Introduction to the Modeling and Analysis of Complex Systems

376CHAPTER 17. DYNAMICAL NETWORKS II: ANALYSIS OF NETWORK TOPOLOGIESSince lim n→∞ (1 + x/n) n = e x , Eq. (17.11) can be further simplified for large n toq = e 〈k〉(q−1) . (17.13)Apparently, q = 1 satisfies this equation. If this equation also has another solution in0 < q < 1, then that means a giant component is possible in the network. Figure 17.3shows the plots of y = q and y = e 〈k〉(q−1) for 0 < q < 1 for several values of 〈k〉.1.0〈k〉 = 0.11.0〈k〉 = 0.51.0〈k〉 = 1.1.0〈k〉 = 1.50.80.80.80.80.60.60.60.6yyyy0.40.40.40.40.20.20.20.20.00.00.00.00.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0qqqqFigure 17.3: Plots of y = q and y = e 〈k〉(q−1) for several values of 〈k〉.These plots indicate that if the derivative of the right hand side of Eq. (17.13) (i.e., theslope of the curve) at q = 1 is greater than 1, then the equation has a solution in q < 1.Therefore,ddq e〈k〉(q−1) ∣∣∣q=1= 〈k〉e 〈k〉(q−1) | q=1 = 〈k〉 > 1, (17.14)i.e., if the average degree is greater than 1, then network percolation occurs.A giant component is a connected component whose size is on the same order ofmagnitude as the size of the whole network. Network percolation is the appearanceof such a giant component in a random graph, which occurs when the average nodedegree is above 1.Exercise 17.2 If Eq. (17.13) has a solution in q < 1/n, that means that all thenodes are essentially included in the giant component, and thus the network ismade of a single connected component. Obtain the threshold of 〈k〉 above whichthis occurs.