Some scaling properties of the topological width function for an infinite population of networks obeying the random model are analyzed. A Monte Carlo procedure is applied to generate width functions according to the hypothesis of topological randomness. The probability distributions of both peak and distance to peak of the topological width functions, conditioned (1) on the network diameter &lgr; and (2) on &lgr; and parameter &bgr;=<log(2&mgr;-1)>/log &lgr;, are studied. The parameter &bgr; can be considered a shape factor of the network; indeed, low &bgr; values describe elongated networks, while high &bgr; values refer to fan-like networks. Scale invariance for both random variables is established in the first case by using &lgr; as a scale parameter. Also in the second case, scale invariance is observed for both the peak and the distance to peak of the topological width function; in particular, the invariance property for the peak involves a scaling function which is directly related to the shape factor &bgr;, allowing determination of the statistical similarity between random networks indexed by the same &bgr;. Then, a coarse-graining procedure is applied to a set of 15,000 width functions with &lgr;=512; a scaling behavior of peaks of the original width function and aggregated ones is observed over a wide range of aggregation scales. Consequently, a statistical self-similarity for the peaks is also observed, which involved the same &bgr;-related scaling function. Finally, possible implication of the present results on the hydrologic response, at the basin scale, is discussed. ¿ 1998 American Geophysical Union