term for a diagram with vertices (points on a piece of paper, in our case people, places and things) and edges (lines connecting these points, or connections between people, places, and things). The distance between any two vertices of a graph is the number of edges in the shortest path connecting the two points. The diameter would then be the maximum distance between any two connected vertices.

Here's the crux of the reason why this problem (figuring the probability that a new connection "shrinks" the universe) may be hard. If you define that the distance between any two disconnected vertices to be infinite, the proof is trivial (and closer to human intuition), but this is a flawed assumption in an unbounded set of verticies (IE people, places and things will always be born, found, and created respectively, and new connections made between them, so the set of vertices between these "noun" vertices must be unbounded). It is more accurate to not consider unconnected people/places/things (vertices), meaning a new connection (edge) between them could concievably decrease the diameter of the universe (if the two people/places/things are already connected) or increase it (if the new edge connects two already well-connected "cliques" of people/places/things, such as the first contact between two cultures).

So keep in mind the connections that you make with other people. You may be making the universe larger or smaller as a result of your interaction. Whether or not that's a good or bad thing all depends on your outlook on life. Does it make you happier that we seem to live in a small world?

Edit: fixed an antecedent error in the first paragraph and removed an identifying marker.


Anittah said...

Buddhism, baby. We are all interconnected!

Leadhyena Inrandomtan said...

If this is the case then more connections are only going to shrink the diameter. Therefore Buddhism believes in a constantly shrinking world.

As a non-committal mathematician I'd rather not take that as an axiom. :)