On a conditionally poissonian graph process

Random (pseudo)graphs G(N) with the following structure are studied: first, independent and identically distributed capacities Lambda(i) are drawn for vertices i = 1,..., N; then, each pair of vertices (i, j) is connected, independently of the other pairs, with E(i, j) edges, where E(i, j) has distribution Poisson(Lambda(i)Lambda(j)/Sigma(N)(k=1) Lambda k. The main result of the paper is that when P(Lambda(1) > x) >= x(-tau+1), where tau is an element of (2, 3), then, asymptotically almost surely, GN has a giant component, and the distance between two randomly selected vertices of the giant component is less than (2 + o(N))(log log N)/(-log (tau - 2)). It is also shown that the cases tau > 3, tau is an element of (2, 3), and tau is an element of (1, 2) present three qualitatively different connectivity architectures.