Solving Color Convergence Problem - Monte's Research

[Monte_Carlo]

Hello,

I'm pondering over this research question.

Let's suppose you've got a bunch of units which can be colored black or white. They're roaming around 2d grid in random walk. Any time a unit meets with another unit, it has an option to change color. It doesn't have to though, depending on its internal logic. The point is to calculate how long will it take until all units over the grid are of the same color.

Anybody knows what this problem is called, and how to model it? Any advice would be appreciated.

I'm thinking of simple strategy like each unit counts the number of black and white units it met with, and as soon as one number exceeds the other, the unit will change color. Any ideas how to math model this?

Thanks,

Monte

 

[phinds]

I'm thinking of simple strategy like each unit counts the number of black and white units it met with, and as soon as one number exceeds the other, the unit will change color.

And I'm thinking you might want to rethink that.

EDIT: snippy, smart-*** comment deleted.

 

[Monte_Carlo]

Phinds,

Got you, but just to make sure we're on the same page here:

if unit color is black and it sees one white unit then 1 > 0 (like you say) and it flips color because it simply assumes the most frequent color amongst units it has seen. Each unit keeps count of white and black units it has seen.

I simulated this approach - it does converge. Question is how to model this mathematically (it's pretty simple to program.)

Also, if you have a simpler approach amenable to modeling it would be great also. Any ideas if this is some canonical (or well known problem)?

Thanks,

Monte

 

[phinds]

Phinds,

Got you, but just to make sure we're on the same page here:

if unit color is black and it sees one white unit then 1 > 0 (like you say) and it flips color because it simply assumes the most frequent color amongst units it has seen. Each unit keeps count of white and black units it has seen.

I simulated this approach - it does converge. Question is how to model this mathematically (it's pretty simple to program.)

Also, if you have a simpler approach amenable to modeling it would be great also. Any ideas if this is some canonical (or well known problem)?

Thanks,

Monte

I don't know this kind of problem, so no help to you there.

It does seem that your approach seems easy to model, but boring. It would be more interesting if the rules were more complex.

Now that I think about it there used to be a "game" that programmers loved to play (and may still for all I know) that is EXACTLY the same kind of thing. Try looking that up on the Internet and see if it helps you out. Try Googling "game of life".

 

[CellCoree]

Hello Monte,

Your research question is quite interesting. The problem you are describing is known as the "Color Convergence Problem" or "Voter Model." It is a well-studied problem in statistical physics and has been used to model various real-world phenomena such as opinion formation and social dynamics.

To model this problem, you can use a mathematical approach known as the "Ising Model." This model is commonly used to study phase transitions in physical systems and can also be applied to the Color Convergence Problem. In this model, each unit on the grid can be represented by a spin (either up or down) and the interactions between units can be described by an energy function. The energy function would determine whether a unit will change its color or not based on the number of black and white units it has encountered.

I would recommend exploring the Ising Model and its applications to the Color Convergence Problem to further your research. You can also consider incorporating other factors such as the size of the grid, the initial distribution of colors, and the behavior of individual units to make your model more realistic.

I wish you the best of luck in your research!