(a) (10 pts) The mean neighbor degree of a node is the average degree of its neighbors. We define MND(G) to be the average mean neighbor degree across all the nodes of a network. In particular, MND(G)...

(a) (10 pts) The mean neighbor degree of a node is the average degree of its neighbors. We define MND(G) to be the average mean neighbor degree across all the nodes of a network. In particular, MND(G) = 1 2m Xn u=1 Xn v=1 kvAuv where n and m are the number of nodes and edges in the network G respectively. Show that this is equivalent to hk 2 i/hki.(b) (15 pts) The friendship paradox occurs when MND(G) is greater than hki. In terms of a social network, this means that, on average, each of your friends have more friends than you. For each node in the UC Irvine student network, determine the average degree of its neighbors and make a scatterplot with node degree on the x-axis and mean neighbor degree divided by node degree on the y-axis. Include a horizontal line separating points that display the friendship paradox and points that do not.(c) (5 pts) Does the UC Irvine student network display the friendship paradox? Why might this paradox not be so surprising? Given part (a), consider what must be true of G for this paradox to be absent.3. (a) (15 pts) Write a function configModel(degSeq) taking a degree sequence as input and returning a graph generated according to the configuration model as described in chapter 4.8 of Network Science. Consider the set of graphs for which all vertices have degree 1 or 3 and n = 104 nodes. Let p1 be the probability that a node is of degree 1 and p3 = 1 − p1 be the probability that a node is of degree 3. Use configModel(degSeq) to make a figure showing the mean fractional size of the largest component for values of p1 from 0 to 1 in steps of 0.01. Estimate the value of p1 at which a phase transition occurs and the giant component disappears.(b) (10 pts) Write a function degPresRand(G) taking a graph G as input and returning a new network with a degree sequence identical to that of G as described in chapter 4.8 of Network Science. Write your function to perform at least m 2 ln(107 ) rewirings as suggested here.(c) (20 pts) Using both configModel(degSeq) and degPresRand(G), explore the degree and distance distributions of the undirected versions of Openflights airport network (2016) and the German highway system network (2002). What patterns do you notice? Do you observe any significant deviations in these networks from what we might expect given a graph with a similar degree sequence?