Consider a model of a growing directed network similar to Prices model described in Section 13.1, but without preferential attachment. That is, nodes are added one by one to the growing network and each has c outgoing edges, but those edges now attach to existing nodes chosen uniformly at random, without regard for degrees or any other node properties.

a) Derive master equations, the equivalent of Eqs. (13.7) and (13.8), that govern the distribution of in-degrees q in the limit of large network size.

b) Hence show that in the limit of large size the in-degrees have an exponential distribution pq=C r^q , where C is a normalization constant and r=c/(c + 1).

Pa = [(9-1+ a)P4-1 - (q + a)pa] for q 2 1, for q = 0. (13.7) (13.8) Po = 1 - 1-ctapo ca . Pa = [(9-1+ a)P4-1 - (q + a)pa] for q 2 1, for q = 0. (13.7) (13.8) Po = 1 - 1-ctapo ca