SCALE-INVARIANCE PROPERTIES OF THE PEAK OF THE WIDTH FUNCTION IN TOPOLOGICALLY RANDOM NETWORKS

Some scaling properties of the topological width function for an infin ite population of networks obeying the random model are analyzed. A Mo nte Carlo procedure is applied to generate width functions according t o the hypothesis of topological randomness. The probability distributi ons of both peak and distance to peak of the topological width functio ns, conditioned (1) on the network diameter lambda and (2) on lambda a nd parameter beta = [log (2 mu -1)]/log lambda, are studied. The param eter beta can be considered a shape factor of the network; indeed, low beta values describe elongated networks, while high beta values refer to fan-like networks. Scale invariance for both random variables is e stablished in the first case by using lambda as a scale parameter. Als o in the second case, scale invariance is observed for both the peak a nd the distance to peak of the topological width function; in particul ar, the invariance property for the peak involves a scaling function w hich is directly related to the shape factor beta, allowing determinat ion of the statistical similarity between random networks indexed by t he same beta. Then, a coarse-graining procedure is applied to a set of 15,000 width functions with lambda = 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-si milarity for the peaks is also observed, which involved the same p-rel ated scaling function. Finally, possible implication of the present re sults on the hydrologic response, at the basin scale, is discussed.