Probability, Random Walk, and Invariant

First, let’s talk a little bit about some words.   

Probability
Everybody knows what a probability is, right? Well, in short, it is a numerical value of a chance that something will happen in an interested situation. We define 1 to be “always happen” and 0 to be “never happen.”

For example, what is the probability for one to pick 5 from set {5,10,15,20}? 
1/4 or 0.25 … easy enough … so let’s move on.

Random Walk
As its name, Random Walk (RW) is a mathematical formalization for random walking. Arr.. it means mathematicians want to study about someone on the street that is randomly walking – something like that.
They study the probability that this guy will walk to Apple Store on 5th Avenue if he start randomly walking (like a drunk man) at the Columbia University. They also study other things about this guy, but we’re gonna skip that this time.

Get an idea?

RW is a huge huge subject in mathematics, so we’re not gonna go through all of them. The one I pick out today is Random Walk on Graphs.

A Sample of Graph

The idea is that we generate some graphs. A graph consists of dots and lines. Dots represent objects, situations, people, or anything we want, and lines represent connections of those dots.
Random Walk on Graphs is a study of walking from dots to dots via those lines randomly.   

For example, we can calculate what is the probability that a drunk man will end up at the top red dot if he start at the green one on the bottom-right by the total of twenty moves.

I think we get an idea. Next…

Invariant
In mathematics, an invariant is something that doesn’t change under a set of transformation. We call “invariance” for the property of being an invariant.
It’s kind of confuse, I know. Let’s see an example then.

Let’s say that we pick any two integers – X and Y. I can say that the odd-evenness of XY(X+Y) is an invariant. It doesn’t matter how we choose X or Y. XY(X+Y) is always an even number.

Never change… invariant … never change … no matter what
I think you get an idea.

So…. what’s this all about?

A sample of a very simple life

Let each decision or situation in my life is a dot. Lines connect each decision or situation that leads to all another possible situation. We write down an arrow to indicate a direction to each line and, if possible, weight each line with probability. It will come out to be something like this.   

Idealistically, I can generate the whole graph of my life. I then can see the path I actually have taken from my birth to today, and see what is the probability of that.

What if we generate two sets of graphs? Then, we combine dots that represent intersections of those two lives together – in all possible way. After that, we generate two random walks, and see what is the probability that today those two walks are on the same dot. 

That’s pretty small, I believe. 

And then we see… to not-so-far future … there’s no such a dot.
And then we wonder … in further future … the system is too complex to see what’s exactly going on out there.

wonder…. if there will be such a dot again

But you know what?

It actually doesn’t matter.

because I know….

something is invariance.