Introduction to the Modeling and Analysis of Complex Systems

374CHAPTER 17. DYNAMICAL NETWORKS II: ANALYSIS OF NETWORK TOPOLOGIESIn this code, we measure the size of the largest connected component in an Erdős-Rényigraph with connection probability p, while geometrically increasing p. The result shown inFig. 17.2 indicates that a percolation transition happened at around p = 10 −2 .10 2size of largest connected component10 110 010 -4 10 -3 10 -2 10 -1pFigure 17.2: Visual output of Code 17.4.Giant components are a special name given to the largest connected components thatappear above this percolation threshold (they are seen in the third and fourth panels ofFig. 17.1), because their sizes are on the same order of magnitude as the size of thewhole network. Mathematically speaking, they are defined as the connected componentswhose size s(n) has the following propertys(n)limn→∞ n= c > 0, (17.3)where n is the number of nodes. This limit would go to zero for all other non-giant components,so at macroscopic scales, we can only see giant components in large networks.Exercise 17.1 Revise Code 17.4 so that you generate multiple random graphsfor each p, and calculate the average size of the largest connected components.Then run the revised code for larger network sizes (say, 1,000) to obtain a smoothercurve.