I recently worked a problem that required counting the number of paths to a node in a lattice. For something to do, I decided to try to produce a visualization of how the number of paths increases with increasing distance from the starting node – the origin. Here is an example of a path from the origin to the the point (5,5,5) created with Mathematica.
The path randomly decides to increase one unit in either the x or the y or the z direction with equal probability. The total number of paths to (5,5,5) is .
I first thought to use gephi, a program that produces graph theory graphs but I couldn’t get the hang of it and managed this graph before I gave up.
This depiction uses YifanHu’s Multilevel scheme. This scheme shows the three dimensional structure but is not fully rotatable.
So I turned to Mathematica to produce my own drawings. A first try produced this.
And then this.
The node size corresponds to the number of paths to that node.
The pictures are not very informative obviously because the number of paths is a factorial expression. So I decided to use the logarithm of the number of paths. Thus this,
And this with color coding for number of paths added.
But now I have lost all sense of the size of the increase since I am using logarithms. I ended my exploration here.
I was thinking of creating an animation but to what end. I have five classes to occupy me now. So just some pretty? pictures.