On a conditionally Poissonian graph process

Random (pseudo)graphs GN with the following structure are studied: first, independent and identically distributed capacities .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(.i .j / .k=1N .k). The main result of the paper is that when P(.1 > x) . x..+1, where . . (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 (. . 2)). It is also shown that the cases . > 3, . . (2, 3), and . . (1, 2) present three qualitatively different connectivity architectures.