Chutes and Ladders, the real deal (part 2)

    Continuing on with the quest to find real-life applications for Chutes and Ladders, we take a look again at the directed graph mentioned in the previous past. As the directed graph for the real game of Chutes and Ladders is essentially no more than a larger version of this one, we should be good to stop here. Now that we have a rigorous basis to comparing other processes to Chutes and Ladders, the next step requires some creativity: we have to think of a real situation that is analogous to Chutes and Ladders based on a comparison of directed graphs. Of course reality is far more complex than Chutes and Ladders is, so finding a perfect analogy is next to impossible, but something close is better than nothing. Back in one of the first posts bringing up applications of graph theory, the process of a forest fire spreading was discussed. Let's see what we can do with that.
   
    Just as Chutes and Ladders progressed along randomly based on dice throws, a forest fire progresses randomly with time. Thus, while a dice throw represented one "iteration" in the Chutes and Ladders Markov Chain, some unit of time, say, a minute, represents an iteration in the forest fire Markov Chain. How might we define our states in this Markov Chain? With Chutes and Ladders, we defined the states based on what square a player was positioned on -- a player on the thirty-fifth square was said to be in the thirty fifth state. However, since a fire can be on multiple trees at once (while a Chutes and Ladders player can only be on one square at a time), it is more sensible to define the states based on the number of trees that have been set ablaze -- if thirty trees are on fire, the Markov Chain will be in state thirty. Moreover, the last state, just like in Chutes and Ladders, is an absorbing state. We do, however, have to assume that the fire travels in a fixed path and that the number of trees than can burn in the time interval is fixed (just like how you always spin between a 1 and a 6 in Chutes and Ladders), but so far this seems to have some potential. Now for the deal breaker: what is the analogy to chutes and ladders? Chutes are a bit difficult, since forest fires don't normally just fizz out when a certain number of trees are burned. However, a "ladder" might be possible if a group of trees are clustered so tightly together that when one burns, the others immediately burn too. The Markov Chain is instantly carried from one state to a higher state, just as what happens when a player lands on a ladder in Chutes and Ladders.
   
    The beginning of the directed graph, assuming that the maximum number of trees that can be burned in one time interval is three and the minimum is one, might look something like this...

   
       

    ...If the "forest" looked something like this:

    (Note that under this model, "red lines" are impossible. Yellow lines will appear if we decrease the number of trees that can be burned in one time interval down to zero).

    Although this forest fire example still seems a bit contrived, I think the similarities it does share with Chutes and Ladders are quite remarkable. With only a very small amount of tweaking, we can get a lot of information about this process with the tools already developed for analyzing Chutes and Ladders.