Suppose a network has a degree distribution that follows the exponential form p_k = Ce^-lambda k,…

Suppose a network has a degree distribution that follows the exponential form p_k = Ce^-lambda k, where C and A are constants. Find C as a function of lambda. Calculate the fraction P of vertices that have degrees greater than or equal to k. Calculate the fraction W of ends of edges that are attached to vertices of degree greater than or equal to k. Hence show that for this degree distribution, the Lorenz curve-the equivalent of Eq. (8.23) in Networks-is given by W = P +1-e^lambda/lambda p ln P. What is the equivalent of the “80-20” rule for such a network with lambda = 1? That is, what fraction of the “richest” nodes in the network have 80% of the “wealth”? Show that the value of W is greater than unity for some values of P in the range 0 lessthanorequalto P lessthanorequalto 1. What is the meaning of these “unphysical” values?

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