(6) [6 points] The structure of contact network plays an important role in determining the rate and range of epidemic spreading. To see this, let us consider a group of 4 people, connected following two possible types of contact networks (instead of the 'tree structure' we discussed in class). The process starts with a person x{A,B,C,D} getting infected. In subsequent waves, each newly infected individual (if any) may transmit the disease to each of their network neighbors, independently with probability p=0.5. Our primary focus in this scenario is to determine q, the probability that the disease will spread to at least one additional person beyond the initially infected individual x. (a) Suppose we are interested in the expected outcome. In particular, x is randomly selected from {A,B,C,D}. What is the average probability that the disease will spread to at least one additional person beyond x ? Calculate the probability for Networks 1 and 2 respectively. (b) Suppose we are interested in the worst outcome. In particular, x is selected from {A,B,C,D} to maximize the probability of interest q (the choice of x might vary for Networks 1 and 2). What is the highest possible probability that the disease will spread to at least one additional person beyond x ? Calculate the probability for Networks 1 and 2 respectively. 4