Introduction to the Modeling and Analysis of Complex Systems

426CHAPTER 18. DYNAMICAL NETWORKS III: ANALYSIS OF NETWORK DYNAMICSThis shows 0 < lim pi →0 r(k) < 1, i.e., the non-zero equilibrium state remains stable evenif p i approaches to zero. In other words, there is no epidemic threshold if the network isscale-free! This profound result was discovered by statistical physicists Romualdo Pastor-Satorras and Alessandro Vespignani in the early 2000s [79], which illustrates an importantfact that, on networks whose topologies are scale-free, diseases can linger around indefinitely,no matter how weak their infectivity is. This is a great example of how complexnetwork topologies can fundamentally change the dynamics of a system, and it also illustrateshow misleading the predictions made using random models can be sometimes.This finding and other related theories have lots of real-world applications, such as understanding,modeling, and prevention of epidemics of contagious diseases in society aswell as those of computer viruses on the Internet.Exercise 18.7 Simulate the dynamics of the network SIS model on Barabási-Albert scale-free networks to check if there is indeed no epidemic threshold asthe theory predicts. In simulations on any finite-sized networks, there is alwaysthe possibility of accidental extinction of diseases, so you will need to make thenetwork size as large as possible to minimize this “finite size” effect. You shouldcompare the results with those obtained from control experiments that use randomnetworks.Finally, I should mention that all the analytical methods discussed above are still quitelimited, because we didn’t consider any degree assortativity or disassortativity, any possiblestate correlations across edges, or any coevolutionary dynamics that couple statechanges and topological changes. Real-world networks often involve such higher-ordercomplexities. In order to better capture them, there are more advanced analytical techniquesavailable, such as pair approximation and moment closure. If you want to learnmore about these techniques, there are some more detailed technical references available[80, 81, 82, 83].Exercise 18.8Consider a network with extremely strong assortativity, so thatP (k ′ |k) ≈{ 1 if k ′ = k,0 otherwise.(18.66)Use this new definition of P (k ′ |k) in Eq. (18.30) to conduct a mean-field approximation,and determine whether this strongly assortative network has an epidemicthreshold or not.