Introduction to the Modeling and Analysis of Complex Systems

18.5. MEAN-FIELD APPROXIMATION ON RANDOM NETWORKS 419means that the disease will never go away from the network. This epidemic threshold isoften written in terms of the infection probability, asp i >p r(n − 1)p e= p r〈k〉 , (18.28)as a condition for the disease to persist, where 〈k〉 is the average degree. It is an importantcharacteristic of epidemic models on random networks that there is a positive lower boundfor the disease’s infection probability. In other words, a disease needs to be “contagiousenough” in order to survive in a randomly connected social network.Exercise 18.5 Modify Code 16.6 to implement a synchronous, simultaneous updatingversion of the network SIS model. Then simulate its dynamics on an Erdős-Rényi random network for the following parameter settings:• n = 100, p e = 0.1, p i = 0.5, p r = 0.5 (p r < (n − 1)p e p i )• n = 100, p e = 0.1, p i = 0.04, p r = 0.5 (p r > (n − 1)p e p i )• n = 200, p e = 0.1, p i = 0.04, p r = 0.5 (p r < (n − 1)p e p i )• n = 200, p e = 0.05, p i = 0.04, p r = 0.5 (p r > (n − 1)p e p i )Discuss how the results compare to the predictions made by the mean-field approximation.As you can see in the exercise above, the mean-field approximation works much betteron random networks than on CA. This is because the topologies of random networks arenot locally clustered. Edges connect nodes that are randomly chosen from the entire network,so each edge serves as a global bridge to mix the states of the system, effectivelybringing the system closer to the “mean-field” state. This, of course, will break down if thenetwork topology is not random but locally clustered, such as that of the Watts-Strogatzsmall-world networks. You should keep this limitation in mind when you apply mean-fieldapproximation.Exercise 18.6 If you run the simulation using the original Code 16.6 with asynchronousupdating, the result may be different from the one obtained with synchronousupdating. Conduct simulations using the original code for the same parametersettings as those used in the previous exercise. Compare the results obtainedusing the two versions of the model, and discuss why they are so different.