Introduction to the Modeling and Analysis of Complex Systems

220 CHAPTER 12. CELLULAR AUTOMATA II: ANALYSISI am well aware that the explanation above is still vague and somewhat cryptic, solet’s discuss the detailed steps of how this method works. Here are typical steps of howrenormalization group analysis is done:1. Define a property of a “portion” of the system you are interested in. This propertymust be definable and measurable for portions of any size, like a material propertyof matter. For analyzing percolation, it is typically defined as the probability for aportion to conduct a fire or a disease from one side to another side through it.2. Calculate the property at the smallest scale, p 1 . This is usually at the single-celllevel, which should be immediately obtainable from a model parameter.3. Derive a mathematical relationship between the property at the smallest scale, p 1 ,and the same property at a one-step larger scale, p 2 = Φ(p 1 ). This derivation isdone by using single cells as building blocks to describe the process at a largerscale (e.g., two-by-two blocks)4. Assume that the relationship derived above can be applied to predict the propertyat even larger scales, and then study the asymptotic behavior of the iterative mapp s+1 = Φ(p s ) when scale s goes to infinity.Let’s see what these steps actually look like with a specific forest fire CA model withMoore neighborhoods. The essential parameter of this model is, as discussed in Section11.5, the density of trees in a space. This tree density was called p in the previous chapter,but let’s rename it q here, in order to avoid confusion with the property to be studied in thisanalysis. The key property we want to measure is whether a portion of the system (a blockof the forest in this context) can conduct fire from one side to another. This probability isdefined as a function of scale s, i.e., the length of an edge of the block (Fig. 12.4). Let’scall this the conductance probability for now.The conductance probability at the lowest level, p 1 , is simply the probability for a treeto be present in a single cell (i.e., a 1×1 block). If there is a tree, the fire will be conducted,but if not, it won’t. Therefore:p 1 = q (12.11)Next, we calculate the conductance probability at a little larger scale, p 2 . In so doing,we use p 1 as the basic property of a smaller-sized building block and enumerate what aretheir arrangements that conduct the fire across a two-cell gap (Fig. 12.5).As you see in the figure, if all four cells are occupied by trees, the fire will definitelyreach the other end. Even if there are only three or two trees within the 4 area, there are