Fractal structure, entropy and energy dissipation in river networks

Informational entropy of river networks, as defined by Fiorentino and Claps (1992a), was shown to be a useful tool to explain several properties exhibited by natural networks. In this paper, selfsimilar properties of river networks are taken as a starting point for investigating how the entropy of fractal plane trees can be used to show analogies and differences between natural networks and geometric fractal trees. Attention is directed particularly to the relations between entropy and Horton order and entropy and topological diameter of sub-networks. Comparison of features of natural and geometric networks suggest that the use of entropy can contribute to clarify important points concerning regularity properties of river networks in the plane. Furthermore, an interesting comparison is shown in the variability of entropy with the magnitude of sub-networks, for both fractal and natural trees. In natural networks this relation is compared to an index of energy expenditure with the basin size, leading to an intriguing connection between plane and altitudinal features found in river basins.