Introduction to the Modeling and Analysis of Complex Systems

186 CHAPTER 11. CELLULAR AUTOMATA I: MODELINGwhere both time and space are discrete. A CA model consists of identical automata (cellsor sites) uniformly arranged on lattice points in a D-dimensional discrete space (usuallyD = 1, 2, or 3). Each automaton is a dynamical variable, and its temporal change is givenbys t+1 (x) = F (s t (x + x 0 ), s t (x + x 1 ), . . . , s t (x + x n−1 )) , (11.1)where s t (x) is the state of an automaton located at x at time t, F is the state-transitionfunction, and N = {x 0 , x 1 , . . . , x n−1 } is the neighborhood. The idea that the same statetransitionfunction and the same neighborhood apply uniformly to all spatial locations isthe most characteristic assumption of CA. When von Neumann and Ulam developed thismodeling framework, researchers generally didn’t have explicit empirical data about howthings were connected in real-world complex systems. Therefore, it was a reasonablefirst step to assume spatial regularity and homogeneity (which will be extended later innetwork models).s t is a function that maps spatial locations to states, which is called a configuration ofthe CA at time t. A configuration intuitively means the spatial pattern that the CA displayat that time. These definitions are illustrated in Fig. 11.1.Neighborhood N is usually set up so that it is centered around the focal cell beingupdated (x 0 = 0) and spatially localized (|x i − x 0 | ≤ r for i = 1, 2, . . . , n − 1), where r iscalled a radius of N. In other words, a cell’s next state is determined locally accordingto its own current state and its local neighbors’ current states. A specific arrangementof states within the neighborhood is called a situation here. Figure 11.2 shows typicalexamples of neighborhoods often used for two-dimensional CA. In CA with von Neumannneighborhoods (Fig. 11.2, left), each cell changes its state according to the states of itsupper, lower, right, and left neighbor cells as well as itself. With Moore neighborhoods(Fig. 11.2, right), four diagonal cells are added to the neighborhood.The state-transition function is applied uniformly and simultaneously to all cells inthe space. The function can be described in the form of a look-up table (as shown inFig. 11.1), some mathematical formula, or a more high-level algorithmic language.If a state-transition function always gives an identical state to all the situations that areidentical to each other when rotated, then the CA model has rotational symmetry. Suchsymmetry is often employed in CA with an aim to model physical phenomena. Rotationalsymmetry is called strong if all the states of the CA are orientation-free and if the rotationof a situation doesn’t involve any rotation of the states themselves (Fig. 11.3, left). Otherwise,it is called weak (Fig. 11.3, right). In CA with weak rotational symmetry, some statesare oriented, and the rotation of a situation requires rotational adjustments of those statesas well. Von Neumann’s original CA model adopted weak rotational symmetry.