Introduction to the Modeling and Analysis of Complex Systems

212 CHAPTER 12. CELLULAR AUTOMATA II: ANALYSISbetween a configuration and its unique ID number (from 0 to 2 L − 1; 511 in our example),arranging the bits in the order of their significance from left to right. For example, theconfiguration [0, 1, 0, 1, 1, 0, 1, 0, 1] is mapped to 2 7 + 2 5 + 2 4 + 2 2 + 2 0 = 181. Thefunction config receives a non-negative integer x and returns a list made of 0 and 1, i.e.,a configuration of the CA that corresponds to the given number. Note that “&” is a logicalAND operator, which is used to check if x’s i-th bit is 1 or not. The function cf_numberreceives a configuration of the CA, cf, and returns its unique ID number (i.e., cf_numberis an inverse function of config).Next, we need to define an updating function to construct a one-step trajectory of theCA model. This is similar to what we usually do in the implementation of CA models:Code 12.2:def update(cf):nextcf = [0] * Lfor x in range(L):count = 0for dx in range(-r, r + 1):count += cf[(x + dx) % L]nextcf[x] = 1 if count > (2 * r + 1) * 0.5 else 0return nextcfIn this example, we adopt the majority rule as the CA’s state-transition function, whichis written in the second-to-last line. Specifically, each cell turns to 1 if more than half ofits local neighbors (there are 2r + 1 such neighbors including itself) had state 1’s, or itotherwise turns to 0. You can revise that line to implement different state-transition rulesas well.Now we have all the necessary pieces for phase space visualization. By plugging theabove codes into Code 5.5 and making some edits, we get the following:Code 12.3: ca-graph-based-phasespace.pyfrom pylab import *import networkx as nxg = nx.DiGraph()r = 2L = 9