Random (pseudo)graphs G
with the following structure are studied: first, independent and identically distributed capacities Λ
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(Λ
). The main result of the paper is that when P(Λ1 > x) ≥ x
−τ+1, where τ ∈ (2, 3), then, asymptotically almost surely, G
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 (τ − 2)). It is also shown that the cases τ > 3, τ ∈ (2, 3), and τ ∈ (1, 2) present three qualitatively different connectivity architectures.