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Suppose that a network has a degree distribution that follows the exponential (or geometric) form pk Cak, where C and a are positive constants and a < 1.
a) Assuming the distribution is properly normalized, find C as a function of a.
b) Calculate the fraction P of nodes that have degree k or greater.
c) Calculate the fraction W of ends of edges that are attached to nodes of degree k orgreater.
d) Hence show that the Lorenz curve—the equivalent of Eq. (10.24) for this degreedistribution—is given byW P − 1 − 1/a P log P.log a
e) Show that the value of W is greater than one for some values of P in the range0 ≤ P ≤ 1. What is the meaning of these “unphysical” values?
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