Introduction to the Modeling and Analysis of Complex Systems

222 CHAPTER 12. CELLULAR AUTOMATA II: ANALYSISSingle cell level:p 1 = q2x2 block level:p 2 = F(p 1 )= p 14+ 4p 13 (1 – p 1 )+ 4p 12 (1 – p 1 ) 2Figure 12.5: How to estimate the conductance probability p 2 at scale s = 2 using p 1 .four different ways to transmit the fire through the area. But if there is only one tree, thereis no way to transmit the fire, so there is no need to consider such cases. This is theexhaustive list of possible situations where the fire will be conducted through a scale-2block. We can calculate the probability for each situation and add them up, to obtain thefollowing relationship between p 1 and p 2 :p 2 = Φ(p 1 ) = p 4 1 + 4p 3 1(1 − p 1 ) + 4p 2 1(1 − p 1 ) 2 (12.12)And this is the place where a very strong assumption/approximation comes in. Let’sassume that the relationship above applies to 4 × 4 blocks (Fig. 12.6), 8 × 8 blocks, etc.,all the way up to infinitely large portions of the forest, so thatp s+1 = Φ(p s ) = p 4 s + 4p 3 s(1 − p s ) + 4p 2 s(1 − p s ) 2 for all s. (12.13)Of course this is not correct, but some kind of approximation needs to be made inorder to study complex systems analytically.Exercise 12.8 Why is it not correct to assume that the relationship between the1×1 blocks and the 2×2 blocks can be applied to 2×2 and 4×4 (and larger)? Look atFig. 12.6 carefully and figure out some configurations that violate this assumption.