Introduction to the Modeling and Analysis of Complex Systems

5.5. VARIABLE RESCALING 77g.add_edge((x, y), ((x * y) % 6, x))ccs = [cc for cc in nx.connected_components(g.to_undirected())]n = len(ccs)w = ceil(sqrt(n))h = ceil(n / w)for i in xrange(n):subplot(h, w, i + 1)nx.draw(nx.subgraph(g, ccs[i]), with_labels = True)show()In this example, a network object, named g, is constructed using NetworkX’s DiGraph(directed graph) object class, because the state transitions have a direction from one stateto another. I also did some additional tricks to improve the result of the visualization. I splitthe network into multiple separate connected components using NetworkX’s connected_components function, and then arranged them in a grid of plots using pylab’s subplot feature.The result is shown in Fig. 5.11. Each of the six networks represent one connectedcomponent, which corresponds to one basin of attraction. The directions of transitions areindicated by thick line segments instead of conventional arrowheads, i.e., each transitiongoes from the thin end to the thick end of a line (although some thick line segments arehidden beneath the nodes in the crowded areas). You can follow the directions of thoselinks to find out where the system is going in each basin of attraction. The attractor maybe a single state or a dynamic loop. Don’t worry if you don’t understand the details of thissample code. We will discuss how to use NetworkX in more detail later.Exercise 5.8 Draw a phase space of the following difference equation within therange 0 ≤ x < 100 by modifying Code 5.5:x t = x x t−1t−1 mod 100 (5.19)5.5 Variable RescalingAside from finding equilibrium points and visualizing phase spaces, there are many othermathematical analyses you can do to study the dynamics of discrete-time models. But